Pythagorean Theorem Calculator

This Pythagorean theorem calculator effortlessly finds the length of any missing side of a right triangle when the other two sides are known. All computations are powered by the foundational Pythagorean theorem.

Directions for use

Enter the known side lengths and click "Calculate." Our right triangle calculator will instantly return the following values:

• Length of the third side.
• Angle values of the non-90° angles in degrees and radians.
• Area of the triangle.
• Perimeter of the triangle.
• Length of the altitude perpendicular to the hypotenuse.

The calculator will also provide a detailed, step-by-step solution, which you can expand by clicking "+ Show Calculation Steps."

For your convenience, the input fields for each side include both a whole number component and a square root component, allowing you to easily enter exact values like 2√3, √3, and so on.

Please note that the lengths of the legs of the triangle (a and b) must be shorter than the length of the hypotenuse (c).

Pythagorean Theorem

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides (the legs or catheti).

The Pythagorean equation can be written as follows:

a² + b² = c²,

Where a and b represent the lengths of the shorter sides, or legs, of a right triangle, and c is the length of the longest side, or hypotenuse. In words, the equation above is commonly read as: a squared plus b squared equals c squared.

Proof of the Pythagorean theorem

We can prove the Pythagorean theorem by comparing the areas of specific geometric shapes.

In the diagram above, a large square with side lengths of (a + b) contains a smaller inner square with a side length of c, surrounded by four identical right triangles with sides a, b, and c. Let's find the total area of this large square using two different methods:

1. The surface area of the larger square with the side length (a + b) can be calculated as (a + b)²:

A = (a + b)²

2. Alternatively, the same total surface area can be found by adding the surface areas of the internal shapes—the area of the inner square with side c, and the areas of the four right triangles with sides a, b, and c. The area of the inner square is calculated as c². The area of each right triangle is calculated as (ab)/2. Therefore,

A = c² + 4 × (ab)/2 = c² + 2ab

Since both of these calculations describe the exact same total surface area, we can equate them:

(a + b)² = c² + 2ab

Expanding the binomial square on the left side of the equation yields:

a² + 2ab + b² = c² + 2ab

Subtracting 2ab from both sides of the equation gives us:

a² + b² = c²

which mathematically proves the Pythagorean theorem.

Calculation algorithms

Finding the sides of a right triangle

If two sides of a right triangle are known, the missing third side can easily be found using the Pythagorean theorem. For example, if sides a and b are given, the length of the hypotenuse c can be calculated as follows:

$$c=\sqrt{a²+b²}$$

Similarly, to find a missing leg:

$$a=\sqrt{c²-b²}$$

and

$$b=\sqrt{c²-a²}$$

Finding the angles of a right triangle

If all three sides of the right triangle are known, the non-90° angles of the triangle can be calculated as follows:

• ∠α = arcsin(a/c) or ∠α = arccos(b/c)
• ∠β = arcsin(b/c) or ∠β = arccos(a/c)

Here, ∠α represents the angle opposite leg 'a', ∠β is the angle opposite leg 'b', and 'c' is the hypotenuse. The choice between using arcsin (inverse sine) and arccos (inverse cosine) depends on which leg you are referencing relative to the angle. The arcsin function uses the leg opposite the angle, while the arccos function uses the adjacent leg. Both trigonometric approaches are entirely valid and will yield accurate angle measurements for any right triangle.

Area of a right triangle

The area of a right triangle is calculated as one-half the product of its two legs:

A = 1/2 × (ab) = (ab)/2

Perimeter of a right triangle

The perimeter of a right triangle is simply the total sum of all its side lengths:

P = a + b + c

Altitude to hypotenuse

When all three sides of a right triangle are known, the altitude to the hypotenuse (h) can be found using this formula:

h = (a × b)/c

Real-life examples

The Pythagorean theorem is widely used in architecture, engineering, and construction to calculate the exact lengths of necessary components and ensure that structures maintain perfectly straight, right-angled corners. Let's look at a practical, real-world example of applying this mathematical theorem.

Fitting objects

Imagine you are moving, and you have rented a moving truck with a length of 4 meters and a height of 3 meters. You don't have many bulky items, but you do own a ladder that is 4.5 meters long. Will your ladder fit inside the truck?

Solution

Since the ladder's length (4.5 meters) exceeds the length of the truck (4 meters), the only way the ladder will fit inside is diagonally. To determine whether that's mathematically possible, we need to use the Pythagorean theorem to calculate the hypotenuse of a triangle where the sides equal the length and height of the truck. Therefore, in our case a = 4, b = 3, and we need to find the hypotenuse c:

$$c=\sqrt{a²+b²}=\sqrt{4²+3²}=\sqrt{16+9}=\sqrt{25}=5$$

The hypotenuse of a right triangle with legs a = 4 and b = 3 is c = 5. This means the longest rigid object that can fit diagonally inside the truck is exactly 5 meters. Since your ladder is 4.5 meters long, it will easily fit!

Answer

Yes, the ladder will fit.

Additional calculations

Our online hypotenuse calculator also computes several additional geometric characteristics of the given right triangle. Let's look at the expanded results for our moving truck triangle with sides a = 4, b = 3, and hypotenuse c = 5.

Area of the triangle:

A = (ab)/2 = (3 × 4)/2 = 12/2 = 6

Perimeter of the triangle:

P = a + b + c = 3 + 4 + 5 = 12

Altitude to hypotenuse:

h = (a × b)/c = (3 × 4)/5 = 12/5 = 2.4

Angle opposite to side a:

∠α = arcsin(a/c) = arcsin(4/5) = arcsin(0.8) = 53.13° = 53°7'48" = 0.9273 rad

Angle opposite to side b:

∠β = arcsin(b/c) = arcsin(3/5) =arcsin(0.6) = 36.87° = 36°52'12" = 0.6435 rad