SOLVING PROBLEMS INVOLVING AREA
Let us recall the formula to find the area of a rectangle.
length × width
To find area of a rectangle, we multiply length and width.
In case we know the area, and only one dimension, we can divide the area by the known dimension to find the other dimension.
The ultimate aim of this section is, how to solve problems using area.
Problem 1 :
The area of a rectangular sandbox is 56 2/3 square feet. The length of the sandbox is 8 1/2 feet. What is the width ?
We know the formula to find area of the rectangle.
Area of the rectangle = length x width
Plug the known values area and length and solve for the unknown value width.
56 2/3 = 8 1/2 x width
170/3 = 17/2 x width
(170/3) x (2/17) = width
20/3 = width
6 2/3 = width
So, the width of the rectangle is 6 2/3 feet.
Problem 2 :
The area of a circle is 154 square cm. Find its perimeter.
To find perimeter of the circle, we have to know the radius.
So, we have to find the radius from the given area.
We know the formula for area of circle.
Area of the circle = π r 2
Substitute π = 22/7, area = 154.
154 = 22/7 x r 2
154 x 7/22 = r 2
49 = r 2
7 2 = r 2
Perimeter of the circle = 2 π r
= 2x (22/7) x 7
= 2 x 22
= 44 cm
So, perimeter of the circle is 44 cm.
Problem 3 :
Mr. Webster is buying carpet for an exercise room where the basement of the room is in the shape of rectangle. The length and width of the room are 18 2/5 feet and 12 1/2 feet respectively. Find the total cost of the carpet, if the price of the carpet is $3 per square feet ?
To know the total cost of the carpet, first we have to know the area of the basement.
Area of the basement = length x width
= 18 2/5 x 12 1/2
= (92/5) x (25/2)
= 46 x 5
= 230 square feet
So, we need 230 square feet of carpet.
The cost each square feet of carpet = $3
Then, the cost of 230 square ft of carpet is
= 3 x 230
So, the total cost of the carpet is $690.
Problem 4 :
If the area of a square is 64 square inches, find the length of its diagonal.
Let 'a' be the length of one side of the square.
We all know the formula for area of square.
Area of square = side x side = a 2
64 = a 2
8 2 = a 2
8 = a
So, the length of side of the square is 8 inches.
In the above picture, we have to find the length of AC, say 'x'.
Using Pythagorean theorem in the right triangle ABC, we get
AC> 2 = AB 2 + BC 2
x 2 = 8 2 + 8 2
x 2 = 64 + 64
x 2 = 128
Take square root on both sides.
√ x 2 = √128
x = √(2x64)
x = 8 √2 inches
The length of its diagonal is 8 √2 inches.
Problem 5 :
If the radii of two circles are in the ratio 2 : 3, then find the ratio between their areas.
From, the given information, we have
Radius of the first circle = 2x
Radius of the second circle = 3x
Area of the first circle = π r 2 = π (2x) 2 = 4 π x 2
Area of the second circle = π r 2 = π (3x) 2 = 9 π x 2
Ratio between their areas is
= 4 π x 2 : 9 π x 2
= 4 : 9
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Area Worksheets | 500+ Printable PDF Worksheets
Equip future architects, aeronauts, coast guards, graphic designers with this meticulously designed assemblage of printable area worksheets to figure out the area of irregular figures, area of 2D shapes like squares, rectangles, triangles, parallelograms, trapezoids, quadrilaterals, rhombus, circles, polygons, kites, mixed and compound shapes using appropriate area formulas. Sample our free worksheets that are exclusively drafted for grade 2 through grade 8 children.
List of Area Worksheets
- Counting Squares
- Area of Squares
- Area of Rectangles
- Area of Rectilinear Figures
- Area of Triangles
- Area of Parallelograms
- Area of Trapezoids
- Area of Rhombus
- Area of Kites
- Area of Quadrilaterals
- Area of Circles
- Area of an Annulus
- Area of Mixed Shapes
- Area of Polygons
- Area of Compound Shapes
- Area of a Segment of a Circle
Explore Area Worksheets in Detail
Counting Squares Worksheets
The children in the 2nd grade and 3rd grade enhance practice with this interesting collection of pdf worksheets on finding the area by counting unit squares. Included here area exercises to count the squares in the irregular figures and rectangular shapes.
