Find the Area of a Square Using the Length of its Diagonal

The most common formula for the area of a square is simple: it's the length of the side squared, or s2. But sometimes you only know the length of the square's diagonal, running between opposite vertices. If you've studied right triangles, you can find a new area formula that uses this diagonal as its only variable.

Steps

Finding the Area from the Diagonal

  1. Draw your square. A square has four equal sides. Let's say each one has a length of "s".
  2. Review the basic formula for a square's area. A square's area is equal to its length times its width. Since each side is s, the formula is Area = s x s = s2. This will be useful later on.
  3. Join any two opposite corners to make a diagonal. Let the measure of this diagonal be d units. This diagonal divides the square into two right-triangles.
  4. Apply the Pythagorean Theorem to one of the triangles. The Pythagorean theorem is a formula for finding the hypotenuse (longest side) of a right triangle: (side one)2 + (side two)2 = (hypotenuse)2, or <math>a^2 + b^2 = c^2</math>. Now that the square is divided in half, you can use this formula on one of the right triangles:
    • The two shorter sides of the triangle are the sides of the square: each one has a length of s.
    • The hypotenuse is the diagonal of the square, d.
    • <math>s^2 + s^2 = d^2</math>
  5. Arrange the equation so s2 is on one side. Remember that we already know the square's area is equal to s2. If you can get s2 alone on side, you'll have a new equation for area:
    • <math>s^2 + s^2 = d^2</math>
    • Simplify: <math>2s^2 = d^2</math>
    • Divide both sides by two: <math>s^2 = \frac{d^2}{2}</math>
    • Area = <math>s^2 = \frac{d^2}{2}</math>
    • Area = <math>\frac{d^2}{2}</math>
  6. Use this formula on an example square. These steps have proven that the formula Area = <math>\frac{d^2}{2}</math> works for all squares. Just plug in the length of the diagonal for d and solve.
    • For example, let's say a square has a diagonal that measures 10 cm.
    • Area = <math>\frac{10^2}{2}</math>
      = <math>\frac{100}{2}</math>
      = 50 square centimeters.

Additional Info

  1. Find the diagonal from the length of a side. The Pythagorean theorem for a square with side s and diagonal d gives you the formula <math>2s^2 = d^2</math>. Solve for d if you know the side lengths and want to find the length of the diagonal:
    • <math>2s^2 = d^2</math>
      <math>\sqrt{2s^2} = \sqrt{d^2}</math>
      <math>s\sqrt{2} = d</math>
    • For example, if a square has sides of 7 inches, its diagonal d = 7√2 inches, or about 9.9 inches.
    • If you don't have a calculator, you can use 1.4 as an estimate for √2.
  2. Find the side length from the diagonal. If you are given the diagonal and you know that the diagonal of a square is <math>s\sqrt{2}</math>, you can divide both sides by <math>\sqrt{2}</math> to get <math>s = \frac{d}{\sqrt{2}}</math>.
    • For example, a square with a diagonal of 10cm has sides with length <math>\frac{10}{\sqrt{2}} = 7.071</math> cm.
    • If you need to find both the side length and the area from the diagonal, you can use this formula first, then quickly square the answer to get the area: Area <math>= s^2 = 7.071^2 = 50</math> square centimeters. This is a bit less accurate, since <math>\sqrt{2}</math> is an irrational number that can lead to rounding errors.
  3. Interpret the area formula. The math checks out for the formula Area = <math>\frac{d^2}{2}</math>, but is there a way to test this directly? Well, <math>d^2</math> is the area of a second square with the diagonal as a side. Since the full formula is <math>\frac{d^2}{2}</math>, you can reason that this second square has exactly twice the area of the original square. You can test this yourself:
    • Draw a square on a piece of paper. Make sure all the sides are equal.
    • Measure the diagonal. Draw a second square using that measurement as the length of the square.
    • Trace a copy of your first square so you have two of them. Cut all three squares out.
    • Cut apart the two smaller squares into any shapes so you can arrange them to fit inside the large square. They should fill the space perfectly, showing that the area of the larger square is exactly twice the area of the smaller square.



Tips

  • This simple equation is used in many fields, including crystallography, chemistry, and art. For example, you can use it to calculate the area of landscape you can see when surveying, or when using perspective in photography or painting, by measuring the distance you've walked and imagining a grid with that distance as the diagonal.
  • If you prefer a more visual approach to math, or want to learn how to use charts and graphs in art, explore the spirallic spin particle path, or browse articles in Microsoft Excel Imagery, Mathematics, Spreadsheets or Graphics.
  • If you don't have a calculator and you need a more precise estimate for the square root of 2, there are ways to Find-a-Square-Root-Without-a-Calculator. The Newton-Raphson method is one example.[1]

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Sources and Citations

  1. ↑ http://www.sosmath.com/calculus/diff/der07/der07.html
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