How to find the hypotenuse of a triangle with one side

Please provide 2 values below to calculate the other values of a right triangle. If radians are selected as the angle unit, it can take values such as pi/3, pi/4, etc.

Right triangle

A right triangle is a type of triangle that has one angle that measures 90°. Right triangles, and the relationships between their sides and angles, are the basis of trigonometry.

In a right triangle, the side that is opposite of the 90° angle is the longest side of the triangle, and is called the hypotenuse. The sides of a right triangle are commonly referred to with the variables a, b, and c, where c is the hypotenuse and a and b are the lengths of the shorter sides. Their angles are also typically referred to using the capitalized letter corresponding to the side length: angle A for side a, angle B for side b, and angle C (for a right triangle this will be 90°) for side c, as shown below. In this calculator, the Greek symbols α (alpha) and β (beta) are used for the unknown angle measures. h refers to the altitude of the triangle, which is the length from the vertex of the right angle of the triangle to the hypotenuse of the triangle. The altitude divides the original triangle into two smaller, similar triangles that are also similar to the original triangle.

If all three sides of a right triangle have lengths that are integers, it is known as a Pythagorean triangle. In a triangle of this type, the lengths of the three sides are collectively known as a Pythagorean triple. Examples include: 3, 4, 5; 5, 12, 13; 8, 15, 17, etc.

Area and perimeter of a right triangle are calculated in the same way as any other triangle. The perimeter is the sum of the three sides of the triangle and the area can be determined using the following equation:

Special Right Triangles

30°-60°-90° triangle:

The 30°-60°-90° refers to the angle measurements in degrees of this type of special right triangle. In this type of right triangle, the sides corresponding to the angles 30°-60°-90° follow a ratio of 1:√3:2. Thus, in this type of triangle, if the length of one side and the side's corresponding angle is known, the length of the other sides can be determined using the above ratio. For example, given that the side corresponding to the 60° angle is 5, let a be the length of the side corresponding to the 30° angle, b be the length of the 60° side, and c be the length of the 90° side.:

Angles: 30°: 60°: 90°

Ratio of sides: 1:√3:2

Side lengths: a:5:c

Then using the known ratios of the sides of this special type of triangle:

As can be seen from the above, knowing just one side of a 30°-60°-90° triangle enables you to determine the length of any of the other sides relatively easily. This type of triangle can be used to evaluate trigonometric functions for multiples of π/6.

45°-45°-90° triangle:

The 45°-45°-90° triangle, also referred to as an isosceles right triangle, since it has two sides of equal lengths, is a right triangle in which the sides corresponding to the angles, 45°-45°-90°, follow a ratio of 1:1:√2. Like the 30°-60°-90° triangle, knowing one side length allows you to determine the lengths of the other sides of a 45°-45°-90° triangle.

Angles: 45°: 45°: 90°

Ratio of sides: 1:1:√2

Side lengths: a:a:c

Given c= 5:

45°-45°-90° triangles can be used to evaluate trigonometric functions for multiples of π/4.

The Pythagorean Theorem

Learning Objective(s)

·         Use the Pythagorean Theorem to find the unknown side of a right triangle.

·         Solve application problems involving the Pythagorean Theorem.

Introduction

A long time ago, a Greek mathematician named Pythagoras discovered an interesting property about right triangles: the sum of the squares of the lengths of each of the triangle’s legs is the same as the square of the length of the triangle’s hypotenuse. This property—which has many applications in science, art, engineering, and architecture—is now called the Pythagorean Theorem.

Let’s take a look at how this theorem can help you learn more about the construction of triangles. And the best part—you don’t even have to speak Greek to apply Pythagoras’ discovery.

The Pythagorean Theorem

Pythagoras studied right triangles, and the relationships between the legs and the hypotenuse of a right triangle, before deriving his theory.

The Pythagorean Theorem

If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, then the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. This relationship is represented by the formula:

In the box above, you may have noticed the word “square,” as well as the small 2s to the top right of the letters in

When you see the equation

Let’s try out all of the Pythagorean Theorem with an actual right triangle.

This theorem holds true for this right triangle—the sum of the squares of the lengths of both legs is the same as the square of the length of the hypotenuse. And, in fact, it holds true for all right triangles.

