Diagonal of a Rectangle

Statement

The diagonal of a rectangle can be calculated using Pythagoras’ theorem. If a rectangle has length \( \ell \) and width \( w \), then the diagonal \( d \) is given by:

\[ d = \sqrt{\ell^2 + w^2} \]

This comes from treating the diagonal as the hypotenuse of a right-angled triangle with sides equal to the rectangle’s length and width.

Why it’s true

• In a rectangle, opposite sides are equal, and each angle is 90°.
• If you draw a diagonal, it forms a right-angled triangle with the length and width.
• By Pythagoras’ theorem, the square of the diagonal equals the sum of the squares of length and width.

Recipe (how to use it)

1. Identify the length and width of the rectangle.
2. Square both values.
3. Add the squares together.
4. Take the square root to get the diagonal.

Spotting it

This formula is used whenever a diagonal across a rectangle or square is required, especially in geometry, coordinate problems, and construction contexts.

Common pairings

• Area of a rectangle \( A = \ell \times w \).
• Perimeter of a rectangle \( P = 2(\ell + w) \).
• Applying trigonometry (using diagonal with sine, cosine, or tangent).

Mini examples

1. Given: \( \ell = 3, w = 4 \). Find: diagonal. Answer: \( d = \sqrt{3^2 + 4^2} = 5 \).
2. Given: \( \ell = 6, w = 8 \). Find: diagonal. Answer: \( d = \sqrt{36 + 64} = 10 \).

Pitfalls

• Forgetting to square both sides before adding.
• Taking the square root too early.
• Mixing up perimeter or area with the diagonal.
• Leaving answers in root form when decimals are required.

Exam strategy

• Label the rectangle clearly with length, width, and diagonal.
• Apply Pythagoras carefully: diagonal squared = length squared + width squared.
• Check whether the question wants exact form (square root) or decimal approximation.

Summary

The diagonal of a rectangle is always found using Pythagoras’ theorem. By squaring the length and width, adding them, and taking the square root, we find the diagonal quickly. This formula connects geometry with algebra and is often tested in GCSE exams to check both understanding and calculation skills.