Area of Squares Worksheets
Give learning a head start with these finding the area of a square worksheets. Figure out the area of squares using the formula, determine the side lengths, find the length of the diagonals and calculate the perimeter using the area as well.
Area of Rectangles Worksheets
Strengthen skills in finding the area of a rectangle with these pdf worksheets featuring topics such as determine the area of rectangles, area of rectilinear shapes, rectangular paths and solve word problems. Recommended for grade 3, grade 4, grade 5 and up.
Area of Rectilinear Figures Worksheets
Augment your practice on finding the area with our area of rectilinear figures worksheets! With two or more non-overlapping rectangles composing them, these rectilinear shapes require adding the areas of those non-overlapping parts to arrive at their area.
Area of Triangles Worksheets
Focusing on finding the area of triangles, this set of worksheets features triangles whose dimensions are given as integers, decimals and fractions involving conversion to specified units as well. Approx. grade levels: 5th grade, 6th grade and 7th grade.
Area of Parallelograms Worksheets
The area of a parallelogram worksheets comprise adequate skills to find the area of a parallelogram, compute the value of the missing dimensions - base or height, practice finding the area by converting to specific units and more. The exercises are presented as geometric illustrations and also in word format.
Area of Trapezoids Worksheets
This collection of area worksheets encompasses a variety of PDFs to find the area of trapezoids whose dimensions are given as integers, fractions and decimals. Determine the missing parameters by substituting the values in the formulas, solve exercises involving unit conversions too.
Area of a Rhombus Worksheets
Emphasizing on how to find the area of a rhombus the worksheets here contain myriad PDFs to practice the same with dimensions presented as integers, decimals and fractions. Find the diagonal lengths, missing parameters, compute the area, learn to convert to a specified unit and much more.
Area of Kites Worksheets
Improve efficiency in finding the area of kites with these printable worksheets comprising illustrations and exercises in word format. Calculate the area of the kite, find the missing diagonal lengths using the area and much more!
Area of Quadrilaterals Worksheets
Calculate the area of quadrilaterals whose dimensions are presented as whole numbers and fractions. The worksheets on area of a quadrilateral consist of exercises on rectangles, trapezoids, kites in the form of illustrations, on grids and in word format. Practice conversion to a specified unit in the process.
Area of Circles Worksheets
Reaffirm the concept of finding the area of a circle by using these practice worksheets. Learn to find the area or circumference using the given radius or diameter, calculate the area and circumference, compute the radius and diameter from the area or circumference given and a lot more.
Area of an Annulus Worksheets
An Annulus is the region enclosed between two concentric circles. How about calculating their area? Plug into this compact set of printable worksheets and practice calculating the area of those circular rings!
Area of Mixed Shapes Worksheets
The children of grade 5, grade 6 and grade 7 can reinforce their skills in finding the area of mixed shapes by practicing this set of printable worksheets.
Area of Polygons Worksheets
Incorporate these area of polygons worksheets comprising examples and adequate exercises to find the area of regular polygons like triangles, quadrilaterals and irregular polygons using the given side lengths, circumradius and apothem. Free worksheets are available for practice.
Area of Compound Shapes Worksheets
The area of compound shapes worksheets consist of a combination of two or more geometric shapes, find the area of the shaded parts by adding or subtracting the indicated areas, calculate the area of rectilinear shapes (irregular figures) and rectangular paths as well. This practice set is ideal for 4th grade through 7th grade.
Area of a Segment of a Circle Worksheets
Develop practice in finding the area of a segment of a circle with these practice pdfs. Adequate exercises in finding the area of the triangle and the area of the sector using one of the parameters given; are sure to help students master calculating the area of the segment in no time.
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Area Math Problems and Tests for Kids
In daily life, you may have to calculate the area of a space when you buy new furniture or carpeting. In school, your child will have to calculate the area of many different types of shapes for tests. To prepare for these situations, you and your child can review and practice finding areas by working the sample problems below.
How Do I Find the Area in Math Problems?
