## Solving Triangles

"Solving" means finding missing sides and angles.

## Six Different Types

If you need to solve a triangle right now choose one of the six options below:

Which Sides or Angles do you know already? (Click on the image or link)

... or read on to find out how you can become an expert triangle solver :

## Your Solving Toolbox

Want to learn to solve triangles?

Imagine you are " The Solver " ... ... the one they ask for when a triangle needs solving!

In your solving toolbox (along with your pen, paper and calculator) you have these 3 equations:

## 1. The angles always add to 180° :

A + B + C = 180°

When you know two angles you can find the third.

## 2. Law of Sines (the Sine Rule):

When there is an angle opposite a side, this equation comes to the rescue.

Note: angle A is opposite side a, B is opposite b, and C is opposite c.

## 3. Law of Cosines (the Cosine Rule):

This is the hardest to use (and remember) but it is sometimes needed to get you out of difficult situations.

It is an enhanced version of the Pythagoras Theorem that works on any triangle.

## Six Different Types (More Detail)

There are SIX different types of puzzles you may need to solve. Get familiar with them:

This means we are given all three angles of a triangle, but no sides.

AAA triangles are impossible to solve further since there are is nothing to show us size ... we know the shape but not how big it is.

We need to know at least one side to go further. See Solving "AAA" Triangles .

This mean we are given two angles of a triangle and one side, which is not the side adjacent to the two given angles.

Such a triangle can be solved by using Angles of a Triangle to find the other angle, and The Law of Sines to find each of the other two sides. See Solving "AAS" Triangles .

This means we are given two angles of a triangle and one side, which is the side adjacent to the two given angles.

In this case we find the third angle by using Angles of a Triangle , then use The Law of Sines to find each of the other two sides. See Solving "ASA" Triangles .

This means we are given two sides and the included angle.

For this type of triangle, we must use The Law of Cosines first to calculate the third side of the triangle; then we can use The Law of Sines to find one of the other two angles, and finally use Angles of a Triangle to find the last angle. See Solving "SAS" Triangles .

This means we are given two sides and one angle that is not the included angle.

In this case, use The Law of Sines first to find either one of the other two angles, then use Angles of a Triangle to find the third angle, then The Law of Sines again to find the final side. See Solving "SSA" Triangles .

This means we are given all three sides of a triangle, but no angles.

In this case, we have no choice. We must use The Law of Cosines first to find any one of the three angles, then we can use The Law of Sines (or use The Law of Cosines again) to find a second angle, and finally Angles of a Triangle to find the third angle. See Solving "SSS" Triangles .

## Tips to Solving

Here is some simple advice:

When the triangle has a right angle, then use it, that is usually much simpler.

When two angles are known, work out the third using Angles of a Triangle Add to 180° .

Try The Law of Sines before the The Law of Cosines as it is easier to use.

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Example 1 : Solve the right triangle shown in Figure (b) if ∠ B = 22°

Because the three angles of a triangle must add up to 180°, ∠ A = 90 ∠ B thus ∠ A = 68°.

The following is an alternate way to solve for sides a and c:

This alternate solution may be easier because no division is involved.

Example 2 : Solve the right triangle shown in Figure (b) if b = 8 and a = 13.

You can use the Pythagorean theorem to find the missing side, but trigonometric relationships are used instead. The two missing angle measurements will be found first and then the missing side.

Example 3: A large airplane (plane A) flying at 26,000 feet sights a smaller plane (plane B) traveling at an altitude of 24,000 feet. The angle of depression is 40°. What is the line of sight distance ( x ) between the two planes?

Figure 3 illustrates the conditions of this problem.

From Figure 3 , you can find the solution by using the sine of 40 °:

Example 4: A ladder must reach the top of a building. The base of the ladder will be 25′ from the base of the building. The angle of elevation from the base of the ladder to the top of the building is 64°. Find the height of the building (h) and the length of the ladder ( m ).

Figure 4 illustrates the conditions of this problem.

Example 5: A woodcutter wants to determine the height of a tall tree. He stands at some distance from the tree and determines that the angle of elevation to the top of the tree is 40°. He moves 30′ closer to the tree, and now the angle of elevation is 50°. If the woodcutter's eyes are 5′ above the ground, how tall is the tree?

Figure 5 can help you visualize the problem.

