Hypotenuse Calculator

Calculate the hypotenuse of a right triangle using the Pythagorean theorem and other formulas. Enter two sides, an angle and side, or area and side to find the hypotenuse and all other triangle measurements.

a b c α β

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c = √(a² + b²)

What is the hypotenuse of a triangle?

A hypotenuse is the longest side of a right triangle. It's the side that is opposite to the right angle (90°). The hypotenuse is always longer than either of the other two sides (called legs) of the right triangle.

The word "hypotenuse" comes from the Ancient Greek hypoteinousa, meaning "stretching under (a right angle)." This, in turn, comes from hypo- "under" and teinein "to stretch."

How to use this hypotenuse calculator?

This calculator offers three different methods to find the hypotenuse of a right triangle:

  1. Select your input type: Choose from "Two sides," "Angle and one side," or "Area and one side" depending on what information you have.
  2. Enter the known values: Input the measurements you have, following the labels in the triangle diagram.
  3. Select your unit: Choose from various units (inches, centimeters, meters, etc.).
  4. Click "Calculate": The tool will instantly calculate the hypotenuse and all other triangle measurements.

The calculator works in reverse too! For example, if you know the hypotenuse is 13 cm and one side is 12 cm, select "Two sides" and enter these values to find that the other side is 5 cm.

Hypotenuse formulas

This hypotenuse calculator uses several formulas to handle different scenarios. Here are the main formulas:

1. Given Two Sides (Pythagorean Theorem)

The most common method uses the Pythagorean theorem to calculate the hypotenuse from the two legs of the right triangle:

c = √(a² + b²)

where c is the hypotenuse, and a and b are the two legs

Example: If a = 3 cm and b = 4 cm, then c = √(3² + 4²) = √(9 + 16) = √25 = 5 cm

2. Given an Angle and One Side (Trigonometry)

Using the law of sines or basic trigonometry:

c = a / sin(α) = b / sin(β)

where α and β are the angles opposite to sides a and b respectively

Example: If a = 5 cm and angle α = 60°, then c = 5 / sin(60°) = 5 / 0.866 = 5.77 cm

3. Given the Area and One Side

Since the area of a right triangle equals a × b / 2, we can derive:

c = √(a² + b²)

where b = (Area × 2) / a or a = (Area × 2) / b

First find the unknown side, then use the Pythagorean theorem

Example: If Area = 6 cm² and a = 3 cm, then b = (6 × 2) / 3 = 4 cm, and c = √(3² + 4²) = 5 cm

Ladder example - How to find the hypotenuse

Let's calculate how long a ladder should be if we want to reach a 10 ft roof. We need to find the ladder length (hypotenuse) from the ground to the edge of the roof.

Scenario: Rescuing a kitten from a roof

  • Known: Roof height (side a) = 10 ft
  • Known: Safe ladder angle α = 75.5°
  • Find: Ladder length (hypotenuse c)

Solution using angle and one side:

  1. Use the formula: c = a / sin(α)
  2. c = 10 / sin(75.5°)
  3. c = 10 / 0.968
  4. c = 10.33 ft

Bonus calculations:

  • Angle β = 90° - 75.5° = 14.5°
  • Distance from wall (side b) = 10 × tan(14.5°) = 2.59 ft

Remember the 4:1 rule: for every four feet of vertical height, the ladder foot should be one foot from the wall for safe positioning!

Special right triangles

3-4-5 Triangle

The 3-4-5 triangle is one of the most famous Pythagorean triples. If the two legs are 3 and 4 units, the hypotenuse will always be exactly 5 units. This ratio can be scaled up (6-8-10, 9-12-15, etc.) and is commonly used in construction to create perfect right angles.

Isosceles Right Triangle (45-45-90)

In an isosceles right triangle, both legs are equal. If each leg has length a, then:

c = a√2

Example: If a = b = 5 cm, then c = 5√2 ≈ 7.07 cm

30-60-90 Triangle

In a 30-60-90 triangle, the sides are in the ratio 1 : √3 : 2. If the shortest side is a:

  • Short leg (opposite 30°) = a
  • Long leg (opposite 60°) = a√3
  • Hypotenuse = 2a

Real-world applications

Construction and Carpentry

  • Roof rafters: Calculate the length of diagonal roof supports
  • Stair stringers: Determine the length of stair supports from rise and run measurements
  • Corner bracing: Find the correct length for diagonal braces in framing

Example: A wall is 8 feet high and needs a diagonal brace to a point 6 feet from the base. The brace length = √(8² + 6²) = √(64 + 36) = 10 feet

Navigation and Distance

  • GPS calculations: Find straight-line distance when given north-south and east-west components
  • Aviation: Calculate actual flight distance when given altitude change and ground distance
  • Marine navigation: Determine the shortest distance between two points