The area of a rectangle is the product of length and width, which is also sometimes called base and height. Written as an equation, it looks like this: A = l x w. The area of an object is labeled as a square unit. For instance, if a rectangle has a length of three inches and a width of four inches, then the area would be 12 square inches. Remind your child to label every answer on tests because sometimes teachers take off points for answers that aren't labeled, even if the answers are correct.
As your child advances through her study of math, she will be introduced to more challenging shapes, like parallelograms, circles and triangles. These shapes require alternate equations to calculate the area. The area of a triangle, for instance, is A = 1/2 (b)(h). For a triangle that has a base of five feet and a height of two feet, the equation would look like this: A = 1/2 (5)(2), which equals 5 sq. ft. For tests, teachers sometimes provide these equations because there are so many to remember.
Area of a rectangle = l x w
Area of a parallelogram = b x h
Area of a circle = pi (r^2)
Area of a triangle = 1/2 (b)(h)
1. Calculate the area of a triangle whose base is five centimeters and whose height is eight centimeters.
2. A parallelogram's base is 70 millimeters (mm) long. Its height is 30 mm high. What is the area of the parallelogram?
3. If a circle has a diameter of four meters, what is its area?
4. A triangle has a base of 11 inches and a height of two inches. A rectangle has a base of four inches and a height of 14 inches. A circle has a radius of seven inches. Which shape has the largest area?
5. Melanie's bedroom is nine feet by 13 feet long. How much carpeting does she need to buy to cover the area of her room?
1. The area of the triangle is 20 sq. cm.
2. The parallelogram has an area of 2,100 sq. mm.
3. To solve this problem, you first have to determine the circle's radius, which is half of its diameter. Because the diameter is four meters, the radius is two meters. Then, plug this number into the equation, like this: A = pi (2^2). The answer is approximately 12.57 sq. m.
4. In order to figure out which shape has the largest area, you must calculate the area for all three shapes. Beginning with the triangle, the equation should look like this: A = 1/2 (11)(2), which equals 11 sq. in. The area of the rectangle is A = (4)(14) = 56 sq. in. Finally, the circle's area is A = pi (7^2) = approximately 154 sq. in. Thus, the circle has the largest area.
5. Melanie must buy 117 sq. ft. of carpeting.
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Supplementing your child's math lessons with a few problems and worksheets of your own will help you become better acquainted with his or her curriculum, and will help your child perform better in the classroom. Here are some tips for creating them.
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Area of Plane Shapes
Area is the size of a surface! Learn more about Area , or try the Area Calculator .
Example: What is the area of this rectangle?
The formula is:
Area = w × h w = width h = height
We know w = 5 and h = 3 , so:
Area = 5 × 3 = 15
Example: What is the area of this circle?
Radius = r = 3
Example: What is the area of this triangle?
Height = h = 12
Base = b = 20
Area = ½ × b × h = ½ × 20 × 12 = 120
A harder example:
Example: Sam cuts grass at $0.10 per square meter
How much does sam earn cutting this area:.
Let's break the area into two parts:
Part A is a square:
Part B is a triangle. Viewed sideways it has a base of 20m and a height of 14m.
So the total area is:
Sam earns $0.10 per square meter
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Videos, worksheets, 5-a-day and much more, area of compound shapes practice questions, click here for questions, click here for answers.
Free Mathematics Tutorials
Area of Triangles Problems with Solutions
A set of problems on how to calculate the area of triangles using different formulas are presented along with detailed solutions .
Formulas for Area of Triangles
We first recall some of the most widely used formulas used to calculate the area of a triangle.
Formula 1 - Base and Height of a triangle are known
Area = (1 / 2) × base × height = (1 / 2) × AB × CD
Formula 2 - Two sides of a triangle and the angle between them are known
Formula 3 - Three sides of a triangle are known
Formula 4 - the three vertices of the triangle are known by their coordinates, problems on areas with solutions.
Problem 4 Find the area of an equilateral triangle wide length equal to 6 cm.
Problem 7 Find the area of the triangle whose vertices are the points of intersection of the lines with equations y = x , y = - x + 4 and y = 2 x + 2.
