Pythagorean Theorem Formula: The Ultimate Guide

1. Introduction

The Pythagorean Theorem is one of the most famous and widely used formulas in all of mathematics. Discovered over two millennia ago, it fundamentally describes the relationship between the three sides of a right triangle.

But why is it so important? This theorem bridges the gap between geometry and algebra. It allows us to calculate distances, design buildings, navigate the globe, and even program modern 3D computer graphics. Whenever you need to find the shortest path between two points, or Find the hypotenuse, this formula is your go-to tool.

2. What Is the Pythagorean Theorem Formula?

Formula Overview:

The formula is written as a² + b² = c², where $a$ and $b$ are the legs of the right triangle, and $c$ is the hypotenuse.

Breaking Down the Formula

• $a$ (Leg 1): One of the sides that form the 90-degree right angle.
• $b$ (Leg 2): The other side that forms the 90-degree right angle.
• $c$ (Hypotenuse): The longest side of the right triangle, located directly opposite the right angle.

Why must $c$ always be the hypotenuse?

In any triangle, the longest side is always opposite the largest angle. Since a right triangle has a 90-degree angle (which must be the largest angle since the sum of all angles is 180 degrees), the side opposite to it is invariably the longest. If you plug the longest side into $a$ or $b$, the math will fail.

[SVG showing a right triangle with a, b, c labeled]

3. When Can You Use the Formula?

When to use it:

You can only use the Pythagorean theorem formula on right triangles (triangles with a 90-degree angle).

Examples of When to Use It:

• A triangle with angles 90°, 45°, and 45° (45-45-90 Triangle).
• A triangle with angles 90°, 30°, and 60° (30-60-90 Triangle).
• Finding the diagonal of a rectangle or square.

Non-Examples (When NOT to Use It):

• Equilateral triangles (all angles are 60°).
• Acute or obtuse triangles that lack a 90° angle. (For non-right triangles, you must use the Law of Sines or Law of Cosines instead!)

4. How to Use the Formula

Step-by-Step Guide:

Identify the known sides, substitute them into $a^2 + b^2 = c^2$, and solve for the unknown variable using basic algebra and square roots.

Finding the Hypotenuse ($c$)

When you know both legs ($a$ and $b$):

1. Square side $a$.
2. Square side $b$.
3. Add the two squared numbers together.
4. Take the square root of the sum to find $c$. (Shortcut: use our Hypotenuse Calculator)

Finding a Missing Leg ($a$ or $b$)

When you know the hypotenuse ($c$) and one leg ($a$):

1. Square side $c$.
2. Square side $a$.
3. Subtract $a^2$ from $c^2$.
4. Take the square root of the result to find $b$. (Shortcut: use our Missing Side Calculator)

Checking if a Triangle is a Right Triangle

If you have all three sides, plug them into the formula. Make sure $c$ is the largest number. If the left side equals the right side, it is a perfect right triangle!

5. Worked Examples

Mastering the Formula:

Review these 13 solved examples to understand how to apply the Pythagorean theorem formula across different difficulty levels and real-world scenarios.

Example 1: Very Easy

Problem: A right triangle has legs $a = 3$ and $b = 4$. Find the missing side $c$.

Known values: $a = 3$, $b = 4$

Formula: $c = \sqrt{a^2 + b^2}$

Calculation: $c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25}$

Answer: $c = 5$

Explanation: Square each leg, add the results, then take the square root. This is the classic 3-4-5 triple, worth memorizing.

Example 2: Very Easy

Problem: A right triangle has legs $a = 5$ and $b = 12$. Find the missing side $c$.

Known values: $a = 5$, $b = 12$

Formula: $c = \sqrt{a^2 + b^2}$

Calculation: $c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169}$

Answer: $c = 13$

Explanation: Squaring 5 and 12 gives 25 and 144. Their sum, 169, is a perfect square, so the hypotenuse is exactly 13.

Example 3: Easy

Problem: A right triangle has hypotenuse $c = 17$ and leg $b = 15$. Find leg $a$.

Known values: $c = 17$, $b = 15$

Formula: $a = \sqrt{c^2 - b^2}$

Calculation: $a = \sqrt{17^2 - 15^2} = \sqrt{289 - 225} = \sqrt{64}$

Answer: $a = 8$

Explanation: To find a leg, rearrange the formula and subtract: square the hypotenuse, subtract the known leg’s square, then take the square root.

Example 4: Easy

Problem: A right triangle has hypotenuse $c = 25$ and leg $b = 24$. Find leg $a$.

