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Diagonals and the Pythagorean Theorem

Author: Sophia
PDF Version

what's covered
In this lesson, you will use the Pythagorean theorem to calculate the length of a diagonal for a rectangle. You will also determine the length of a diagonal for the rectangular face of a real-world object. You'll continue to use your mathematical knowledge to build your productivity skill. Specifically, this lesson will cover:

Table of Contents

1. Calculating Diagonals

A diagonal connects two non-adjacent vertices in an enclosed shape. Below is an example of a diagonal of a rectangle:

A dotted line runs diagonally across a rectangle, connecting two opposite corners and dividing the rectangle into two congruent triangles. Each of the two remaining opposite corners of the rectangle contains a square.

Notice that the diagonal of the rectangle connects two opposite corners. It also creates two congruent triangles. Congruent means of equal measure, so the two triangles are the same size, and take up the same amount of space. We should also point out that the triangles are right triangles, because one of their angles is a 90 degree angle (taken from the 90 degree angles of the rectangle).

Let's take a closer look at the rectangle and the two triangles that the diagonal created. The sides of the rectangle correspond to the vertical and horizontal legs of the right triangle. What about the diagonal? We can refer to the diagonal as the hypotenuse of the right triangle (the hypotenuse is always opposite of the right angle).

A right triangle with a small square drawn at one of the corners indicates a right angle. The longest side of the triangle is labeled ‘Hypotenuse’.

2. The Pythagorean Theorem

To calculate the length of the diagonal, we can use the Pythagorean theorem to calculate the length of the hypotenuse. The Pythagorean theorem uses the side lengths of the other legs of the right triangle in order to find the length of the hypotenuse:

formula to know
The Pythagorean Theorem
table attributes columnalign left end attributes row cell a squared plus b squared equals c squared end cell row cell c equals square root of a squared plus b squared end root end cell end table

So, if we take the sum of the squares of the side lengths, this equals the square of the hypotenuse leg. We'll just need to take the square root of the sum in order to express the length of the hypotenuse.

EXAMPLE

What is the length of the diagonal below?

A dotted line runs diagonally across a rectangle, connecting two opposite corners and dividing the rectangle into two congruent triangles. Each of the two remaining opposite corners of the rectangle contains a square. The length of the triangles is labeled ‘8 ft’ and the width is labeled ‘3.5 ft’.

We can substitute 3.5 ft and 8 ft into a and b, respectively, and apply the Pythagorean theorem:

a squared plus b squared equals c squared The Pythagorean theorem
open parentheses 3.5 space ft close parentheses squared plus open parentheses 8 space ft close parentheses squared equals c squared Substitute the measurements of the legs: a equals 3.5 space ft comma space b equals 8 space ft
12.25 space ft squared plus 64 space ft squared equals c squared Square 3.5 ft and 8 ft.
76.25 space ft squared equals c squared Add 12.25 ft2 and 64 ft2.
square root of 76.25 space ft squared end root equals square root of c squared end root Apply a square root to both sides.
8.73 space ft equals c Calculate the square root and round to the hundredths place

Another method to solving is to rewrite the Pythagorean theorem, isolating c on one side of the equation. Then, we can substitute a and b into the equation, and calculate the length of the diagonal.

a squared plus b squared equals c squared The Pythagorean theorem
square root of a squared plus b squared end root equals c Take the square root of both sides.
square root of open parentheses 3.5 space ft close parentheses squared plus open parentheses 8 space ft close parentheses squared end root equals c Substitute 3.5 ft for a and 8 ft for b.
square root of 76.25 space ft squared end root equals c Square 3.5 and 8, then add together.
8.73 space ft equals c Calculate the square root and round to the hundredths place

3. Applications

We can apply the Pythagorean theorem to find the diagonal length for squares and rectangles found in the real world.

did you know
Television manufacturers state the screen size of each TV based on the length of its diagonal. If you have a 32-inch television at home, your TV screen measures 32 inches across its diagonal. This same convention is used for screens on computer monitors and smartphones.

try it
Rudy measures the height and width of the television screen in his dorm room. The screen is 19.6 inches high and 34.9 inches long.
Which standard size TV does Rudy have?
Using the Pythagorean theorem, you will find that Rudy has a standard 40-inch television. The calculations are as follows:

a squared plus b squared equals c squared The Pythagorean theorem
square root of open parentheses 19.6 space in close parentheses squared plus open parentheses 34.9 space in close parentheses squared end root equals c Substitute 19.6 for a and 34.9 for b.
square root of 1 comma 602.17 space in squared end root equals c Square 19.6 and 34.9, then add together.
40.0 space in equals c Calculate the square root and round to the tenths place

Productivity: Apply Your Skill
On the way to work, you normally cut diagonally across a square park to save time. However, the park is closed due to an event. If it normally takes 10 minutes to walk across the park, how much longer will it take to get to work if you walk along the outside edge? Solving problems such as this in your everyday life will increase your productivity.

summary
The diagonal of a rectangle or square corresponds to the hypotenuse of a right triangle. To calculate the diagonal, we can use the Pythagorean theorem, which we use for right triangles, to calculate the diagonal of a rectangle or square. There are two different forms of the Pythagorean theorem. Remember that c in our formula has to be the hypotenuse, but a and b can be either of the legs of our right triangle. Calculating diagonals has applications in the real world. For example, television and computer monitor manufacturers establish screen size in terms of diagonal length.

Best of luck in your learning!

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Formulas to Know
Pythagorean Theorem

a squared plus b squared equals c squared</p> <p>c equals square root of a squared plus b squared end root

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