# Top 7 real-life Pythagorean theorem word problems

- Author: Noreen Niazi
- Last Updated on: August 22, 2023

SECTIONS

ToggleDo you know about the Pythagorean theorem?

Are you familiar with how to solve the Pythagorean theorem?

As a math student, the Pythagorean theorem is one of the first concepts you learn in high school. The theorem is a mathematical formula used to find the length of one side of a right triangle if the other two sides are known. However, solving Pythagorean theorem word problems can take time and effort. In this article, I will share the top seven Pythagorean theorem word problems you might encounter in real life and some tips for solving them.

## Introduction to the Pythagorean Theorem

The Pythagorean theorem is a mathematical formula used to find the length of one side of a right triangle if the other two sides are known. The theorem states that the square of the hypotenuse (the longest side of the triangle) equals the sum of the squares of the other two sides. The Pythagorean theorem formula is written as a² + b² = c², where “a” and “b” are the two shorter sides of the triangle, and “c” is the hypotenuse[1].

## Steps to solve Pythagorean theorem word problems

Here are the steps you can take to solve Pythagorean theorem word problems:

- Read the problem carefully and identify what is given and what needs to be found.
- Identify the sides of the triangle that are known and unknown.
- Write the formula for the Pythagorean theorem and substitute known values into the formula.
- Solve the equation to find the unknown value.
- Check your answer and make sure it makes sense with the problem.

## Tips for Solving Pythagorean Theorem Word Problems

Here are some tips you can follow to solve Pythagorean theorem word problems more easily:

- Draw a diagram of the triangle and label the known and unknown sides.
- Be consistent with the measurement units used in the problem.
- Check that the triangle is a right triangle before using the Pythagorean theorem formula.
- If the value found for the length of one side of the triangle is not an integer, round to the nearest decimal number or answer in radical form.

Get more details about the Different math Problem solving strategies in Math

19 Best Math Problem-Solving Strategies For Elementary Students – LearnAboutMath

## What are Pythagorean theorem word problems?

In word problems employing the Pythagorean theorem, the length of one side of a right triangle is determined using the formula. Finding the length of a side that is not the hypotenuse is common in these issues. Word problems are based on the Pythagorean Theorem in many facets of life, including science, engineering, and architecture.

Learn about the divison word probelms.

Mastering Long Division Word Problems: Tips And Tricks – LearnAboutMath

## Understanding the Fundamentals of the Pythagorean Theorem Word Problems

To comprehend Pythagorean theorem word problems, you must first understand the theorem itself. According to the theorem, the hypotenuse’s square equals the sum of the squares of the triangle’s other two sides. In a triangle, the hypotenuse, opposite the right angle, is the longest side. The legs are the other two sides of the triangle.

## Real Life Pythagorean Theorem Word Problem - Problem 1

An architect is designing a house and needs to calculate the length of a hallway that connects two rooms. He knows that the first room has a wall that is 12 feet long, and the second room has a wall that is 9 feet long. The hallway is a right triangle. What is the length of the hallway?

**Solution:**

To solve this problem, we need to use the Pythagorean theorem formula.

- We know that the first wall is 12 feet long, and the second is 9 feet long.
- Let’s call the unknown length of the corridor “x.”
- We can write the equation as

$$12² + 9² = x²$$

Solving the equation, we find that x² = 225, which means x = 15 feet.

Therefore, the length of the corridor is 15 feet.

## Real Life Pythagorean Theorem Word Problem - Problem 2

An engineer is building a bridge and needs to calculate the cable length between two towers. He knows that the horizontal distance between the towers is 200 feet and the vertical distance is 75 feet. What is the cable length?

**Solution:**

To solve this problem, we need to use the Pythagorean theorem formula.

- We know the horizontal distance between the towers is 200 feet, and the vertical distance is 75 feet.
- Let’s call the unknown cable length “x.”
- We can write the equation as 200² + 75² = x².
- Solving the equation, we find that x² = 45.625, which means x = 213.8 feet (rounding to the nearest decimal number).

Therefore, the cable length is 213.8 feet.

## Real Life Pythagorean Theorem Word Problem - Problem 3

A scientist is studying a volcano and needs to calculate the distance between his observation point and the volcano’s base.

- He knows that his position is 500 feet above sea level and that the volcano’s base is 1000 feet above sea level.
- He also knows that the horizontal distance between his position and the volcano’s base is 2000 feet.
- How far is your observation point from the base of the volcano?

**Solution:**

To solve this problem, we need to use the Pythagorean theorem formula.

- We know that the height of the observation point is 500 feet above sea level, and the volcano’s base is 1000 feet above sea level.
- Let’s call the unknown distance between the observation point and the volcano’s base “x.”
- The horizontal distance between the observation point and the volcano’s base is 2000 feet.
- We can write the equation as 500² + 2000² = (x + 1000)².
- Solving the equation, we find that x² = 3,500,000, which means that x = 1870 feet (rounding to the nearest decimal number).

Therefore, the distance between the observation point and the volcano’s base is 1870 feet

## Real Life Pythagorean Theorem Word Problem - Problem 4

A farmer needs to build a fence around a rectangle to keep his animals safe.

- He knows that one side of the rectangle is 30 feet and that the diagonal of the rectangle is 50 feet.
- What is the length of the other side of the rectangle?

**Solution:**

To solve this problem, we need to use the Pythagorean theorem formula.

