How to Find the Length of a Diagonal Line Through a Rectangle

You can find the length of a diagonal line through a rectangle by applying the Pythagorean theorem. This fundamental geometric principle is used because a diagonal divides a rectangle into two right-angled triangles.

Understanding the Concept

A rectangle has four right angles. When you draw a diagonal across a rectangle, it forms the hypotenuse of two right-angled triangles. The sides of the rectangle (its width and height) become the legs of these triangles.

The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides (legs).

Steps to Calculate the Diagonal Length

To calculate the length of a rectangle's diagonal, you need its width and height.

1. Identify the Width (w) and Height (h): Measure or identify the given dimensions of the rectangle.
2. Square Each Dimension: Calculate the square of the width ($w^2$) and the square of the height ($h^2$).
3. Sum the Squares: Add the squared width and squared height together ($w^2 + h^2$).
4. Take the Square Root: The diagonal equals the square root of the width squared plus the height squared. This final step gives you the length of the diagonal ($d$).

The formula for calculating the diagonal ($d$) is:

$d = \sqrt{w^2 + h^2}$

For a deeper dive into the Pythagorean theorem, you can explore resources like Wikipedia's article on the Pythagorean theorem.

Practical Example

Let's say you have a rectangular piece of land and you need to find the distance across it diagonally.

Given:

• Width = 8 feet
• Height = 6 feet

Calculation:

1. Square the width: $8^2 = 64$
2. Square the height: $6^2 = 36$
3. Sum the squares: $64 + 36 = 100$
4. Take the square root: $\sqrt{100} = 10$

Therefore, the length of the diagonal is 10 feet.

Example Table

Here's another example illustrating the calculation:

This method is universally applicable for finding the diagonal length of any rectangle, provided you know its width and height.