**To find the hypotenuse of a right triangle, you can use the Pythagorean theorem: c² = a² + b², where ‘c’ is the hypotenuse, and ‘a’ and ‘b’ are the lengths of the other two sides. Simply plug in the values of ‘a’ and ‘b’ into the formula and calculate ‘c’ by taking the square root of the sum of their squares.**

## Right Triangle Hypotenuse Calculator

Topic | Details |
---|---|

Definition of Hypotenuse | – Longest side in a right triangle – Opposite the right angle |

Formulas | – Pythagorean Theorem: a2 + b2 = c2 a and b = legs c = hypotenuse |

– Pythagorean Triples: 3-4-5 triangle: 32 + 42 = 52 5-12-13 triangle: 52 + 122 = 132 | |

Steps to Find Hypotenuse | 1. Identify known legs (a and b) 2. Square each leg 3. Add squared legs 4. Take square root of sum 5. Result is hypotenuse (c) |

For Pythagorean Triples | – Recognize triple sides – Longest side is hypotenuse |

## Finding the Hypotenuse of a Right Triangle

The hypotenuse is the longest side in a right triangle, located opposite the right angle. Determining the hypotenuse length is a common geometry challenge. Luckily, there are simple formulas and methods for finding the hypotenuse of any right triangle.

## What is the Hypotenuse?

In a right triangle, the hypotenuse is always the longest side and is opposite the 90 degree angle. The other two sides are called legs. Understanding right triangle definitions provides a solid foundation for hypotenuse calculation.

## Pythagorean Theorem

The most direct way to calculate the hypotenuse is using the Pythagorean theorem:

a2 + b2 = c2

Where c is the hypotenuse and a and b are the legs. By squaring the legs and adding them, you can find the square of the hypotenuse. Taking the square root reveals the actual hypotenuse length.

## Using the Example Triangle:

Leg a = 3 units Leg b = 4 units c2 = a2 + b2 = 32 + 42 = 9 + 16 = 25

Taking the square root of 25 gives the hypotenuse √25 = c = 5

So in this triangle, the hypotenuse is 5 units long.

## Pythagorean Triple Shortcuts

Some right triangles have whole number side lengths that make the Pythagorean theorem work nicely. These are called Pythagorean triples.

The most common is the 3-4-5 triangle. Since 32 + 42 = 52, you immediately know a 3-4-5 triangle has a hypotenuse of 5 units. Recognizing triples makes solving some triangles quicker.

Understanding how to calculate hypotenuse lengths empowers you to solve geometry problems. Using the Pythagorean theorem or memorizing common Pythagorean triples provides simple, straightforward techniques. With a hypotenuse formula at your fingertips, you can find missing side lengths, calculate distances, and more!

## FAQs

**What is the formula for finding the hypotenuse of a right triangle?** The formula for finding the hypotenuse of a right triangle is given by the Pythagorean theorem: c² = a² + b², where ‘c’ represents the length of the hypotenuse, and ‘a’ and ‘b’ represent the lengths of the other two sides.

**How do you find the hypotenuse of a right triangle with two sides?** You can find the hypotenuse of a right triangle with two sides (‘a’ and ‘b’) by applying the Pythagorean theorem and using the formula c = √(a² + b²).

**How to find the hypotenuse of a right triangle using the Pythagorean theorem?** To find the hypotenuse of a right triangle using the Pythagorean theorem, square the lengths of the other two sides (‘a’ and ‘b’), add them together, and then take the square root of the sum: c = √(a² + b²).

**How do you find the missing side of a right triangle?** To find the missing side of a right triangle, you can use the Pythagorean theorem if you have the lengths of the other two sides. Alternatively, you can use trigonometric functions like sine, cosine, or tangent when you have an angle and one side.

**How to find the hypotenuse of a right angle triangle with one side?** If you have one side (‘a’) and one acute angle (‘θ’) in a right triangle, you can use trigonometric functions. To find the hypotenuse (‘c’), use the formula c = a / cos(θ).

**How do you find the hypotenuse and opposite of a right triangle?** The hypotenuse (‘c’) can be found using the Pythagorean theorem (c = √(a² + b²)), while the opposite side (‘b’) can be found using trigonometric functions (b = a * tan(θ)), where ‘a’ is the adjacent side and ‘θ’ is an acute angle.

**How do you find the hypotenuse when adjacent and opposite?** If you have the lengths of the adjacent side (‘a’) and the opposite side (‘b’) in a right triangle, you can find the hypotenuse (‘c’) using the Pythagorean theorem: c = √(a² + b²).

**What is the formula for a right angle triangle?** The Pythagorean theorem (c² = a² + b²) is the key formula for right triangles, where ‘c’ is the hypotenuse, and ‘a’ and ‘b’ are the other two sides.

**How do you find the hypotenuse of a right triangle quizlet?** To find the hypotenuse of a right triangle, you can apply the Pythagorean theorem (c² = a² + b²), where ‘c’ is the hypotenuse, and ‘a’ and ‘b’ are the lengths of the other two sides.

**Is the hypotenuse formula only for right triangles?** Yes, the hypotenuse formula, based on the Pythagorean theorem, is specifically for right triangles. It does not apply to other types of triangles.

**How to find the other sides of a right triangle with one side?** To find the other sides of a right triangle with one side, you typically need additional information, such as an angle or the length of the hypotenuse. You can use trigonometric functions or the Pythagorean theorem depending on the given information.

**How to find the missing side of a right triangle without a calculator?** You can find the missing side of a right triangle without a calculator by using basic algebraic techniques to manipulate the Pythagorean theorem formula and solve for the unknown side.

**How to find sides of a right triangle when only given one side?** To find the sides of a right triangle when given only one side, you’ll also need an angle or the length of the hypotenuse. Trigonometric functions like sine, cosine, or tangent can help you find the missing sides.

**How to find a side of a right triangle with one side and one angle?** If you have one side and one acute angle in a right triangle, you can use trigonometric functions to find the other side. For example, you can use sine, cosine, or tangent depending on the known information.

**Is the hypotenuse the side opposite the 90-degree angle in a right triangle?** Yes, the hypotenuse is indeed the side opposite the 90-degree angle in a right triangle. It’s the longest side in a right triangle.

**What formula is hypotenuse over opposite?** The formula that relates the hypotenuse to the opposite side in a right triangle is the sine function: sin(θ) = opposite / hypotenuse.

**How do you find the diagonal length of a right-angled triangle?** To find the diagonal length of a right-angled triangle, you can apply the Pythagorean theorem to the triangle formed by the two legs of the right triangle. The diagonal serves as the hypotenuse of this smaller triangle.

**How do you find the third side of a right triangle?** To find the third side of a right triangle, you can use the Pythagorean theorem (c² = a² + b²), provided you have the lengths of the other two sides (‘a’ and ‘b’). If you have an angle and one side, you can use trigonometric functions to find the third side.

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