From the small right triangle and from the large right triangle, the following relationships are evident:

Substituting the first equation in the second yields:

Note that 5′ must be added to the value of x to get the height of the tree, or 90.06′ tall.

Example 6: Using Figure 6, find the length of sides x and y and the area of the large triangle.

Because this is an isosceles triangle, and equal sides are opposite equal angles, the values of x and y are the same. If the triangle is divided into two right triangles, the base of each will be 6. Therefore,

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## Right Triangle Questions

Multiple choice questions right triangle problems related to trigonometry with answers at the bottom of the page.

## Questions with their Answers

Question 10, question 11, question 12, more references and links.

## Chapter 4.2: Right Triangle Trigonometry

Using right triangle trigonometry to solve applied problems, using trigonometric functions.

In previous examples, we evaluated the sine and cosine in triangles where we knew all three sides. But the real power of right-triangle trigonometry emerges when we look at triangles in which we know an angle but do not know all the sides.

## How To: Given a right triangle, the length of one side, and the measure of one acute angle, find the remaining sides.

- For each side, select the trigonometric function that has the unknown side as either the numerator or the denominator. The known side will in turn be the denominator or the numerator.
- Write an equation setting the function value of the known angle equal to the ratio of the corresponding sides.
- Using the value of the trigonometric function and the known side length, solve for the missing side length.

## Example 5: Finding Missing Side Lengths Using Trigonometric Ratios

Find the unknown sides of the triangle in Figure 11.

We know the angle and the opposite side, so we can use the tangent to find the adjacent side.

We rearrange to solve for [latex]a[/latex].

We can use the sine to find the hypotenuse.

Again, we rearrange to solve for [latex]c[/latex].

A right triangle has one angle of [latex]\frac{\pi }{3}[/latex] and a hypotenuse of 20. Find the unknown sides and angle of the triangle.

Right-triangle trigonometry has many practical applications. For example, the ability to compute the lengths of sides of a triangle makes it possible to find the height of a tall object without climbing to the top or having to extend a tape measure along its height. We do so by measuring a distance from the base of the object to a point on the ground some distance away, where we can look up to the top of the tall object at an angle. The angle of elevation of an object above an observer relative to the observer is the angle between the horizontal and the line from the object to the observer’s eye. The right triangle this position creates has sides that represent the unknown height, the measured distance from the base, and the angled line of sight from the ground to the top of the object. Knowing the measured distance to the base of the object and the angle of the line of sight, we can use trigonometric functions to calculate the unknown height. Similarly, we can form a triangle from the top of a tall object by looking downward. The angle of depression of an object below an observer relative to the observer is the angle between the horizontal and the line from the object to the observer’s eye.

## How To: Given a tall object, measure its height indirectly.

- Make a sketch of the problem situation to keep track of known and unknown information.
- Lay out a measured distance from the base of the object to a point where the top of the object is clearly visible.
- At the other end of the measured distance, look up to the top of the object. Measure the angle the line of sight makes with the horizontal.
- Write an equation relating the unknown height, the measured distance, and the tangent of the angle of the line of sight.
- Solve the equation for the unknown height.

## Example 6: Measuring a Distance Indirectly

To find the height of a tree, a person walks to a point 30 feet from the base of the tree. She measures an angle of [latex]57^\circ [/latex] between a line of sight to the top of the tree and the ground, as shown in Figure 13. Find the height of the tree.

We know that the angle of elevation is [latex]57^\circ [/latex] and the adjacent side is 30 ft long. The opposite side is the unknown height.

The trigonometric function relating the side opposite to an angle and the side adjacent to the angle is the tangent. So we will state our information in terms of the tangent of [latex]57^\circ [/latex], letting [latex]h[/latex] be the unknown height.

The tree is approximately 46 feet tall.

How long a ladder is needed to reach a windowsill 50 feet above the ground if the ladder rests against the building making an angle of [latex]\frac{5\pi }{12}[/latex] with the ground? Round to the nearest foot.

- Precalculus. Authored by : OpenStax College. Provided by : OpenStax. Located at : http://cnx.org/contents/fd53eae1-fa23-47c7-[email protected]:1/Preface . License : CC BY: Attribution
- Example: Determine the Length of a Side of a Right Triangle Using a Trig Equation. Authored by : Mathispower4u. Located at : https://youtu.be/8jU2R3BuR5E . License : All Rights Reserved . License Terms : Standard YouTube License

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## How to Solve Triangles Problems? (+FREE Worksheet!)

