Pythagorean Theorem: Formula, Examples, and Applications

The Pythagorean Theorem is one of the most important and widely used concepts in mathematics. Named after the ancient Greek mathematician Pythagoras, its applications stretch far beyond the classroom, impacting fields like architecture, engineering, physics, and computer science.

In this guide, we’ll dive into the history, proof, applications, and significance of the Pythagorean Theorem, giving you a deeper understanding of this essential mathematical principle.

What is the Pythagorean theorem?

The Pythagorean Theorem is a fundamental principle in geometry that relates to right-angled triangles. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In simpler terms, it provides a way to calculate the length of one side of a right triangle if the lengths of the other two sides are known. This theorem is not only a cornerstone of geometry but also has real-world applications in fields such as architecture, navigation, and physics.

​Pythagorean theorem formula

In mathematics, the Pythagorean theorem or Pythagoras’ theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. The formula for the Pythagoras theorem is:

a²+b²=c²

Where:

• a and b are the lengths of the two legs (the sides forming the right angle).
• c is the length of the hypotenuse (the longest side, opposite the right angle).

This equation allows you to find the missing side of a right-angled triangle abc, provided you know the lengths of the other two sides. The theorem has been proved numerous times by many different methods – possibly the most for any mathematical theorem.

We can rearrange the Pythagoras theorem formula to solve for the missing side:

• To find the length of Side A: a² = c² – b²
• To find the length of Side B: b² = c² – a²
• To find the length of Side C: c² = a² + b²

Pythagoras Theorem Proofs

There are a multitude of Pythagoras theorem proofs, possibly even the greatest number of any mathematical theorem.

Here are two algebraic proofs of the Pythagoras theorem. These proofs use algebraic manipulation and geometric principles to demonstrate why the Pythagorean equation a²+b²=c² holds for a right-angled triangle with legs a and b, and hypotenuse c.

Proof 1: Using Area and Rearrangement

1. Start with a square: Consider a large square with side length a+b. Inside this square, place four same triangles, each with legs a and b, and hypotenuse c. Arrange the triangles so that they form a smaller square in the center with side length c.
2. Calculate the area of the large square: (a+b)²
3. Express the area in terms of the triangles and the smaller square: The area of the large square can also be expressed as the sum of the areas of the four triangles and the smaller square:

Area of large square = 4 × (1/2 ab) + c²

(a + b)² = 2ab + c²

a² + 2ab + b² = 2ab + c²

a² + b² = c²

Proof 2: Using Similar Triangles

1. Start with a right-angled triangle: Consider a right-angled triangle with legs a and b, and hypotenuse c. Draw an altitude from the right angle to the hypotenuse, dividing the hypotenuse into two segments of lengths p and q, where p+q=c.
2. Identify similar triangles:
   • The original triangle is similar to the two smaller triangles formed by the altitude. This means the ratios of corresponding sides are equal: a/p = c/a and b/q = c/b​
3. Express a² and b² in terms of p and q:
   • From the first similarity: a²=p⋅c
   • From the second similarity: b²=q⋅c
4. Add the two equations: a²+b² = p⋅c+q⋅c = c(p+q)
5. Substitute p+q=c: a²+b² = c⋅c = c²

These two algebraic proofs demonstrate the versatility of the Pythagorean Theorem and how it can be derived using different algebraic and geometric approaches.

How to use the Pythagorean theorem

The Pythagorean Theorem is a powerful tool that helps us solve problems involving right-angled triangles. To use it effectively, follow these steps:

1. Identify the Right Triangle

The first step in using the Pythagorean Theorem is to make sure you have a right-angled triangle. This means one of the angles must be exactly 90 degrees. The hypotenuse, the longest side, is always opposite the right angle.

2. Label the Sides

Next, label the sides of a right triangle. In any right triangle:

• The two sides that form the right angle are called the legs (often denoted as a and b).
• The longest side, opposite the right angle, is called the hypotenuse (denoted as c).

3. Choose the Formula Based on What You Know

Once you’ve labeled the sides, you’ll need to decide which side you want to find. There are three possible scenarios:

• To find the length of the hypotenuse (c): Use the Pythagoras theorem formula:

c = √(a² + b²)

• To find the length of one leg (a or b): If you know the hypotenuse and the length of one leg, rearrange the Pythagoras theorem formula to solve for the missing leg:

a = √(c² – b²) or b = √(c² – a²)

4. Plug in the Known Values

Substitute the values you know into the Pythagoras theorem formula. Make sure the sides are labeled correctly and that you’re using the appropriate formula.

5. Solve for the Missing Side

Now, do the math! For a more straightforward calculation:

• Square the known values (e.g., a² or b²).
• Add or subtract the squared values (depending on the formula you’re using).
• Take the square root of the result to find the length of the unknown side.

6. Double-Check Your Units

Make sure you’re consistent with your units. For example, if the sides are measured in centimeters, the answer will also be in centimeters. Also, remember to check that the triangle is right-angled—this method only works for right triangles!

Example 1: Finding the Hypotenuse

Suppose you have a right triangle where the legs measure 3 cm and 4 cm. To find the hypotenuse, use the formula:

c = √(a² + b²)

Substitute the known values:

c = √(3² + 4²) = √(9 + 16) = √(25) = 5 cm

So, the hypotenuse is 5 cm.