The Pythagorean Theorem can also be represented in terms of area. In any right triangle, the area of the square drawn from the hypotenuse is equal to the sum of the areas of the squares that are drawn from the two legs. You can see this illustrated below in the same 3-4-5 right triangle.

Note that the Pythagorean Theorem only works with right triangles.

Finding the Length of the Hypotenuse

You can use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle if you know the length of the triangle’s other two sides, called the legs. Put another way, if you know the lengths of a and b, you can find c.

In the triangle above, you are given measures for legs a and b: 5 and 12, respectively. You can use the Pythagorean Theorem to find a value for the length of c, the hypotenuse.

	The Pythagorean Theorem.
	Substitute known values for a and b.
	Evaluate.
	Simplify. To find the value of c, think about a number that, when multiplied by itself, equals 169. Does 10 work? How about 11? 12? 13? (You can use a calculator to multiply if the numbers are unfamiliar.)
13 = c	The square root of 169 is 13.

Using the formula, you find that the length of c, the hypotenuse, is 13.

In this case, you did not know the value of c—you were given the square of the length of the hypotenuse, and had to figure it out from there. When you are given an equation like

Finding a square root takes some practice, but it also takes knowledge of multiplication, division, and a little bit of trial and error. Look at the table below.

It is a good habit to become familiar with the squares of the numbers from 0‒10, as these arise frequently in mathematics. If you can remember those square numbers—or if you can use a calculator to find them—then finding many common square roots will be just a matter of recall.

Finding the Length of a Leg

You can use the same formula to find the length of a right triangle’s leg if you are given measurements for the lengths of the hypotenuse and the other leg. Consider the example below.

Example
Problem	Find the length of side a in the triangle below. Use a calculator to estimate the square root to one decimal place.
	a = ? b = 6 c = 7	In this right triangle, you are given the measurements for the hypotenuse, c, and one leg, b. The hypotenuse is always opposite the right angle and it is always the longest side of the triangle.
		To find the length of leg a, substitute the known values into the Pythagorean Theorem.
		Solve for a2. Think: what number, when added to 36, gives you 49?
		Use a calculator to find the square root of 13. The calculator gives an answer of 3.6055…, which you can round to 3.6. (Since you are approximating, you use the symbol .)
Answer

Which of the following correctly uses the Pythagorean Theorem to find the missing side, x?

A)

B) x + 8 = 10

C)

D)

Using the Theorem to Solve Real World Problems

The Pythagorean Theorem is perhaps one of the most useful formulas you will learn in mathematics because there are so many applications of it in real world settings. Architects and engineers use this formula extensively when building ramps, bridges, and buildings. Look at the following examples.

Example
Problem	The owners of a house want to convert a stairway leading from the ground to their back porch into a ramp. The porch is 3 feet off the ground, and due to building regulations the ramp must start 12 feet away from the base of the porch. How long will the ramp be? Use a calculator to find the square root, and round the answer to the nearest tenth.
	To solve a problem like this one, it often makes sense to draw a simple diagram showing where the legs and hypotenuse of the triangle lie.
	a = 3 b = 12 c = ?	Identify the legs and the hypotenuse of the triangle. You know that the triangle is a right triangle since the ground and the raised portion of the porch are perpendicular—this means you can use the Pythagorean Theorem to solve this problem. Identify a, b, and c.
		Use the Pythagorean Theorem to find the length of c.
	12.4 = c	Use a calculator to find c. The square root of 153 is 12.369…, so you can round that to 12.4.
Answer	The ramp will be 12.4 feet long.

Example
Problem	A sailboat has a large sail in the shape of a right triangle. The longest edge of the sail measures 17 yards, and the bottom edge of the sail is 8 yards. How tall is the sail?
		Draw an image to help you visualize the problem. In a right triangle, the hypotenuse will always be the longest side, so here it must be 17 yards. The problem also tells you that the bottom edge of the triangle is 8 yards.
		Setup the Pythagorean Theorem.
	a = 15	15 • 15 = 225, so a = 15.
Answer	The height of the sail is 15 yards.

Summary

The Pythagorean Theorem states that in any right triangle, the sum of the squares of the lengths of the triangle’s legs is the same as the square of the length of the triangle’s hypotenuse. This theorem is represented by the formula

How do you find the missing hypotenuse of a triangle?

How do you find the hypotenuse of a right triangle with one leg?