Learn how to solve mathematics problems related to triangles area and angles using common triangle formulas.

## Related Topics

- How Calculate the Area of Trapezoids
- How to Find the volume and surface area of Rectangular Prisms
- How to Find the Area and Circumference of Circles
- How to Find Volume and Surface Area of Cubes
- How to Find the Perimeter of Polygons

## Step by step guide to solve Triangles

- In any triangle the sum of all angles is \(180\) degrees.

## Triangles – Example 1:

Use the are formula: Area \(= \color{blue}{\frac{1}{2 }(base \ × \ height)}\) base \(=10\) and height \(=6\) Area \(= \frac{1}{2} (10×6)=\frac{1}{2} (60)=30\)

## Triangles – Example 2:

What is the missing angle of the following triangle?

All angles in a triangle sum up to \(180\) degrees. Then: : \(45+60+x=180 → 105+x=180 → x=180-105=75\)

## Triangles – Example 3:

What is the area of the following triangle?

Use the are formula: Area \(= \color{blue}{\frac{1}{2 }(base \ × \ height)}\) base \(=12\) and height \(=8\) Area \(= \frac{1}{2} (12×8)= \frac{1}{2} (96)=48 \)

## Triangles – Example 4:

What is the area of a triangle with base \(6\) and height \(5\)?

Use the are formula: Area \(= \color{blue}{\frac{1}{2 }(base \ × \ height)}\) base \(=6\) and height \(=5\) Area \(= \frac{1}{2 } (5×6)= \frac{1}{2} (30)=15 \)

## Exercises for Solving Triangles

Find the measure of the unknown angle in each triangle., download triangles worksheet.

- \(\color{blue}{45^{\circ}}\)
- \( \color{blue}{ 15^{\circ}}\)
- \( \color{blue}{ 40^{\circ}}\)

by: Reza about 4 years ago (category: Articles , Free Math Worksheets )

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## Right Triangle Side and Angle Calculator

How to find the sides of a right triangle, how to find the angle of a right triangle, how do you solve a right angle triangle with only one side, how to find the missing side of a right triangle how to find the angle example.

Finding the missing side or angle couldn't be easier than with our great tool – right triangle side and angle calculator. Choose two given values, type them into the calculator, and the calculator will determine the remaining unknowns in a blink of an eye! If you are wondering how to find the missing side of a right triangle, keep scrolling, and you'll find the formulas behind our calculator.

There are a few methods of obtaining right triangle side lengths. Depending on what is given, you can use different relationships or laws to find the missing side:

1. Given two sides

If you know two other sides of the right triangle, it's the easiest option; all you need to do is apply the Pythagorean theorem:

a² + b² = c²

If leg a is the missing side, then transform the equation to the form where a is on one side and take a square root:

a = √(c² - b²)

If leg b is unknown, then:

b = √(c² - a²)

For hypotenuse c missing, the formula is:

c = √(a² + b²)

🙋 Our Pythagorean theorem calculator will help you if you have any doubts at this point.

2. Given an angle and the hypotenuse

Apply the law of sines or trigonometry to find the right triangle side lengths:

a = c × sin(α) or a = c × cos(β)

b = c × sin(β) or b = c × cos(α)

🙋 Refresh your knowledge with Omni's law of sines calculator !

3. Given an angle and one leg

Find the missing leg using trigonometric functions:

a = b × tan(α)

b = a × tan(β)

4. Given the area and one leg

As we remember from basic triangle area formula , we can calculate the area by multiplying the triangle height and base and dividing the result by two. A right triangle is a special case of a scalene triangle, in which one leg is the height when the second leg is the base, so the equation gets simplified to:

area = a × b / 2

For example, if we know only the right triangle area and the length of the leg a , we can derive the equation for the other sides:

- b = 2 × area / a
- c = √(a² + (2 × area / a)²)

🙋 For this type of problem, see also our area of a right triangle calculator .

