Pythagorean Theorem

Here you will learn about the Pythagorean Theorem, including how to find side measurements of a right triangle and using Pythagoras’ theorem to check and see if a triangle has a right angle or not.

Students will first learn about Pythagorean Theorem as part of geometry in 8 th grade and continue to use it in high school.

What is the Pythagorean Theorem?

The Pythagorean Theorem states that the square of the longest side of a right triangle (called the hypotenuse) is equal to the sum of the squares of the other two sides.

Pythagorean Theorem formula shown with triangle ABC is:

a^2+b^2=c^2

Side c is known as the hypotenuse. The hypotenuse is the longest side of a right triangle. Side a and side b are known as the adjacent sides. They are adjacent, or next to, the right angle.

You can only use the Pythagorean Theorem with right triangles.

For example,

Let’s look at this right triangle:

Above, three square grids have been drawn next to each of the sides of the triangle.

The area of the side of length 3=3 \times 3=3^2=9

The area of the side of length 4=4 \times 4=4^2=16

The area of the side of length 5=5 \times 5=5^2=25

The sum of the areas of the squares on the two shorter sides is equal to the area of the square on the longest side.

9+16=25

When you square the sides of the two shorter sides of a right triangle and add them together, you get the square of the longest side.

3^2+4^2=5^2

3, 4, 5 is known as a Pythagorean triple.

There are other Pythagorean triples such as 5, 12, 13 and 8, 15, 17.

If you know two lengths of a right triangle, you can use Pythagorean Theorem to work out the length of the third side.

Pythagorean Theorem Proof

Use the drawing of a square with a smaller square shown inside with the proof below.

The area of each triangle is \cfrac{1}{2} \, a b and the area of the smaller square is c^2.

There are two ways to find the area of the larger square.

1. Combine the area of the four congruent triangles and the smaller square:
   
   \begin{aligned}& =\cfrac{1}{2} \, a b+\cfrac{1}{2} \, a b+\cfrac{1}{2} \, a b+\cfrac{1}{2} \, a b+c^2 \\\\ & =2 a b+c^2\end{aligned}
   
   
2. Multiply the side lengths of the larger square together:
   
   \begin{aligned}& =(a+b)(a+b) \\\\ & =a^2+2 a b+b^2\end{aligned}

Now set the two expressions equal to each other to prove a^2+b^2=c^2 \text{:}

\begin{aligned}a^2 +2 a b+b^2 &=2 a b+c^2 \\\\ -2 a b \hspace{0.3cm} & \hspace{0.3cm} -2 a b \\\\ a^2+b^2&=c^2\end{aligned}

Since the triangles formed by the vertices of a square will also be right triangles, the proof above shows that a^2+b^2=c^2 will always be true for the sides of a right triangle.

Common Core State Standards

How does this relate to 8 th grade math?

• Grade 8 – Geometry (8.G.B.6)
  Explain a proof of the Pythagorean Theorem and its converse.

• Grade 8 – Geometry (8.G.B.7)
  Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

How to use Pythagorean Theorem

In order to use Pythagorean Theorem:

1. Label the sides of the triangle.
2. Write down the formula and apply the numbers.
3. Record the answer.

Pythagorean Theorem examples

Example 1: missing length of the hypotenuse c

Find x and answer to the nearest hundredth.

1. Label the sides of the triangle.

Label the hypotenuse (the longest side) with c. The adjacent sides, next to the right angle can be labeled a and b (either way – they are interchangeable).

2Write down the formula and apply the numbers.

\begin{aligned}& a^2+b^2=c^2 \\\\ & 3^2+8^2=x^2 \\\\ & 9+64=x^2 \\\\ & 73=x^2 \\\\ & \sqrt{73}=x\end{aligned}

An alternative method of rearranging the formula and to put one calculation into a calculator will also work.

\begin{aligned}& a^2+b^2=c^2 \\\\ & c^2=a^2+b^2 \\\\ & c=\sqrt{a^2+b^2} \\\\ & x=\sqrt{3^2+8^2}\end{aligned}

3Record the answer.

Make sure you give your final answer in the correct form by calculating the square root value, including units where appropriate.

x=\sqrt{73}=8.5440037…

The final answer to the nearest hundredth is:

x=8.54 \mathrm{~cm}

Example 2: missing length c

Find x and answer to the nearest tenth.

Label the sides of the triangle.

Label the hypotenuse (the longest side) with c. The adjacent sides, next to the right angle can be labeled a and b (either way – they are interchangeable).

Example 3: finding an adjacent side (a short side)

Find x and write your answer to the nearest hundredth.

Label the sides of the triangle.

Label the hypotenuse (the longest side) with c. The adjacent sides, next to the right angle can be labeled a and b (either way – they are interchangeable).

Example 4: finding an adjacent side (a short side)

Find x and write your answer to the nearest tenth.

Label the sides of the triangle.

Label the hypotenuse (the longest side) with c. The adjacent sides, next to the right angle can be labeled a and b (either way – they are interchangeable).

Example 5: checking if a triangle has a right angle

Is the triangle below a right triangle?

Label the sides of the triangle.

Label the longest side with c. The adjacent sides, next to the right angle can be labeled a and b (either way – they are interchangeable).

Example 6: checking if a triangle has a right angle

Is the triangle below a right triangle?

Label the sides of the triangle.

Label the longest side with c. The adjacent sides, next to the right angle can be labeled a and b (either way – they are interchangeable).

Teaching tips for Pythagorean Theorem

• In the beginning, give examples that are on grids and have the sides of the right triangle as positive integers. This allows students to draw the corresponding square for each side of the triangle and test to see that a^2+b^2=c^2.

• Give students a chance to try and prove the Pythagorean Theorem on their own. Then give students examples of a few different theorems and challenge them to find the one that makes the most sense for them.

• The Pythagorean Theorem has connections outside of math classes. If time allows, students can explore the history of this theorem.

Easy mistakes to make

• Not correctly identifying the hypotenuse
  It is very important to make sure that the hypotenuse, the long side, is correctly identified and labeled c.

• Forgetting that right triangles may be in different orientations
  The triangles can be drawn in different orientations.
  For example,
  These are examples of the same right triangle ADB in different orientations.
  
  

• Thinking the lengths of sides can only be whole numbers
  Lengths can be decimals, fractions or even irrational numbers such as, \sqrt{2}.

• Not knowing the Pythagorean theorem can also be applied to other shapes
  An application of Pythagorean theorem is to extend it to work on other shapes such as a trapezoid.
  
  

• Rounding too early
  If you need to use Pythagorean Theorem in a question with multiple steps, do not round until the very end of the question or you will lose accuracy. For example, you may need to find the height of a triangle, and then use that height to find its area.

The next lessons are

• Trigonometry
• Circle math
• Sectors, arcs and segments

Still stuck?

At Third Space Learning, we specialize in helping teachers and school leaders to provide personalized math support for more of their students through high-quality, online one-on-one math tutoring delivered by subject experts.

Each week, our tutors support thousands of students who are at risk of not meeting their grade-level expectations, and help accelerate their progress and boost their confidence.

Find out how we can help your students achieve success with our math tutoring programs.