Known values: $c = 25$, $b = 24$

Formula: $a = \sqrt{c^2 - b^2}$

Calculation: $a = \sqrt{25^2 - 24^2} = \sqrt{625 - 576} = \sqrt{49}$

Answer: $a = 7$

Explanation: Here the two known sides are almost equal, so the answer is small: $625 - 576 = 49$, and $\sqrt{49} = 7$.

Example 5: Medium

Problem: A right triangle has legs $a = 14$ and $b = 48$. Find the missing side $c$.

Known values: $a = 14$, $b = 48$

Formula: $c = \sqrt{a^2 + b^2}$

Calculation: $c = \sqrt{14^2 + 48^2} = \sqrt{196 + 2304} = \sqrt{2500}$

Answer: $c = 50$

Explanation: Notice this is just the 7-24-25 triple doubled: $14 = 2 \times 7$, $48 = 2 \times 24$, $50 = 2 \times 25$. Scaling a triple keeps it a triple.

Example 6: Medium

Problem: A right triangle has legs $a = 9$ and $b = 40$. Find the missing side $c$.

Known values: $a = 9$, $b = 40$

Formula: $c = \sqrt{a^2 + b^2}$

Calculation: $c = \sqrt{9^2 + 40^2} = \sqrt{81 + 1600} = \sqrt{1681}$

Answer: $c = 41$

Explanation: A larger pair of legs still lands on a perfect integer: $81 + 1600 = 1681 = 41^2$, the 9-40-41 triple.

Example 7: Hard

Problem: A right triangle has hypotenuse $c = 29$ and leg $b = 21$. Find leg $a$.

Known values: $c = 29$, $b = 21$

Formula: $a = \sqrt{c^2 - b^2}$

Calculation: $a = \sqrt{29^2 - 21^2} = \sqrt{841 - 441} = \sqrt{400}$

Answer: $a = 20$

Explanation: A quick sanity check: the answer 20 is less than the hypotenuse 29, as it must be. This is the 20-21-29 triple.

Example 8: Hard

Problem: A right triangle has hypotenuse $c = 37$ and leg $b = 35$. Find leg $a$.

Known values: $c = 37$, $b = 35$

Formula: $a = \sqrt{c^2 - b^2}$

Calculation: $a = \sqrt{37^2 - 35^2} = \sqrt{1369 - 1225} = \sqrt{144}$

Answer: $a = 12$

Explanation: When the known leg is almost as long as the hypotenuse, the missing leg is short. Here $1369 - 1225 = 144$, so $a = 12$.

Example 9: Word Problem

Problem: A ladder is 13 feet long and leans against a wall. The base of the ladder is 5 feet away from the wall. How high up the wall does the ladder reach?

Known values: hypotenuse $c = 13$, leg $a = 5$

Formula: $b = \sqrt{c^2 - a^2}$

Calculation: $b = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12$

Answer: The height is 12.

Explanation: Drawing a diagram reveals a right triangle. Applying the Right Triangle Formula gives the missing dimension.

Example 10: Construction Example

Problem: You are building a rectangular deck that is 8 meters wide and 15 meters long. To ensure the corners are perfectly square, what should the diagonal measurement be?

Known values: leg $a = 8$, leg $b = 15$

Formula: $c = \sqrt{a^2 + b^2}$

Calculation: $c = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17$

Answer: The diagonal (hypotenuse) is 17.

Explanation: Drawing a diagram reveals a right triangle. Applying the Right Triangle Formula gives the missing dimension.

Example 11: Engineering Example

Problem: A suspension bridge needs a support cable. The tower is 40 meters tall, and the cable anchors to the ground 9 meters from the base of the tower. How long is the cable?

Known values: leg $a = 40$, leg $b = 9$

Formula: $c = \sqrt{a^2 + b^2}$

Calculation: $c = \sqrt{9^2 + 40^2} = \sqrt{81 + 1600} = \sqrt{1681} = 41$

Answer: The cable (hypotenuse) is 41.

Explanation: Drawing a diagram reveals a right triangle. Applying the Right Triangle Formula gives the missing dimension.

Example 12: Navigation Example

Problem: A ship sails 24 miles South, then turns exactly West and sails 7 miles. How far is the ship from its starting point in a straight line?

Known values: leg $a = 24$, leg $b = 7$

Formula: $c = \sqrt{a^2 + b^2}$

Calculation: $c = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25$

Answer: The distance (hypotenuse) is 25.

Explanation: Drawing a diagram reveals a right triangle. Applying the Right Triangle Formula gives the missing dimension.