Example: A plane flies 300 miles east and 400 miles north. The direct distance from the starting point = √(300² + 400²) = 500 miles

Sports and Recreation

  • Baseball diamonds: Calculate the distance from home plate to second base (diagonal of the square)
  • Screen sizes: Determine TV or monitor diagonal from width and height
  • Hiking: Calculate actual distance traveled when climbing or descending

Example: A 90-foot baseball diamond has a distance from home to second base of √(90² + 90²) = 90√2 ≈ 127.3 feet

Engineering and Design

  • Cable runs: Calculate cable length for diagonal installations
  • Antenna masts: Determine guy wire lengths for supporting towers
  • Conveyor systems: Find belt length for inclined conveyors

Example: A 50-foot antenna tower needs guy wires anchored 30 feet from the base. Guy wire length = √(50² + 30²) = √(2500 + 900) ≈ 58.3 feet

Frequently Asked Questions

How do I find hypotenuse with sin?

  1. Identify the angle you're using (not the right angle)
  2. Perform the sin operation on that angle
  3. Divide the length of the side opposite the angle by the result from step 2
  4. The result is the hypotenuse

Formula: c = opposite side / sin(angle)

Is the hypotenuse always the longest side?

Yes, the hypotenuse is always the longest side, but only for right-angled triangles. For isosceles triangles, the two equal sides are called legs. In an equilateral triangle, all sides are simply called sides since they're all equal.

How do I find the hypotenuse adjacent and opposite?

  1. Find the longest side and label it the hypotenuse
  2. Choose one of the non-right angles (less than 90°)
  3. The adjacent side is the side that forms the chosen angle along with the hypotenuse
  4. The opposite side is the side that does not form the chosen angle

How do I find the hypotenuse of an isosceles right triangle?

  1. Find the length of one of the equal sides (legs)
  2. Square that length
  3. Double the result from step 2
  4. Take the square root of step 3

Formula: c = leg × √2

Example: If each leg is 5 inches, then c = 5√2 ≈ 7.07 inches

Can I solve a right triangle with only the hypotenuse?

No, you cannot solve a right-angled triangle with only the hypotenuse. The other two sides and angles remain undefined, so the triangle can have many different forms. You need at least one additional piece of information (another side or an angle) to fully solve the triangle.

What is the hypotenuse angle theorem?

The hypotenuse angle theorem (HA theorem) is a method of testing if two right-angled triangles are congruent. It states that if two right-angled triangles have a hypotenuse and an acute angle that are the same, then the triangles are congruent (identical in shape and size).

How do you find the altitude of a hypotenuse?

  1. Draw the altitude from the right angle to the hypotenuse, creating two smaller similar triangles
  2. Use the formula: altitude = (leg₁ × leg₂) / hypotenuse
  3. Or use: altitude = leg₁ × sin(angle opposite to leg₂)

Example: In a 3-4-5 triangle, altitude to hypotenuse = (3 × 4) / 5 = 2.4 units

What are Pythagorean triples?

Pythagorean triples are sets of three positive integers (a, b, c) that satisfy the equation a² + b² = c². Common examples include:

  • 3-4-5 (and multiples: 6-8-10, 9-12-15, etc.)
  • 5-12-13
  • 8-15-17
  • 7-24-25
  • 20-21-29

These are useful for quick calculations and creating exact right angles in practical applications.

Step-by-step examples

Example 1: Using Two Sides

Problem: Find the hypotenuse when a = 6 cm and b = 8 cm

Solution:

  1. Use the Pythagorean theorem: c = √(a² + b²)
  2. Substitute values: c = √(6² + 8²)
  3. Calculate squares: c = √(36 + 64)
  4. Add: c = √100
  5. Answer: c = 10 cm

Example 2: Using Angle and Side

Problem: Find the hypotenuse when side a = 5 m and angle α = 30°

Solution:

  1. Use the formula: c = a / sin(α)
  2. Substitute values: c = 5 / sin(30°)
  3. Calculate sin(30°) = 0.5
  4. Divide: c = 5 / 0.5
  5. Answer: c = 10 m

Example 3: Using Area and Side

Problem: Find the hypotenuse when Area = 24 ft² and side a = 6 ft

Solution:

  1. First, find side b using: Area = (a × b) / 2
  2. Rearrange: b = (Area × 2) / a
  3. Substitute: b = (24 × 2) / 6 = 48 / 6 = 8 ft
  4. Now use Pythagorean theorem: c = √(a² + b²)
  5. c = √(6² + 8²) = √(36 + 64) = √100
  6. Answer: c = 10 ft

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