Solutions to the Above Problems
Solution to Problem 2 Triangle CDB has a base CD of length 20. Its height is the side AB of the triangle because it start from the vertex B opposite the base CD is perpendicular to AC and therefore to the base DC. Hence the use of formula 1 to find the area of triangle. Area = (1/2) × 20 × 80 = 40 unit 2
Solution to Problem 3 We are given two sides and the angle opposite one of the sides. One way to find the area is to find angle B and use formula 2. Use the sine law to write sin(A) / BC = sin(C) / AB which gives sin(A) = 6 sin(55°) / 5 A = sin -1 (6 sin(55°) / 5) = 79.4 ° We now determine angle B using that fact that the sum of all angles in a triangle to be 180 °. B = 180 - 55 - 79.4 = 45.6 ° Use formula 2 to find the area Area = (1 / 2) × 5 × 6 × sin(45.6 °) = 10.7 unit 2
Solution to Problem 8 The shape whose are is to be calculated is made up of two triangles ADC and ACB with a common side AC. The area of triangle ACB can be calculated using formula 2 (we know two sides and the angle between them). For triangle ADC ,we know two sides. We can find side AC using the cosine law . in triangle ACB as follows AC 2 = AB 2 + CB 2 - 2 (AB)(AC) cos (B) Substitute by the numerical values and calculate AC AC = √( 400 2 + 800 2 - 2 (400)(800) cos (45°) ) = 589.5 We may use Heron's formula since we know the three sides of triangle ADC Let s = (1/2)(AD + DC + CA) = (1/2)(245 + 432 + 589.5) = 633.25 Area of triangle ADC = √( s × (s - AD) × (s - DC) × (s - CA) ) = √( 633.25 × (633.25 - 245) × (633.25 - 432) × (633.25 - 589.5) ) = 46526 ft 2 Area of triangle ACB = (1 / 2) BA × BC × sin (B) = (1 / 2) 400 × 800 × sin (45°) = 113137 ft 2 Total area of the given shape is obtained by adding the areas of triangles ADC and ACB Area = 46526 + 113137 = 158663 ft 2
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Module 2: Limits
The area problem and integral calculus, learning outcomes.
- Describe the area problem and how it was solved by the integral
- Explain how the idea of a limit is involved in solving the area problem
- Recognize the total cost as the area under the marginal cost curve
We now turn our attention to a classic question from calculus. Many quantities in physics—for example, quantities of work—may be interpreted as the area under a curve. This leads us to ask the question: How can we find the area between the graph of a function and the [latex]x[/latex]-axis over an interval (Figure 7)?
Figure 7. The Area Problem: How do we find the area of the shaded region?
As in the answer to our previous questions on velocity, we first try to approximate the solution. We approximate the area by dividing up the interval [latex][a,b][/latex] into smaller intervals in the shape of rectangles. The approximation of the area comes from adding up the areas of these rectangles (Figure 8). Recall that the area of a rectangle can be found simply by taking the length times the width.
Figure 8. The area of the region under the curve is approximated by summing the areas of thin rectangles.
As the widths of the rectangles become smaller (approach zero), the sums of the areas of the rectangles approach the area between the graph of [latex]f(x)[/latex] and the [latex]x[/latex]-axis over the interval [latex][a,b][/latex]. Once again, we find ourselves taking a limit. Limits of this type serve as a basis for the definition of the definite integral. Integral calculus is the study of integrals and their applications.
Example: Estimation Using Rectangles
Estimate the area between the [latex]x[/latex]-axis and the graph of [latex]f(x)=x^2+1[/latex] over the interval [latex][0,3][/latex] by using the three rectangles shown in Figure 9.
Figure 9. The area of the region under the curve of [latex]f(x)=x^2+1[/latex] can be estimated using rectangles.
Watch the following video to see the worked solution to Example: Estimation Using Rectangles
Estimate the area between the [latex]x[/latex]-axis and the graph of [latex]f(x)=x^2+1[/latex] over the interval [latex][0,3][/latex] by using the three rectangles shown here:
Use Figure 9 as a guide.
Other Aspects of Calculus
So far, we have studied functions of one variable only. Such functions can be represented visually using graphs in two dimensions; however, there is no good reason to restrict our investigation to two dimensions. Suppose, for example, that instead of determining the velocity of an object moving along a coordinate axis, we want to determine the velocity of a rock fired from a catapult at a given time, or of an airplane moving in three dimensions. We might want to graph real-value functions of two variables or determine volumes of solids of the type shown in Figure 11. These are only a few of the types of questions that can be asked and answered using multivariable calculus. Informally, multivariable calculus can be characterized as the study of the calculus of functions of two or more variables. However, before exploring these and other ideas, we must first lay a foundation for the study of calculus in one variable by exploring the concept of a limit.