If you know one angle apart from the right angle, the calculation of the third one is a piece of cake:

Given β : α = 90 - β

Given α : β = 90 - α

However, if only two sides of a triangle are given, finding the angles of a right triangle requires applying some basic trigonometric functions:

- sin(α) = a / c so α = arcsin(a / c) (inverse sine);
- cos(α) = b / c so α = arccos(b / c) (inverse cosine);
- tan(α) = a / b so α = arctan(a / b) (inverse tangent);
- cot(α) = b / a so α = arccot(b / a) (inverse cotangent);

and for β :

- sin(β) = b / c so β = arcsin(b / c) (inverse sine);
- cos(β) = a / c so β = arccos(a / c) (inverse cosine);
- tan(β) = b / a so β = arctan(b / a) (inverse tangent);
- cot(β) = a / b so β = arccot(a / b) (inverse cotangent).

To solve a triangle with one side, you also need one of the non-right angled angles . If not, it is impossible:

If you have the hypotenuse , multiply it by sin(θ) to get the length of the side opposite to the angle.

Alternatively, multiply the hypotenuse by cos(θ) to get the side adjacent to the angle.

If you have the non-hypotenuse side adjacent to the angle, divide it by cos(θ) to get the length of the hypotenuse .

Alternatively, multiply this length by tan(θ) to get the length of the side opposite to the angle.

If you have an angle and the side opposite to it, you can divide the side length by sin(θ) to get the hypotenuse .

Alternatively, divide the length by tan(θ) to get the length of the side adjacent to the angle.

Let's show how to find the sides of a right triangle with this tool:

Assume we want to find the missing side given area and one side. Select the proper option from a drop-down list . It's the third one.

Type in the given values . For example, the area of a right triangle is equal to 28 in² and b = 9 in.

Our right triangle side and angle calculator displays missing sides and angles! Now we know that:

- a = 6.222 in
- c = 10.941 in

Now, let's check how finding the angles of a right triangle works:

Refresh the calculator. Pick the option you need . Assume that we have two sides, and we want to find all angles. The default option is the right one.

Enter the side lengths . Our right triangle has a hypotenuse equal to 13 in and a leg a = 5 in.

Missing side and angles appear . In our example, b = 12 in, α = 22.62° and β = 67.38°.

## How many lines of symmetry does a right triangle have?

If a right triangle is isosceles (i.e., its two non-hypotenuse sides are the same length), it has one line of symmetry . Otherwise, the triangle will have no lines of symmetry .

## Can a right angled triangle have equal sides?

No, a right triangle cannot have all 3 sides equal , as all three angles cannot also be equal. One has to be 90° by definition. A right triangle can, however, have its two non-hypotenuse sides equal in length. This would also mean the two other angles are equal to 45°.

## Are all right triangles similar?

Not all right-angled triangles are similar , although some can be. They are similar if all their angles are the same length, or if the ratio of two of their sides is the same.

## Equilateral triangle area

Place value.

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Trigonometry 4 units · 36 skills. Unit 1 Right triangles & trigonometry. Unit 2 Trigonometric functions. Unit 3 Non-right triangles & trigonometry. Unit 4 Trigonometric equations and identities. Course challenge. Test your knowledge of the skills in this course. Start Course challenge. Math.

Problem : Solve the following right triangle, in which C = 90 o: A = 40, B = 50. This triangle cannot be solved. Three angles is not sufficient information to determine a unique triangle. Problem : Solve the following right triangle, in which C = 90 o: b = 6, B = 72 o. A = 90 o - B = 18 o.

Lesson 6: Solving for a side in a right triangle using the trigonometric ratios. Solving for a side in right triangles with trigonometry. Solving for a side in right triangles with trigonometry. Solve for a side in right triangles. Math >. High school geometry >. Right triangles & trigonometry >.

The trigonometric ratio that contains both of those sides is the sine. [I'd like to review the trig ratios.] Step 2: Create an equation using the trig ratio sine and solve for the unknown side. sin ( B) = opposite hypotenuse Define sine. sin ( 50 ∘) = A C 6 Substitute. 6 sin ( 50 ∘) = A C Multiply both sides by 6. 4.60 ≈ A C Evaluate with ...

This page titled 5.2: Solution of Right Triangles is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Henry Africk (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon ...

See Solving "SSS" Triangles . Tips to Solving. Here is some simple advice: When the triangle has a right angle, then use it, that is usually much simpler. When two angles are known, work out the third using Angles of a Triangle Add to 180°. Try The Law of Sines before the The Law of Cosines as it is easier to use.

