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 aa (base) and bb (height) and hypotenuse cc.

Build a square on each side: the square on leg aa has area a2a^2, the square on leg bb has area b2b^2, and the square on the hypotenuse has area c2c^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.
b²a²c²bac
Each square's area equals the corresponding side squared: a2a^2 (blue), b2b^2 (red), c2c^2 (purple).

2. The Formula

The Pythagorean theorem states:
a2+b2=c2a^2 + b^2 = c^2

where aa and bb are the two perpendicular legs and cc 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=a2+b2c = \sqrt{a^2 + b^2}
• Find a missing leg: a=c2−b2a = \sqrt{c^2 - b^2} or b=c2−a2b = \sqrt{c^2 - a^2}

3. Proof of the Theorem

Start with a right-angled triangle with legs aa and bb and hypotenuse cc. Arrange 4 congruent copies of this triangle around a tilted inner square. The result is a large outer square with side (a+b)(a+b).

We can compute the total area in two ways:

• Directly: Abig=(a+b)2=a2+2ab+b2A_{\text{big}} = (a + b)^2 = a^2 + 2ab + b^2
• By parts: Abig=4×ab2+c2=2ab+c2A_{\text{big}} = 4 \times \dfrac{ab}{2} + c^2 = 2ab + c^2
abbabaabc²(a+b)²
The big square has side (a+b)(a+b). The 4 green triangles are congruent to our original triangle. The purple inner square has side cc.
Setting the two expressions equal:
a2+2ab+b2=2ab+c2a^2 + 2ab + b^2 = 2ab + c^2

Subtracting 2ab2ab from both sides:
a2+b2=c2\boxed{a^2 + b^2 = c^2}

This proves the Pythagorean theorem! ✓

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

aac = a√245°45°
Both legs equal: a = b
When both legs are equal (a=ba = b), we substitute into the theorem:
c2=a2+a2=2a2c^2 = a^2 + a^2 = 2a^2

c=a2c = a\sqrt{2}


This means the hypotenuse of an isosceles right triangle is always 2\sqrt{2} times the leg length. For example:
• If a=b=3a = b = 3, then c=32≈4.24c = 3\sqrt{2} \approx 4.24
• If a=b=5a = b = 5, then c=52≈7.07c = 5\sqrt{2} \approx 7.07

This also gives us the diagonal of any square with side ss: the diagonal d=s2d = s\sqrt{2}.

Mastering SealMath: Entering Square Roots

Many Pythagorean answers involve square roots like 50\sqrt{50} or 525\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 525\sqrt{2}, type 5 then sqrt then 2 and close the root.

Frequently Asked Questions

What is the Pythagorean theorem?

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides (a² + b² = c²). It is used to find a missing side length when the other two are known.

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.

Pythagorean Theorem Practice | SealMath | SealMath