Figure 11. We can use multivariable calculus to find the volume between a surface defined by a function of two variables and a plane.
- 2.1 A Preview of Calculus. Authored by : Ryan Melton. License : CC BY: Attribution
- Calculus Volume 1. Authored by : Gilbert Strang, Edwin (Jed) Herman. Provided by : OpenStax. Located at : https://openstax.org/details/books/calculus-volume-1 . License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike . License Terms : Access for free at https://openstax.org/books/calculus-volume-1/pages/1-introduction
The following are calculators to evaluate the area of seven common shapes. The area of more complex shapes can usually be obtained by breaking them down into their aggregating simple shapes, and totaling their areas. This calculator is especially useful for estimating land area.
Use the Triangle Calculator to determine all three edges of the triangle given other parameters.
Related Surface Area Calculator | Volume Calculator
Area is a quantity that describes the size or extent of a two-dimensional figure or shape in a plane. It can be visualized as the amount of paint that would be necessary to cover a surface, and is the two-dimensional counterpart of the one-dimensional length of a curve, and three-dimensional volume of a solid. The standard unit of area in the International System of Units (SI) is the square meter, or m 2 . Provided below are equations for some of the most common simple shapes, and examples of how the area of each is calculated.
A rectangle is a quadrilateral with four right angles. It is one of the simplest shapes, and calculating its area only requires that its length and width are known (or can be measured). A quadrilateral by definition is a polygon that has four edges and vertices. In the case of a rectangle, the length typically refers to the longer two edges of the quadrilateral, while the width refers to the shorter of the two edges. When the length and width of a rectangle are equal, the shape is a special case of a rectangle, called a square. The equation for calculating the area of a rectangle is as follows:
area = length × width
The Farmer and his Daughter – Unsold Land
Imagine a farmer trying to sell a piece of land that happens to be perfectly rectangular. Because he owns some cows that he did not want frolicking freely, he fenced the piece of land and knew the exact length and width of each edge. The farmer also lives in the United States, and being unfamiliar with the use of SI units, still measures his plot of land in terms of feet. The foot was defined to be exactly 0.3048 meters in 1959 after having changed over an extensive period of time, as historically, the human body was often used to provide a basis for units of length, and unsurprisingly, was inconsistent based on time and location. Tangent aside, the farmer's plot of land has a length of 220 feet, and a width of 99 feet. Using this information:
area = 220 × 99 = 21780 sq ft
The farmer's plot of land, which has an area of 21,780 square feet, equates to half an acre, where an acre is defined as the area of 1 chain by 1 furlong, which is defined by something else, and so on, and is why SI now exists. Unfortunately for the farmer, he lives in an area predominated by foreign investors with smaller feet, who felt that they should be getting more square feet for their money, and his land remains unsold today.
There are many equations for calculating the area of a triangle based on what information is available. As mentioned in the calculator above, please use the Triangle Calculator for further details and equations for calculating the area of a triangle, as well as determining the sides of a triangle using whatever information is available. Briefly, the equation used in the calculator provided above is known as Heron's formula (sometimes called Hero's formula), referring to the Hero of Alexandria, a Greek mathematician and engineer considered by some to be the greatest experimenter of ancient times. The formula is as follows:
area = √ s(s - x)(s - y)(s - z)
The Farmer and his Daughter - Triangle Daze
At this point in time, through extreme effort and perseverance, the farmer has finally sold his 21,780 sq ft plot of land and has decided to use some of the money earned to build a pool for his family. Unfortunately for the farmer, he does not consider the fact that the maintenance costs of a pool for one year alone could likely pay for his children to visit any pool or water theme park for years to come. Even more unfortunately for the farmer, his 7-year-old daughter who has recently traveled to Egypt vicariously through Dora the Explorer, has fallen in love with triangles, and insists that the pool not only be triangular in shape, but also that the measurements must only include the number 7, to represent her age and immortalize this point of her life in the form of a triangular pool. Being a doting father, the farmer acquiesces to his daughter's request and proceeds to plan the construction of his triangular pool. The farmer must now determine whether he has sufficient area in his backyard to house a pool. While the farmer has begun to learn more about SI units, he is as yet uncomfortable with their use and decides that his only viable option is to construct a pool in the form of an equilateral triangle with sides 77 ft in length, since any other variation would either be too large or small. Given these dimensions, the farmer determines the necessary area as follows:
area = √ 115.5 × (115.5 - 77) 3 = 2567.33 sq ft
Since the longest distance between any two points of an equilateral triangle is the length of the edge of the triangle, the farmer reserves the edges of the pool for swimming "laps" in his triangular pool with a maximum length approximately half that of an Olympic pool, but with double the area – all under the watchful eyes of the presiding queen of the pool, his daughter, and the disapproving glare of his wife.
