How to Calculate the Perimeter of a Square Given Its Diagonal

Understanding the relationship between the diagonal and perimeter of a square is a fundamental concept in geometry. This article will guide you through the process of finding the perimeter of a square when given its diagonal length. We will use the Pythagorean theorem and a straightforward algebraic method to find the solution.

Understanding the Geometry of a Square

A square is a polygon with four equal sides and four right angles. The relationship between the side length, s, and the diagonal, d, of a square can be expressed through the Pythagorean theorem. In a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In this case, the diagonal of the square acts as the hypotenuse, and the two sides of the square are the other two sides of the right-angled triangle.

Using the Pythagorean Theorem

Given the diagonal of a square is 65 cm, we can set up the following equation using the Pythagorean theorem:

d s√2

Substituting the given value of the diagonal:

65 s√2

Now, solve for s by dividing both sides by √2:

s 65/√2

Thus, the side length of the square is approximately 45.96 cm.

Calculating the Perimeter

The perimeter of a square is the sum of the lengths of all four sides. Since all sides of a square are equal, the perimeter can be calculated as:

P 4s

Substituting the value of s that we found:

P 4 × (65/√2)

By simplifying, we get:

P 4 × 45.962

P ≈ 183.85 cm

Therefore, the perimeter of the square is approximately 183.85 cm.

Alternative Methods

There are alternative methods to solve this problem, as demonstrated by other solutions. One such method is directly applying the Pythagorean theorem to find the side length:

x2 x2 652

2x2 4225

x2 2112.5

x √2112.5 ≈ 45.962 cm

Then, the perimeter is:

P 4x 4 × 45.962 ≈ 183.85 cm

This confirms our earlier result.

Summary

Given the diagonal of a square, the side length can be found using the Pythagorean theorem. Once the side length is known, the perimeter of the square can be calculated by multiplying the side length by 4. This approach provides a clear and systematic way to solve problems involving the relationships between the diagonal and perimeter of a square.

If you have any further questions or need additional assistance with similar problems, feel free to ask!