Lesson 10: Applications of the Pythagorean Theorem

Let’s explore some applications of the Pythagorean Theorem.

10.1: Closest Estimate: Square Roots

Which estimate is closest to the actual value of the expression? Explain your reasoning. 

  1. $\sqrt{24}$

    • 4

    • 4.5

    • 5

  2. $\sqrt7$

    • 2

    • 2.5

    • 3

  3. $\sqrt{42}$

    • 6

    • 6.5

    • 7

  4. $\sqrt{10}$ + $\sqrt{97}$

    • 13

    • 13.25

    • 13.5

10.2: Cutting Corners

Mai and Tyler were standing at one corner of a large rectangular field and decided to race to the opposite corner. Since Mai had a bike and Tyler did not, they thought it would be a fairer race if Mai rode along the sidewalk that surrounds the field while Tyler ran the shorter distance directly across the field. The field is 100 meters long and 80 meters wide. Tyler can run at around 5 meters per second, and Mai can ride her bike at around 7.5 meters per second.

A rectangle with a horizontal base and vertical height bolded. A dashed line is drawn from the top left vertex to the bottom right vertex.
  1. Before making any calculations, who do you think will win? By how much? Explain your thinking.

  2. Who wins? Show your reasoning.

10.3: Internal Dimensions

Here are two rectangular prisms:

A rectangular prism labeled “K” and a cube labeled “L” are indicated. The rectangular prism has a side with a length of 5 units, a side with length of 4 units, and a vertical height with a length of 6 units. A diagonal is drawn from the bottom left vertex to the top right vertex of the prism. The cube has all side lengths of 5 units. A diagonal is drawn from the bottom left vertex to the top right vertex of the cube.
  1. Which figure do you think has the longer diagonal? Note that the figures are not drawn to scale.
  2. Calculate the lengths of both diagonals. Which one is actually longer?


The Pythagorean Theorem can be used to solve any problem that can be modeled with a right triangle where the lengths of two sides are known and the length of the other side needs to be found. For example, let’s say a cable is being placed on level ground to support a tower. It’s a 17-foot cable, and the cable should be connected 15 feet up the tower. How far away from the bottom of the tower should the other end of the cable connect to the ground?

It is often very helpful to draw a diagram of a situation, such as the one shown here:

A right triangle, with the vertical leg labeled tower has a length of 15 units. The hypotenuse, labeled cable has a length of 17 units. The length of the other leg is horizontal and represented by the letter a.

It’s assumed that the tower makes a right angle with the ground. Since this is a right triangle, the relationship between its sides is $a^2+b^2=c^2$, where $c$ represents the length of the hypotenuse and $a$ and $b$ represent the lengths of the other two sides. The hypotenuse is the side opposite the right angle. Making substitutions gives $a^2+15^2=17^2$. Solving this for $a$ gives $a=8$. So, the other end of the cable should connect to the ground 8 feet away from the bottom of the tower.

Practice Problems ▶