A trapezoid is a simple convex quadrilateral that has at least one pair of parallel sides. The property of being convex means that a trapezoid's angle does not exceed 180° (in contrast, a concave quadrilateral would), while being simple reflects that trapezoids are not self-intersecting, meaning two non-adjacent sides do not cross. In a trapezoid, the parallel sides are referred to as the bases of the trapezoid, and the other two sides are called the legs. There exist more distinctions and classifications for different types of trapezoids, but their areas are still calculated in the same manner using the following equation:
The Farmer and his Daughter – Ramping Endeavors
Two years have passed since the farmer's pool was completed, and his daughter has grown and matured. While her love for triangles still persists, she eventually came to the realization that no matter how well-"triangled" she was, triangles alone cannot make the world go round, and that Santa's workshop could not plausibly balance on the North Pole, were the world a pyramid rather than a sphere. Slowly, she has begun to accept other shapes into her life and pursues her myriad different interests – currently freestyle BMX. As such, she requires a ramp, but unfortunately for the farmer, not just any ramp. The ramp must be comprised of only shapes that can be formed using multiple triangles, since, like her rap idol B.o.B, the farmer's daughter still has difficulty accepting the reality of curved surfaces. It must, of course, also only use the number 9 in its measurements to reflect her age. The farmer decides that his best option is to build a ramp comprised of multiple rectangles, with the side face of the ramp being in the shape of a trapezoid. As the farmer has now become more comfortable with SI, he is able to be more creative with his use of units, and can build a more reasonably sized ramp while adhering to his daughter's demands. He decides to build a ramp with a trapezoidal face with a height of 9 ft, a bottom base of length 29.528 ft (9 m), and a top base of 9 ft. The area of the trapezoid is calculated as follows:
A circle is a simple closed shape formed by the set of all points in a plane that are a given distance from a given center point. This distance from the center to any point on the circle is called the radius. More detail can be found regarding circles on the Circle Calculator page, but to calculate the area, it is only necessary to know the radius, and understand that values in a circle are related through the mathematical constant π . The equation for calculating the area of a circle is as follows:
area = πr 2
The Farmer and his Daughter – Circle of Li(f)es
Another six years have passed, and his daughter has grown into a strong, beautiful, powerful, confident 15-year-old ingrate solely focused on seeking external validation from acquaintances and strangers on social media while wholeheartedly ignoring genuine support from immediate family and friends. Having had an argument with her father about her excessive use of social media, she decides to prey on her father's fear of the unknown, and belief in the supernatural in order to prank him. Not knowing where to start, she walks around town talking to a variety of strangers all of whom seemingly have endless founts of wisdom and advice, where she learns about crop circles and their association with aliens and unidentified flying objects as well as many other topics that ignore all scientific and logical explanations. Having finally been convinced of the spherical nature of the Earth, deleted all her past social media posts relating to B.o.B, and expanded her love of triangles to an acceptance of other shapes, she decides to make a basic crop circle consisting of a number of concentric circles, and wants to determine the area necessary to create a crop circle with an outer radius of 15 ft. She does so using the following equation:
area = π × 15 2 = 706.858 sq ft
Unfortunately for the farmer, not only is he terrified of the crop circle that appeared overnight on the night that his daughter told him she was at a slumber party with her friends, that for some odd reason did not result in superfluous Instagram posts (he was of course his daughter's first follower), but the number of "circle investigators" and "cereologists" showing up on his farm to examine, and subsequently confirm the authenticity of the crop circle as an alien construction, cost him significant damages to his crops.
