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

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.

How do you calculate the area of a general triangle?

The area of any general triangle is calculated as half the base times the height (A = (1/2) × b × h or A = bh/2), where the height is the perpendicular distance from the base to the opposite vertex.

How do you calculate the area of a circle?

The area of a circle is calculated as π times the radius squared (A = πr²). Since π (pi) is an irrational number, the area of a circle with a rational radius will always be an irrational number.

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.

What is the area ratio of inscribed circles and squares?

For a square inscribed in a circle, the area ratio of the square to the circle is always exactly 2/π ≈ 0.637. For a circle inscribed in a square, the area ratio of the circle to the square is always exactly π/4 ≈ 0.785. These ratios are constant regardless of the shapes' actual sizes.

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.

What are the main types of special triangles?

The three main types are: isosceles (two equal sides and two equal base angles), equilateral (all sides and angles equal — each angle is 60°), and the 30-60-90 right triangle (fixed side ratios of 1 : √3 : 2).

Why does 1 cm² equal 100 mm² and not just 10 mm²?

Because area is two-dimensional (length × length), the conversion factor must be applied twice. Since 1 cm = 10 mm, we have 1 cm² = 1 cm × 1 cm = 10 mm × 10 mm = 100 mm². In general, if 1 unit = k sub-units, then 1 unit² = k² sub-units². The same logic applies to all other area unit conversions.

Pythagorean Theorem Practice | SealMath | SealMath