Example 2: Finding a Missing Leg

If you know the hypotenuse is 10 cm and one of the legs is 6 cm, you can find the missing leg using the rearranged formula:

a = √(c² – b²)

Substitute the known values:

a = √(10² – 6²) = √(100 – 36) = √(64) = 8 cm

So, the missing leg is 8 cm.

Applications of the Pythagorean Theorem

The Pythagorean Theorem is widely used in many real-world situations, such as:

• Architecture and construction: To ensure buildings are level and walls are perpendicular.
• Navigation: To calculate the shortest distance between two points, especially in coordinate geometry.
• Physics: In problems involving forces, velocity, or motion, where right-angled triangles often appear.

By following these steps and practicing a few problems, you’ll be able to apply the Pythagoras theorem confidently in both academic and real-world contexts!

Solved examples

Let’s explore some practical examples of how to apply the Pythagoras Theorem in different scenarios. These examples will help you understand the steps involved in using the formula and how it works in real-world situations.

Example 1: Finding the Hypotenuse

Problem: You have a right-angled triangle with the lengths of the two legs measuring 6 cm and 8 cm. What is the length of the hypotenuse?

Solution:

To find the hypotenuse (c), use the Pythagorean Theorem formula:

c = √(a² + b²)

Substitute the known values for a=6 and b=8:

c = √(6² + 8²)

= √(36 + 64) = √100 = 10

So, the length of the hypotenuse is 10 cm.

Example 2: Finding a Missing Leg

Problem: You know the hypotenuse of a right-angled triangle is 13 cm, and one of the legs is 5 cm. What is the length of the other leg?

Solution:

To find the missing leg (let’s call it b), we can rearrange the Pythagorean Theorem formula:

b = √(c² – a²)

Substitute the known values for c=13 and a=5:

b = √(13² – 5²)

= √(169 – 25) = √144 = 12

So, the length of the missing leg is 12 cm.

Example 3: Real-World Application (Ladder Problem)

Problem: A ladder is leaning against a wall. The distance from the bottom of the ladder to the wall is 6 meters, and the length of the ladder is 10 meters. How high up the wall does the ladder reach?

Solution:

In this case, the ladder forms a right-angled triangle with the wall, where the ladder is the hypotenuse, the distance from the wall is one leg, and the height up the wall is the other leg. To find the height, we can use the Pythagorean Theorem:

h = √(l² – d²)

Where:

• l = 10 meters (the length of the ladder, hypotenuse),
• d=6 meters (the distance from the wall, one leg),
• h is the height up the wall (the other leg).

Substitute the known values:

h = √(10² – 6²)

= √(100 – 36) = √64 = 8

So, the ladder reaches 8 meters up the wall.

Example 4: Finding the Diagonal of a Square

Problem: A square has a side length of 5 cm. What is the length of the diagonal?

Solution:

The diagonal of a square forms a right-angled triangle with two sides of the square. Since both sides are equal in length, the legs of the triangle are both 5 cm. To find the diagonal, we use the Pythagorean Theorem:

d = √(s² + s²)

Where:

• d is the diagonal,
• s=5 cm (the side length of the square).

Substitute the known values:

d = √(5² + 5²)

= √(25 + 25) = √50 ≈ 7.07

So, the length of the diagonal is approximately 7.07 cm.

Pythagoras theorem calculator

A Pythagorean Theorem calculator simplifies the process of finding the lengths of the sides of a right triangle. Whether you’re a student learning geometry or a professional needing quick calculations, this tool provides instant results, making it easier to visualize and understand the relationships in right triangles.

URL: https://www.calculator.net/pythagorean-theorem-calculator.html

Conclusion

The Pythagorean Theorem has stood the test of time as a fundamental concept in mathematics. From its roots in ancient Greece to its applications in today’s technology, it continues to be a vital tool for solving problems and understanding the world around us.

Whether you’re a student learning geometry, a teacher inspiring your class, or a professional using it in your work, the theorem opens doors to new insights and practical solutions. Mastering this simple yet powerful principle can sharpen your problem-solving skills and deepen your appreciation for mathematics.

Next time you work with a right-angled triangle, take a moment to reflect on the Pythagorean Theorem and its remarkable contributions to science, engineering, and daily life.

Frequently Asked Questions

Q1: What is the full Pythagoras formula?

Pythagoras’ theorem states that, in a right triangle ABC, the square of the hypotenuse is equal to the sum of the square of the other two sides. The full formula for the Pythagorean Theorem is: c² = a² + b². Here, c is the hypotenuse (the longest side of a right triangle), and a and b are the two shorter sides.

Q2: Does the Pythagorean theorem work on only right triangles?

The Pythagorean theorem states that in a right triangle ABC, the square of the hypotenuse is equal to the sum of the squares of the other two legs.

Yes, the Pythagorean Theorem applies exclusively to right triangles. It works because the relationship between the squares of the sides holds only when one angle of the triangle is exactly 90 degrees.

​Q3: What is the Pythagorean theorem in simple terms?

In simple terms, the Pythagorean Theorem states that in a right triangle, the square of the longest side (hypotenuse) is equal to the sum of the squares of the other two sides. For example, if you know the lengths of two sides, you can use this rule to find the third side.