Perimeter of Diagonal
2026-02-28 08:36 Diff
https://brightchamps.com/en-us/math/measurement/perimeter-of-diagonal

150 Learners

Last updated on September 8, 2025

The perimeter of a shape involves the total length of its boundary. While the term "perimeter of diagonal" might not be standard, it can refer to the idea of calculating the total distance around a rectangle or square, including its diagonals. This concept is useful in various practical applications such as construction, design, and more. In this topic, we will explore the perimeter of a diagonal in the context of rectangles and squares.

What is the Perimeter of Diagonal?

The term "perimeter of diagonal" is not typically used in standard geometry; however, it can be understood as the total distance around a rectangle or square, including its diagonals.

For a rectangle, if you imagine walking along all four sides and its two diagonals, you would be covering the perimeter plus the diagonals.

The formula for the perimeter of a rectangle is 𝑃 = 2(𝑎 + 𝑏), where a and b are the lengths of the rectangle's sides.

The length of the diagonal d can be found using the formula 𝑑 = √(𝑎² + 𝑏²).

For instance, if a rectangle has sides a = 6 and b = 8, then its perimeter is p = 2(6 + 8) = 28, and the diagonal is d = √(6² + 8²) = 10.

Formula for Perimeter and Diagonal - 𝑃 = 2(𝑎 + 𝑏), 𝑑 = √(𝑎² + 𝑏²)

Let’s consider another example of a rectangle with side lengths 𝑎 = 8 and 𝑏 = 10.

The perimeter of the rectangle will be 𝑃 = 2(𝑎 + 𝑏) = 2(8 + 10) = 36.

The diagonal length will be 𝑑 = √(8² + 10²) = √(64 + 100) = √164, approximately 12.81.

How to Calculate the Perimeter Including Diagonals

To find the perimeter including the diagonals of a rectangle, first calculate the perimeter of the rectangle using 𝑃 = 2(𝑎 + 𝑏). Then, add the length of the diagonal calculated using 𝑑 = √(𝑎² + 𝑏²).

For example, if a rectangle has sides of a = 6, b = 6, and you wish to include the diagonals, first find the perimeter = 2(6 + 6) = 24. Then, calculate the diagonal using 𝑑 = √(6² + 6²) = √72 ≈ 8.49.

Example Problem on Perimeter including Diagonals 

For finding the perimeter including the diagonal, use the formulas, 𝑃 = 2(𝑎 + 𝑏) and 𝑑 = √(𝑎² + 𝑏²). For example, let’s say, a = 5 cm, b = 4 cm.

Now, the perimeter = 2(5 + 4) = 18 cm, and the diagonal = √(5² + 4²) = √41 ≈ 6.4 cm.

Therefore, the total distance including the diagonal is 18 + 6.4 ≈ 24.4 cm.

Explore Our Programs

Tips and Tricks for Calculating Perimeter Including Diagonals

Learning some tips and tricks makes it easier for children to calculate the perimeter including diagonals. Here are some tips and tricks given below:

  • Always remember that the perimeter of a rectangle is simply 2 times the sum of the lengths of its two sides. For that, use the formula, 𝑃 = 2(𝑎 + 𝑏).
  • Calculating the diagonal involves using the Pythagorean theorem where the diagonal acts as the hypotenuse. The formula is: 𝑑 = √(𝑎² + 𝑏²).
  • To reduce confusion, specifically arrange the indicated side lengths if you need the perimeter of a group of rectangles. After that, apply the formula to each shape.
  • To avoid mistakes when adding the perimeter and diagonals, make sure the side lengths and diagonal lengths are precise for common uses like architecture and design. If you are given the diagonal and one side, you can use the Pythagorean theorem to find the other side.

Common Mistakes and How to Avoid Them in Calculating Perimeter Including Diagonals

Did you know that while working with the perimeter including diagonals, children might encounter some errors or difficulties? We have many solutions to resolve these problems. Here are some given below:

Problem 1

A rectangular-shaped garden has a perimeter of 48 meters and a diagonal measuring 17 meters. If the length of one side is 14 meters, what is the length of the other side?