A sector of a circle is essentially a proportion of the circle that is enclosed by two radii and an arc. Given a radius and an angle, the area of a sector can be calculated by multiplying the area of the entire circle by a ratio of the known angle to 360° or 2π radians, as shown in the following equation:
The Farmer and his Daughter – Sectioning Family
The farmer and his family are facing their most significant dilemma to date. One year has passed, and the farmer's daughter is now 16 years old and as part of her birthday celebration, her mother baked her favorite dessert, blackberry pie. Unfortunately for the farmer's daughter, blackberry pie also happens to be a favorite food of their pet raccoon, Platypus, as evidenced by 180° worth of the pie being missing with telltale signs of the culprit in the form of crumbs leading towards the overindulgent raccoon. Initially, the pie would easily have been split between three people and one raccoon, but now, half the pie has to be divided between three people as a chagrined, but satiated Platypus watches from a distance. Given that each person will receive 60° worth of the pie with a radius of 16 inches, the area of pie that each person receives can be calculated as follows:
area= 60°/360° × π × 16 2 = 134.041 in 2
As a result of Platypus' inconsideration, each person gets one-third less pie, and the daughter contemplatively recalls American history class, where she learned about the Battle of the Alamo and the portrayal of the folk hero Davy Crockett and his coonskin hat.
An ellipse is the generalized form of a circle, and is a curve in a plane where the sum of the distances from any point on the curve to each of its two focal points is constant, as shown in the figure below, where P is any point on the ellipse, and F 1 and F 2 are the two foci.
When F 1 = F 2 , the resulting ellipse is a circle. The semi-major axis of an ellipse, as shown in the figure that is part of the calculator, is the longest radius of the ellipse, while the semi-minor axis is the shortest. The major and minor axes refer to the diameters rather than radii of the ellipse. The equation for calculating the area of an ellipse is similar to that for calculating the area of a circle, with the only difference being the use of two radii, rather than one (since the foci are in the same location for a circle):
area = πab where a and b are the semi-major and semi-minor axes
The Farmer and his Daughter – Falling out of Orbit
Two years have passed since the mysterious disappearance of the family pet, Platypus, and the farmer's daughter's fortuitous winning of a furry accessory through the school lottery that helped fill the void of the loss of their beloved pet. The farmer's daughter is now 18 and is ready to escape rural Montana for a college life replete with freedom and debauchery, and of course some learning on the side. Unfortunately for the farmer's daughter, she grew up in an environment brimming with positive reinforcement, and subsequently, the mentality that one should "shoot for the moon [since] even if you miss, you'll land among the stars," as well as the assertion from everyone around her that she could do absolutely anything she put her mind to! As such, with her suboptimal grades, lack of any extracurricular activities due to her myriad different interests consuming all of her free time, zero planning, and her insistence on only applying to the very best of the best universities, the shock that resulted when she was not accepted to any of the top-tier universities she applied to could be reasonably compared to her metaphorically landing in deep space, inflating, freezing, and quickly suffocating when she missed the moon and landed among the stars. Along with her lungs, her dream of becoming an astrophysicist was summarily ruptured, at least for the time being, and she was relegated to calculating the elliptical area necessary in her room to build a human sized model of Earth's near elliptical orbit around the sun, so she could gaze longingly at the sun in the center of her room and its personification of her heart, burning with passion, but surrounded by the cold vastness of space, with the Earth's distant rotation mockingly representing the distance between her dreams, and solid ground.