Right triangles problems are solved and detailed explanations are included. Example - Problem 1: Find sin(x) and cos(x) in the right triangle shown below. Solution to Problem 1: First use the Pythagorean theorem to find the hypotenuse h of the right triangle. h = √(6 2 + 8 2) = √(36 + 64) = 10 ; In a right triangle, using trigonometric ratio for sin(x) we write. sin(x) = 8 / 10 = 0.8

Solve for an angle in right triangles. Round your answer to the nearest hundredth. Stuck? Review related articles/videos or use a hint. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class ...

Therefore, you can solve the right triangle if you are given the measures of two of the three sides or if you are given the measure of one side and one of the other two angles. Drawing for Example 1. Example 1: Solve the right triangle shown in Figure (b) if ∠ B = 22°. Because the three angles of a triangle must add up to 180°, ∠ A = 90 ...

Solving Right Triangles Worksheets. This Algebra 1 Trigonometry Worksheet will produce problems for solving right triangles. You may select the orientation of the triangles, as well as the side of the triangle to solve for. This worksheet is a great resource for the 5th Grade, 6th Grade, 7th Grade, and 8th Grade.

Use the sides of the triangle and your calculator to find the value of ∠A. Round your answer to the nearest tenth of a degree. Figure 1.4.3. Solution. In reference to ∠A, we are given the opposite leg and the adjacent leg. This means we should use the tangent ratio. tanA = 20 25 = 4 5. So, tan − 14 5 = m∠A.

Right triangle trigonometry word problems. Google Classroom. You might need: Calculator. Bugs Bunny was 33 meters below ground, digging his way toward Pismo Beach, when he realized he wanted to be above ground. He turned and dug through the dirt diagonally for 80 meters until he was above ground. What is the angle of elevation, in degrees, of ...

Practice each skill in the Homework Problems listed. 1 Solve a right triangle #1-16, 63-74. 2 Use inverse trig ratio notation #17-34. 3 Use trig ratios to find an angle #17-22, 35-38. 4 Solve problems involving right triangles #35-48. 5 Know the trig ratios for the special angles #49-62, 75-78.

Right Triangle Trigonometry Special Right Triangles ... so x 3 12.= Solving for x and rationalizing the denominator, we obtain 12 12 3123 3333 x ==⋅= =43. The length of the ... Find the value of x in each of the triangles below. Round answers to the nearest ten-thousandth. 1. Solution:

Right triangle questions, on solving for sides and angles, are presented along with their with answers. Right Triangle Questions. Multiple choice questions right triangle problems related to trigonometry with answers at the bottom of the page. ... In a right triangle, the measure of one of the angles is 49° and the hypotenuse has a length of ...

Solving Right Triangles Date_____ Period____ Find the missing side. ... Write a new problem that is similar to the others on this worksheet. Solve the question you wrote.-2- ... Solve the question you wrote. Many answers.-2-Create your own worksheets like this one with Infinite Geometry. Free trial available at KutaSoftware.com

Solve right triangles. Find the area of any triangle using trigonometry. Solve real-world problems that require you to solve a right triangle. Find angle measures using inverse trigonometric functions. ... This will give us the most accurate answer because we are not using approximations. Now we know the lengths of all three sides of this triangle.

Solve the following right triangle for x : Step 1: The adjacent side is the unknown side. The opposite side has a length of 4 units. Step 2: The trigonometric function that relates the opposite ...

How To: Given a right triangle, the length of one side, and the measure of one acute angle, find the remaining sides. For each side, select the trigonometric function that has the unknown side as either the numerator or the denominator. The known side will in turn be the denominator or the numerator. Write an equation setting the function value ...

Exercises for Solving Triangles Area of Triangles Find the measure of the unknown angle in each triangle. Download Triangles Worksheet \(\color{blue}{45^{\circ}}\)

For example, the area of a right triangle is equal to 28 in² and b = 9 in. Our right triangle side and angle calculator displays missing sides and angles! Now we know that: a = 6.222 in. c = 10.941 in. α = 34.66°. β = 55.34°. Now, let's check how finding the angles of a right triangle works: Refresh the calculator.

Step 1: Draw out a simple graphic to represent the word problem, making sure to include a right triangle. Step 2: Label the sides and/or angles of the right triangle that were given in the word ...

Lesson 3: Special right triangles. Special right triangles proof (part 1) Special right triangles proof (part 2) Special right triangles. 30-60-90 triangle example problem. Area of a regular hexagon. Special right triangles review. Math >. High school geometry >.