What is the key idea in pythagorean theorem examples?

For a right triangle with legs a and b and hypotenuse c, a^2+b^2=c^2. The hypotenuse is always opposite the right angle and is the longest side.

How should I practice pythagorean theorem examples?

Study one worked example, cover the solution, reproduce it independently, and then complete mixed practice while checking each result.

Geometry · Step-by-step guide

Pythagorean Theorem Examples

Use the Pythagorean theorem to find missing sides and solve distance problems with step-by-step examples.

By Tyler Blovat·Updated September 22, 2026·9 min read

The short answer

Start with the central idea

For a right triangle with legs a and b and hypotenuse c, $\displaystyle a^2+b^2=c^2$ . The hypotenuse is always opposite the right angle and is the longest side.

\displaystyle a^2+b^2=c^2

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

Identify c first

Never choose c by position on the page; choose the side opposite the right-angle marker.

Square before combining

Substitute lengths, square them, and isolate the remaining squared variable before taking a square root.

Use exact and approximate forms

A radical is exact; a decimal can be useful for measurement. Round only at the end.

See the structure

A picture makes the relationships easier to remember.

The labeled lengths and right-angle marker connect the diagram to the triangle formulas in the guide.

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Find the hypotenuse

Legs are 6 and 8.

$\displaystyle 6^2+8^2=c^2$ .
$\displaystyle 36+64=100$ .
Take the positive square root.

Answer $\displaystyle c=10$

Example 2Find a leg

Hypotenuse 13, one leg 5.

$\displaystyle 5^2+b^2=13^2$ .
$\displaystyle b^2=169-25=144$ .
Use the positive length.

Answer $\displaystyle b=12$

Example 3Coordinate distance

Find the distance from $\displaystyle (1,2)$ to $\displaystyle (7,10)$ .

Horizontal change is 6; vertical change is 8.
Those changes are perpendicular legs.
$\displaystyle d=\sqrt{6^2+8^2}$ .

Answer $\displaystyle d=10$

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

Using a leg as c.
Adding lengths before squaring.
Keeping a negative square root for a geometric length.

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Pythagorean Theorem

Practice missing-side, coordinate, and application problems.

Preview the worksheet

✓

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

Use the labeled right triangle to find the hypotenuse c.

Show hint

Substitute the two leg lengths into $\displaystyle a^2+b^2=c^2$ .

Show worked solution

$\displaystyle 6^2+8^2=c^2$ .
$\displaystyle 36+64=100$ .
$\displaystyle c=10$ because a length is positive.

Answer: $\displaystyle 10$

Use the labeled right triangle to find the missing leg a.

Show hint

Subtract the known leg square from the hypotenuse square.

Show worked solution

$\displaystyle a^2+5^2=13^2$ .
$\displaystyle a^2=169-25=144$ .
$\displaystyle a=12$ .

Answer: $\displaystyle 12$

Use the rectangle diagram to find diagonal d.

Show hint

The diagonal is the hypotenuse of a right triangle.

Show worked solution

$\displaystyle d^2=9^2+12^2$ .
$\displaystyle d^2=81+144=225$ .
$\displaystyle d=15$ .

Answer: $\displaystyle 15$

Use the coordinate diagram to find the distance d between P and Q.

Show hint

Use the horizontal and vertical changes as legs.

Show worked solution

The horizontal change is Δx = 6, and the vertical change is Δy = 8.
$\displaystyle d=\sqrt{6^2+8^2}=\sqrt{100}$ .

Answer: $\displaystyle 10$

Use the labeled triangle to decide whether it is a right triangle.

Show hint

Test the longest side as c.

Show worked solution

$\displaystyle 7^2+24^2=49+576=625$ .
$\displaystyle 25^2=625$ .
The values match.

Answer: Yes, it is a right triangle.

Frequently asked questions

Quick answers before you move on.

What should I remember first?

For a right triangle with legs a and b and hypotenuse c, $\displaystyle a^2+b^2=c^2$ . The hypotenuse is always opposite the right angle and is the longest side.

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

Start with the central idea

How it works

Identify c first

Square before combining

Use exact and approximate forms

See the structure

Worked examples

Legs are 6 and 8.

Hypotenuse 13, one leg 5.

Find the distance from (1,2)\displaystyle (1,2)(1,2) to (7,10)\displaystyle (7,10)(7,10).

Common mistakes—and how to avoid them

Turn the explanation into a skill

Pythagorean Theorem

Practice problems with worked solutions

Frequently asked questions

Special Right Triangles

Similar Triangles

Area and Volume Formulas

Find the distance from $\displaystyle (1,2)$ to $\displaystyle (7,10)$ .