Area: Pythagorean Theorem

Learn and practice the Pythagorean theorem: find hypotenuses, missing legs, and diagonals of squares and rectangles with step-by-step exercises.

Pythagorean Theorem

Use the workspace below. Write equations like A = 30 to solve for the area.

Learning Topics

📖 Area Study Guide

1. The Pythagorean Theorem and Square Areas

Consider a right-angled triangle with legs

a

(base) and

b

(height) and hypotenuse

c

.

Build a square on each side: the square on leg

a

has area

a^2

, the square on leg

b

has area

b^2

, and the square on the hypotenuse has area

c^2

.

The Pythagorean theorem states that the area of the hypotenuse square exactly equals the combined area of the two leg squares. This is the geometric reason why side lengths are related by squares — and since area is related to the side², we can derive the length of any side if we know the other two.

Each square's area equals the corresponding side squared:

a^2

(blue),

b^2

(red),

c^2

(purple).

2. The Formula

The Pythagorean theorem states:

a^2 + b^2 = c^2

where

a

and

b

are the two perpendicular legs and

c

is the hypotenuse (the side opposite the right angle — always the longest side).

We can rearrange it to solve for any side:
• Find the hypotenuse:

c = \sqrt{a^2 + b^2}

• Find a missing leg:

a = \sqrt{c^2 - b^2}

b = \sqrt{c^2 - a^2}

3. Proof of the Theorem

Start with a right-angled triangle with legs

a

and

b

and hypotenuse

c

. Arrange 4 congruent copies of this triangle around a tilted inner square. The result is a large outer square with side

(a+b)

.

We can compute the total area in two ways:

• Directly:

A_{\text{big}} = (a + b)^2 = a^2 + 2ab + b^2

• By parts:

A_{\text{big}} = 4 \times \dfrac{ab}{2} + c^2 = 2ab + c^2

The big square has side

(a+b)

. The 4 green triangles are congruent to our original triangle. The purple inner square has side

c

Setting the two expressions equal:

a^2 + 2ab + b^2 = 2ab + c^2

Subtracting

2ab

from both sides:

\boxed{a^2 + b^2 = c^2}

This proves the Pythagorean theorem! ✓

4. Special Case: Isosceles Right Triangle ( $a = b$ )

Both legs equal: a = b

When both legs are equal (

a = b

), we substitute into the theorem:

c^2 = a^2 + a^2 = 2a^2

c = a\sqrt{2}

This means the hypotenuse of an isosceles right triangle is always

\sqrt{2}

times the leg length. For example:
• If

a = b = 3

, then

c = 3\sqrt{2} \approx 4.24

• If

a = b = 5

, then

c = 5\sqrt{2} \approx 7.07

This also gives us the diagonal of any square with side

s

: the diagonal

d = s\sqrt{2}

Mastering SealMath: Entering Square Roots

Many Pythagorean answers involve square roots like

\sqrt{50}

5\sqrt{2}

. To enter a square root in the math input:
• Keyboard shortcut: Type sqrt in the input box — MathLive creates

\sqrt{\square}

instantly. Then type your number inside.
• Virtual keyboard: Click the ⌨️ keyboard icon, go to the 123 tab, and press the √□ button.

For answers like

5\sqrt{2}

, type 5 then sqrt then 2 and close the root.

Frequently Asked Questions

How is the area of a shape defined?

The area of a shape is defined by how many unit squares of 1 by 1 fit inside it. For example, if a rectangle can be divided exactly into 30 squares of 1 by 1, its area is 30.

How do you calculate the area of a rectangle and a square?

The area of a rectangle is calculated as width × height (A = w × h). A square is a special type of rectangle where all sides are equal (w = h = s). Thus, the area of a square is side × side, or side squared (A = s²).

How do you calculate the area of a right-angled triangle?

The area of a right-angled triangle is calculated by multiplying its two perpendicular legs and dividing by 2 (A = ab / 2). This is because a right-angled triangle is exactly half of a rectangle with the same width and height.

Can the area of a shape be an irrational number?

Yes, if the side lengths are irrational numbers (such as √2), the resulting area can be either rational or irrational. You can learn more about these classifications in our Number Sets - Real & Complex topic.