Okay, lets begin

Length of the other side = 10 meters.

Explanation

Let ‘b’ be the length of the other side. And the given perimeter = 48 meters. One side length = 14 meters.

Perimeter of rectangle = 2(14 + b) = 48 28 + 2b = 48 2b = 48 - 28 = 20 b = 10

Therefore, the length of the other side is 10 meters.

Well explained 👍

Problem 2

A wire with a total length of 297 cm is reshaped into a square. Find the length of each side of the square and verify its diagonal.

Okay, lets begin

74.25 cm

Explanation

Given that the total length of the wire is reshaped into a square, here is the solution:

Perimeter of a square = 4 × a 297 = 4 × a 297 ÷ 4 = 74.25 a = 74.25

The diagonal of the square is 𝑑 = a√2 = 74.25√2 ≈ 105.03 cm.

Therefore, each side of the square is 74.25 cm, and the diagonal is approximately 105.03 cm.

Well explained 👍

Problem 3

Find the perimeter of a square with sides of 10 cm, and calculate the diagonal.

Okay, lets begin

40 cm

Explanation

Perimeter of square = 4 × a P = 4 × 10 = 40 cm

The diagonal is 𝑑 = 10√2 ≈ 14.14 cm.

Therefore, the perimeter is 40 cm, and the diagonal length is approximately 14.14 cm.

Well explained 👍

Problem 4

Annie is designing a rectangular floor plan for her new room. The sides of the room measure: Side A = 12 meters Side B = 9 meters How long is the diagonal of the room?

Okay, lets begin

The diagonal is 15 meters.

Explanation

The diagonal is calculated using the formula: 𝑑 = √(𝑎² + 𝑏²) 𝑑 = √(12² + 9²) = √(144 + 81) = √225 = 15 meters

Well explained 👍

Problem 5

Calculate the perimeter including the diagonals of a rectangular book cover with sides a = 10 cm and b = 8 cm.

Okay, lets begin

Perimeter = 36 cm, Diagonal = 12.81 cm

Explanation

The perimeter of the rectangle is calculated as: Perimeter = 2(a + b) = 2(10 + 8) = 36 cm

The diagonal is calculated as: Diagonal = √(10² + 8²) = √(100 + 64) = √164 ≈ 12.81 cm

Well explained 👍

FAQs on Perimeter Including Diagonals

1.Evaluate the perimeter of a rectangle if its sides are 3 cm and 4 cm, and calculate its diagonal.

Perimeter of rectangle = 2(a + b), Hence P = 2(3 + 4) = 14 cm. Diagonal, d = √(3² + 4²) = √9 + 16 = √25 = 5 cm.

2.What is meant by the perimeter including diagonals?

It refers to the total length around a rectangle or square, including the measurement of its diagonals.

3.What are the types of polygons related to diagonals?

Polygons related to diagonals in this context are primarily rectangles and squares, as these shapes have diagonal lines connecting opposite corners.

4.Which shape has equal diagonals?

A square has equal diagonals. All sides and angles are equal, making the diagonals equal as well.

5.What is the formula for finding a rectangle's diagonal?

The formula for finding the diagonal of a rectangle is 𝑑 = √(𝑎² + 𝑏²), where a and b are the lengths of the rectangle's sides.

Important Glossaries for Perimeter Including Diagonals

  • Perimeter: The total length of the sides of a shape.
  • Diagonal: A line connecting two non-adjacent vertices in a polygon.
  • Rectangle: A polygon with four sides and four right angles.
  • Square: A polygon with four equal sides and four right angles.
  • Pythagorean Theorem: A mathematical formula used to find the length of the sides in a right-angled triangle, often used to calculate diagonals.

What Is Measurement? 📏 | Easy Tricks, Units & 🎯 Fun Learning for Kids | ✨BrightCHAMPS Math

▶

Seyed Ali Fathima S

About the Author

Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.

Fun Fact

: She has songs for each table which helps her to remember the tables