area = π × 18 ft × 20 ft= 1130.97 sq ft
A parallelogram is a simple quadrilateral which has two pairs of parallel sides, where the opposite sides and angles of the quadrilateral have equal lengths and angles. Rectangles, rhombuses, and squares are all special cases of parallelograms. Remember that the classification of a "simple" shape means that the shape is not self-intersecting. A parallelogram can be divided into a right triangle and a trapezoid, which can further be rearranged to form a rectangle, making the equation for calculating the area of a parallelogram essentially the same as that for calculating a rectangle. Instead of length and width however, a parallelogram uses base and height, where the height is the length of the perpendicular between a pair of bases. Based on the figure below, the equation for calculating the area of a parallelogram is as follows:
area = b × h
The Farmer and his Daughter – Diamond in the Sky
Another two years have passed in the life of the farmer and his family, and though his daughter had been a cause for intense worry, she has finally bridged the distance between the blazing sun that is her heart, and the Earth upon which society insists she must remain grounded. Through the struggles that ensued from her self-imposed isolation, surrounded by imagined, judgmental eyes presuming her failure from all directions, the farmer's daughter emerged from the pressures of the Earth like a diamond, shining brightly and firm in her resolve. Despite all its drawbacks, she decides that there is little choice but to persist through the asteroid field of life in hopes that a Disney fairy tale ending exists. At long last, fortunately for the farmer's daughter and her family, hope does appear, but not in the form of a Prince Charming, but rather as a sign from the supposed heavens. Through all of her metaphorical musings and tribulations involving space, it almost becomes believable that the farmer's daughter somehow influenced the massive octahedral diamond asteroid falling squarely, but safely upon their farmland, which she interprets as representing her journey, formation, and eventual homecoming. The farmer's daughter proceeds to measure the area of one of the rhomboidal faces of her newly found symbol of life:
area = 20 ft × 18 ft = 360 sq ft
Unfortunately for the farmer's daughter, the appearance of the enormous diamond drew attention from all over the world, and after sufficient pressure, she succumbs to the human within her, and sells the diamond, the very representation of her life and soul, to a wealthy collector, and proceeds to live the rest of her life in lavish indulgence, abandoning her convictions, and losing herself within the black hole of society.
Common Area Units
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How to Find Area
Last Updated: May 14, 2023 Fact Checked
This article was co-authored by David Jia . David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. There are 11 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 80,697 times.
Area is a measurement of the amount of space inside a two-dimensional figure. Sometimes, finding area can be as simple as simply multiplying two numbers, but oftentimes it can be more complicated. Read this article for a brief overview for the following shapes: quadrilaterals, triangles, circles, surface areas of pyramids and cylinders, and the area under an arc.
- For more extensive instructions, check out How to Find the Area of a Quadrilateral
- This works because a square is simply a special rectangle that has equal width and length. So, in solving k=b*h, b and h are both the same value. So, you end up squaring a single number in order to find the area.
- If the side opposite to the base is not long enough that the perpendicular line crosses it, extend the side along the line until it intersects the perpendicular line.
- For more extensive instructions, check out How to Find the Area of a Parallelogram
- For more extensive instructions, check out How to Calculate the Area of a Trapezoid
- For more extensive instructions, check out How to Calculate the Area of a Triangle
- For more extensive instructions, check out How to Find the Area of Regular Polygons
- For more extensive instructions, check out How to Calculate the Area of a Circle
Surface Area of a Pyramid
Surface Area of a Cylinder
- For more extensive instructions, check out How to Find the Surface Area of Cylinders
The Area Under a Function
Say you want to find the area under a curve and above the x-axis modeled by function f(x) in the domain interval x within [a,b]. This method requires knowledge of integral calculus. If you have not taken an introductory calculus course, this method may not make any sense.
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- ↑ https://www.khanacademy.org/math/cc-third-grade-math/imp-geometry/imp-multiply-to-find-area/a/area-rectangles-review
- ↑ David Jia. Academic Tutor. Expert Interview. 7 January 2021.
- ↑ https://www.cuemath.com/measurement/area-of-square/
- ↑ https://www.omnicalculator.com/math/square-area
- ↑ https://www.bbc.com/bitesize/guides/z2mtyrd/revision/5
- ↑ https://www.khanacademy.org/math/cc-sixth-grade-math/x0267d782:cc-6th-plane-figures/cc-6th-parallelogram-area/a/area-of-parallelogram
- ↑ https://www.mathopenref.com/trapezoidarea.html
- ↑ https://www.varsitytutors.com/isee_middle_level_math-help/how-to-find-the-area-of-a-trapezoid
- ↑ https://www.omnicalculator.com/math/triangle-area
- ↑ https://www.mathsisfun.com/geometry/circle-area.html
- ↑ https://www.omnicalculator.com/math/surface-area-of-cylinder
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