How to Calculate the Diagonal of a Rectangle

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Rectangle Diagonal Calculator

Diagonal Length: units

function calculateDiagonal() { var length = parseFloat(document.getElementById('rectLength').value); var width = parseFloat(document.getElementById('rectWidth').value); var resultDiv = document.getElementById('calcResult'); var output = document.getElementById('diagonalOutput'); var steps = document.getElementById('calculationSteps'); if (isNaN(length) || isNaN(width) || length <= 0 || width <= 0) { alert("Please enter valid positive numbers for both length and width."); resultDiv.style.display = "none"; return; } // Formula: d = sqrt(l^2 + w^2) var squareSum = (length * length) + (width * width); var diagonal = Math.sqrt(squareSum); output.innerHTML = diagonal.toLocaleString(undefined, {minimumFractionDigits: 2, maximumFractionDigits: 4}); steps.innerHTML = "Calculation: √(" + length + "² + " + width + "²) = √(" + (length * length) + " + " + (width * width) + ") = √" + squareSum; resultDiv.style.display = "block"; }

How to Calculate the Diagonal of a Rectangle

Calculating the diagonal of a rectangle is a fundamental geometric task used in various fields, from construction and interior design to computer science and television manufacturing. By definition, the diagonal is a straight line connecting two opposite corners of a rectangle.

The Mathematical Formula

Because a rectangle is composed of two right-angled triangles joined at the diagonal, we use the Pythagorean Theorem to find the diagonal length. The formula is:

d = √(l² + w²)

  • d = Diagonal length
  • l = Length of the rectangle
  • w = Width of the rectangle

Step-by-Step Calculation Guide

To find the diagonal manually, follow these three simple steps:

  1. Square the dimensions: Multiply the length by itself and the width by itself.
  2. Sum the results: Add the two squared numbers together.
  3. Find the square root: Calculate the square root of that sum to find the final diagonal distance.

Practical Example

Suppose you have a rectangular room that is 12 feet long and 9 feet wide. You want to know the distance from one corner to the opposite corner to hang a string of lights.

  • Step 1: Square the values (12² = 144) and (9² = 81).
  • Step 2: Add them together (144 + 81 = 225).
  • Step 3: Find the square root (√225 = 15).

The diagonal of the room is exactly 15 feet.

Common Use Cases

Understanding rectangle diagonals is essential for several real-world applications:

  • TV and Monitor Sizes: Screen sizes are always marketed by their diagonal length, not their width or height.
  • Squaring a Foundation: Builders check if a structure is perfectly rectangular by measuring both diagonals; if they are equal, the corners are exactly 90 degrees.
  • Furniture Fitting: Determining if a large sofa can pivot through a rectangular doorway often involves understanding the diagonal clearance.
  • Art and Framing: Choosing the right frame or canvas size often requires knowing the corner-to-corner distance for structural integrity.

Frequently Asked Questions

Are both diagonals in a rectangle the same?
Yes. In any standard rectangle, both diagonals are congruent (equal in length) and bisect each other.

What if I only have the area?
You cannot find the diagonal with only the area. You need at least one side length (length or width) and the area to determine the other side, then apply the Pythagorean theorem.

Can this formula be used for squares?
Yes! A square is a special rectangle where length equals width. The formula simplifies to d = s × √2, where s is the length of one side.

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