diagonal of a rectangle — Definition, Formula & Examples
The diagonal of a rectangle is a line segment that connects one corner to the opposite corner, cutting across the interior. Every rectangle has two diagonals, and they are always equal in length.
In a rectangle with length and width , a diagonal is a segment joining two non-adjacent vertices. By the Pythagorean theorem, each diagonal has length . The two diagonals bisect each other—they cross at their shared midpoint—but unlike a square, they do not generally intersect at right angles.
Key Formula
Where:
- = Length of the diagonal
- = Length of the rectangle
- = Width of the rectangle
How It Works
A diagonal splits the rectangle into two congruent right triangles. The length and width of the rectangle become the two legs, and the diagonal becomes the hypotenuse. That is why the Pythagorean theorem gives you the diagonal directly. If you know both side lengths, substitute them into the formula, square each, add the results, and take the square root.
Worked Example
Problem: Find the diagonal of a rectangle with length 12 cm and width 5 cm.
Apply the formula: Substitute the given values into the diagonal formula.
Square each side: Compute the squares of the length and width.
Take the square root: Evaluate the square root to find the diagonal.
Answer: The diagonal is 13 cm.
Why It Matters
Screen sizes for TVs, monitors, and phones are measured along the diagonal, so this formula tells you the actual screen dimension from the width and height. In construction and woodworking, measuring the diagonal of a rectangular frame is the standard way to check that corners are square.
Common Mistakes
Mistake: Adding the length and width instead of using the Pythagorean theorem (writing ).
Correction: The diagonal is the hypotenuse of a right triangle, not a sum of sides. You must square each dimension, add, then take the square root: .