Example 13: Word Problem

Problem: A flat-screen TV is sold by its diagonal length. If a TV is 60 inches wide and 11 inches tall, what is the diagonal measurement advertised?

Known values: leg $a = 60$, leg $b = 11$

Formula: $c = \sqrt{a^2 + b^2}$

Calculation: $c = \sqrt{11^2 + 60^2} = \sqrt{121 + 3600} = \sqrt{3721} = 61$

Answer: The diagonal (hypotenuse) is 61.

Explanation: Drawing a diagram reveals a right triangle. Applying the Right Triangle Formula gives the missing dimension.

6. Pythagorean Triples

What are Pythagorean Triples?

Pythagorean triples are sets of three positive integers $(a, b, c)$ that perfectly fit the formula $a² + b² = c²$. Memorizing them saves immense calculation time!

The Famous 3-4-5 Triangle

The 3-4-5 triangle is the most fundamental triple. If a triangle’s sides are 3, 4, and 5, it is guaranteed to be a right triangle. Builders use this constantly to square corners.

The 5-12-13 Triangle

The 5-12-13 triangle is another extremely common integer ratio found in math exams and architectural designs.

[Infographic of Pythagorean triples]

Table of 30 Common Pythagorean Triples

7. Real Life Applications

How is the formula used in real life?

From construction to game development, the Pythagorean theorem is everywhere.

Construction and Architecture

• Roof Pitch: Carpenters calculate the diagonal rafter lengths using the rise and run of the roof.
• Stairs & Ladders: Determining the exact length of a stringer or ladder needed to reach a certain height.
• Squaring Rooms: Using the 3-4-5 rule to ensure walls meet at a perfect 90-degree angle.

Engineering and Surveying

• Surveying: Calculating the steepness of slopes and the distance between topological points.
• Civil Engineering: Designing support cables for suspension bridges.

Tech and Navigation

• Navigation & GPS: Calculating the shortest straight-line distance between two coordinates.
• Computer Graphics & Gaming: Rendering 3D environments, calculating distances between entities, and ray tracing.
• Astronomy & Physics: Measuring distances between celestial bodies and calculating vector magnitudes.

[Illustration of roof pitch]

Application Table

8. Common Mistakes

Avoiding Pitfalls:

Even advanced students make mistakes. Here are 20 common errors to watch out for.

9. Practice Questions

Test your knowledge with these 30 practice questions!

Q1: In a right triangle, $a = 8$ and $b = 10$. What is $c$? Answer: $c = \sqrt{8^2 + 10^2} = \sqrt{164} \approx 12.81$.

Q2: In a right triangle, $a = 5$ and $b = 15$. What is $c$? Answer: $c = \sqrt{5^2 + 15^2} = \sqrt{250} \approx 15.81$.

Q3: In a right triangle, $a = 13$ and $b = 10$. What is $c$? Answer: $c = \sqrt{13^2 + 10^2} = \sqrt{269} \approx 16.4$.

Q4: In a right triangle, $a = 4$ and $b = 10$. What is $c$? Answer: $c = \sqrt{4^2 + 10^2} = \sqrt{116} \approx 10.77$.

Q5: In a right triangle, $a = 8$ and $b = 13$. What is $c$? Answer: $c = \sqrt{8^2 + 13^2} = \sqrt{233} \approx 15.26$.

Q6: In a right triangle, $a = 3$ and $b = 3$. What is $c$? Answer: $c = \sqrt{3^2 + 3^2} = \sqrt{18} \approx 4.24$.

Q7: In a right triangle, $a = 9$ and $b = 7$. What is $c$? Answer: $c = \sqrt{9^2 + 7^2} = \sqrt{130} \approx 11.4$.

Q8: In a right triangle, $a = 6$ and $b = 7$. What is $c$? Answer: $c = \sqrt{6^2 + 7^2} = \sqrt{85} \approx 9.22$.

Q9: In a right triangle, $a = 3$ and $b = 14$. What is $c$? Answer: $c = \sqrt{3^2 + 14^2} = \sqrt{205} \approx 14.32$.

Q10: In a right triangle, $a = 8$ and $b = 7$. What is $c$? Answer: $c = \sqrt{8^2 + 7^2} = \sqrt{113} \approx 10.63$.

Q11: In a right triangle, $a = 14$ and $b = 6$. What is $c$? Answer: $c = \sqrt{14^2 + 6^2} = \sqrt{232} \approx 15.23$.

Q12: In a right triangle, $a = 11$ and $b = 14$. What is $c$? Answer: $c = \sqrt{11^2 + 14^2} = \sqrt{317} \approx 17.8$.