- We know that one side of the rectangle is 30 feet and that the diagonal of the rectangle is 50 feet.
- Let’s call the unknown length of the other side of the rectangle “x”.
- We can write the equation as 30² + x² = 50².
- Solving the equation, we find that x² = 1600, which means x = 40 feet.

Therefore, the length of the other side of the rectangle is 40 feet.

## Real Life Pythagorean Theorem Word Problem - Problem 5

A pilot flying straight from one city to another must avoid a mountain.

- He knows that the mountain’s altitude is 1500 feet, and his flight altitude is 10,000 feet.
- He also knows the horizontal distance between the two cities is 50 miles.
- How far does he need to dodge to avoid the mountain?

**Solution:**

To solve this problem, we need to use the Pythagorean theorem formula.

- We know the mountain’s altitude is 1500 feet, and the pilot’s flight altitude is 10,000 feet.
- Let’s call the unknown distance the pilot needs to deviate “x.”
- We know that the horizontal distance between the two cities is 50 miles.
- We can convert this distance to feet by multiplying by 5280, which means the horizontal distance is 264,000 feet.
- We can write the equation as 1500² + x² = 10,000².
- Solving the equation, we find that x² = 98,551,000, which means that x = 9,927 feet (rounding to the nearest decimal number).

Therefore, the pilot needs to deviate 9,927 feet to avoid the mountain.

## Real Life Pythagorean Theorem Word Problem - Problem 6

A diver is diving in a lake and must swim to a buoy 30 feet away.

- He knows that the depth of the lake is 20 feet.
- How far does he have to swim to reach the buoy?

**Solution:**

To solve this problem, we need to use the Pythagorean theorem formula.

- We know the lake is 20 feet deep, and the buoy is 30 feet away.
- The unknown distance the diver needs to swim “x”.
- We can write the equation as 20² + 30² = x².
- Solving the equation, we find that x² = 1300, which means x = 36.06 feet (rounding to the nearest decimal number).

Therefore, the diver needs to swim 36.06 feet to reach the buoy.

## Real Life Pythagorean Theorem Word Problem - Problem 7

An archaeologist is measuring the height of an ancient pyramid.

- He knows that the base of the pyramid is 100 feet long and that the pyramid is 75 feet high.
- What is the length of a straight line from the top of the pyramid to the ground?

**Solution:**

To solve this problem, we need to use the Pythagorean theorem formula.

- We know that the base of the pyramid is 100 feet long and that the pyramid is 75 feet high.
- Let’s call the unknown length of the straight line from the top of the pyramid to the ground “x”.
- We can write the equation as 75² + (50)² = x².
- Solving the equation, we find that x² = 7.625, which means x = 87.3 feet (rounding to the nearest decimal number).

Therefore, the straight line measurement from the top of the pyramid to the ground is 87.3 feet.

## 'Conclusion and Tips to Solve Pythagorean Theorem Word Problems

The Pythagorean theorem makes finding a right triangle’s missing side or the angle between two sides easier. According to this rule, the square of the hypotenuse, the longest side, equals the sum of the squares of the other two sides. The right angle must be located, the sides must be denoted by the letters a, b, and c, and the known values must be entered into the equation $a2 + b2 = c2$ to apply the theorem. We can then use trigonometric functions to get the angle or solve for the unknown side.

The following are some pointers for resolving word problems with the Pythagorean theorem:

– Sketch a circumstance diagram, labeling the sides and angles as necessary.

– Determine if the triangle is right-angled by looking for hints in the problem or applying the Pythagorean theorem’s opposite.

– To get the omitted side or angle, apply the Pythagorean theorem or one of its variations, such as $$a2 + b2 – 2ab cos C = c2$$.

– Your response should be rounded to the proper degree of precision and, if necessary, include units.

– Put your solution back into the formula or use another approach to verify it.

## FAQs: Pythagorean Theorem Word Problems

The following actions must be taken in order to use the Pythagorean theorem to solve word problems:

- Name the sides of the right triangle in the given situation as a, b, and c.
- Choose the sides that you have been provided and that you need to find.
- Solve for the unknown side by substituting the provided values into the Pythagorean theorem.
- Round off your response to the required degree of precision.

The Pythagorean theorem can be applied in a variety of circumstances to determine the shortest distance, sound wave speed, square angles, and the missing side lengths of right triangles.

Here is an real life example of Pythagorean theorem.

**Navigation:** The Pythagorean theorem can be used to determine the quickest route and the direction to take if you are sailing or flying and wish to reach a destination that is not directly north, south, east, or west of your current location.

The Pythagorean theorem, for instance, can be used to determine that the shortest distance and direction are approximately 500 miles and 53 degrees west of north, respectively, if you are at sea and travelling to a position that is 300 miles north and 400 miles west of your present location.

Finding the length of a right triangle’s third side given the measurements of its other two sides is an illustration of the Pythagorean theorem in mathematics. A triangle with a single angle that measures 90 degrees is referred to as a right triangle. The hypotenuse is the side of a right triangle that is the longest and sits across from the right angle.

The Pythagorean theorem can be used to determine the length of the hypotenuse, for instance, if a right triangle has legs that are 3 cm and 4 cm long:

Replace the supplied values in the equation as follows: Simplify: 3 + 4 = c 2. 9 + 16 = c 2.2. Add: 25 = c^2. Determine the square root of each side: c = 25 = 5.

The hypotenuse is 5 cm long as a result.