Q13: In a right triangle, $a = 12$ and $b = 15$. What is $c$? Answer: $c = \sqrt{12^2 + 15^2} = \sqrt{369} \approx 19.21$.

Q14: In a right triangle, $a = 3$ and $b = 7$. What is $c$? Answer: $c = \sqrt{3^2 + 7^2} = \sqrt{58} \approx 7.62$.

Q15: In a right triangle, $a = 9$ and $b = 11$. What is $c$? Answer: $c = \sqrt{9^2 + 11^2} = \sqrt{202} \approx 14.21$.

Q16: In a right triangle, $a = 10$ and $b = 5$. What is $c$? Answer: $c = \sqrt{10^2 + 5^2} = \sqrt{125} \approx 11.18$.

Q17: In a right triangle, $a = 6$ and $b = 15$. What is $c$? Answer: $c = \sqrt{6^2 + 15^2} = \sqrt{261} \approx 16.16$.

Q18: In a right triangle, $a = 6$ and $b = 9$. What is $c$? Answer: $c = \sqrt{6^2 + 9^2} = \sqrt{117} \approx 10.82$.

Q19: In a right triangle, $a = 3$ and $b = 15$. What is $c$? Answer: $c = \sqrt{3^2 + 15^2} = \sqrt{234} \approx 15.3$.

Q20: In a right triangle, $a = 10$ and $b = 15$. What is $c$? Answer: $c = \sqrt{10^2 + 15^2} = \sqrt{325} \approx 18.03$.

Q21: In a right triangle, $a = 3$ and $b = 13$. What is $c$? Answer: $c = \sqrt{3^2 + 13^2} = \sqrt{178} \approx 13.34$.

Q22: In a right triangle, $a = 8$ and $b = 4$. What is $c$? Answer: $c = \sqrt{8^2 + 4^2} = \sqrt{80} \approx 8.94$.

Q23: In a right triangle, $a = 6$ and $b = 12$. What is $c$? Answer: $c = \sqrt{6^2 + 12^2} = \sqrt{180} \approx 13.42$.

Q24: In a right triangle, $a = 5$ and $b = 9$. What is $c$? Answer: $c = \sqrt{5^2 + 9^2} = \sqrt{106} \approx 10.3$.

Q25: In a right triangle, $a = 15$ and $b = 9$. What is $c$? Answer: $c = \sqrt{15^2 + 9^2} = \sqrt{306} \approx 17.49$.

Q26: In a right triangle, $a = 6$ and $b = 3$. What is $c$? Answer: $c = \sqrt{6^2 + 3^2} = \sqrt{45} \approx 6.71$.

Q27: In a right triangle, $a = 5$ and $b = 5$. What is $c$? Answer: $c = \sqrt{5^2 + 5^2} = \sqrt{50} \approx 7.07$.

Q28: In a right triangle, $a = 5$ and $b = 4$. What is $c$? Answer: $c = \sqrt{5^2 + 4^2} = \sqrt{41} \approx 6.4$.

Q29: In a right triangle, $a = 10$ and $b = 11$. What is $c$? Answer: $c = \sqrt{10^2 + 11^2} = \sqrt{221} \approx 14.87$.

Q30: In a right triangle, $a = 14$ and $b = 14$. What is $c$? Answer: $c = \sqrt{14^2 + 14^2} = \sqrt{392} \approx 19.8$.

Related Right Triangle Calculators

To make solving right triangles as easy as possible, we have built a suite of free, highly accurate calculators tailored to every specific problem type. Check them out below:

General Triangles & Sides

• Ultimate Right Triangle Solver
• Missing Side Calculator
• Missing Leg Calculator (from Hypotenuse & Leg)
• Hypotenuse Calculator
• Pythagorean Theorem Calculator
• Pythagorean Triples Calculator
• Similar Right Triangles Calculator

Angles & Trigonometry

• Missing Angle Calculator
• Sine Ratio Calculator
• Cosine Ratio Calculator
• Tangent Ratio Calculator
• Angle from Sine Calculator
• Angle from Cosine Calculator
• Angle from Tangent Calculator

Area, Perimeter & Advanced Properties

• Right Triangle Area Calculator
• Right Triangle Perimeter Calculator
• Right Triangle Semiperimeter Calculator
• Right Triangle Altitude Calculator
• Right Triangle Inradius Calculator
• Right Triangle Circumradius Calculator

Special Right Triangles

• 45-45-90 Triangle Calculator
• 30-60-90 Triangle Calculator
• 3-4-5 Triangle Calculator
• 5-12-13 Triangle Calculator

10. Frequently Asked Questions