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The most common formula for the area of a square is simple: it's the length of the side squared, or s^{2}.^{[1] X Research source } 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
Part 1
Part 1 of 2:Finding the Area from the Diagonal

1Draw your square. A square has four equal sides.^{[2] X Research source } Let's say each one has a length of "s".

2Review 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 = s^{2}. This will be useful later on.Advertisement

3Join any two opposite corners to make a diagonal. Let the measure of this diagonal be d units. This diagonal divides the square into two righttriangles.

4Apply the Pythagorean Theorem to one of the triangles. The Pythagorean theorem^{[3] X Research source } is a formula for finding the hypotenuse (longest side) of a right triangle: (side one)^{2} + (side two)^{2} = (hypotenuse)^{2}, or . 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.

5Arrange the equation so s^{2} is on one side. Remember that we already know the square's area is equal to s^{2}. If you can get s^{2} alone on side, you'll have a new equation for area:
 Simplify:
 Divide both sides by two:
 Area =
 Area =

6Use this formula on an example square. These steps have proven that the formula Area = 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 =
=
= 50 square centimeters.
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Part 2
Part 2 of 2:Additional Info

1Find the diagonal from the length of a side.^{[4] X Research source } The Pythagorean theorem for a square with side s and diagonal d gives you the formula . Solve for d if you know the side lengths and want to find the length of the diagonal:

 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.


2Find the side length from the diagonal. If you are given the diagonal and you know that the diagonal of a square is , you can divide both sides by to get .
 For example, a square with a diagonal of 10cm has sides with length 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 square centimeters. This is a bit less accurate, since is an irrational number that can lead to rounding errors.

3Interpret the area formula. The math checks out for the formula Area = , but is there a way to test this directly? Well, is the area of a second square with the diagonal as a side. Since the full formula is , 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.
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Community Q&A

QuestionA square has side lengths of 26.2 m, 21.4m, 27m, and 24.3m. How can I get the diagonal of square?Community AnswerThis shape is not a square, since it does not have equal sides. It is an irregular quadrilateral, either a trapezoid or an irregular shape with no name. Draw a right triangle inside the shape with the diagonal as the hypotenuse. If you have enough information to calculate the height and base of this right triangle (usually with trigonometry), you can use Pythagoras' Theorem to find the diagonal. The two diagonals will be different lengths, since the shape is not symmetrical.

QuestionWhat is the area of a school lawn with a width of 69 meters?DonaganTop AnswererAssuming it is a square or rectangle, you have to know the other dimension, too.

QuestionHow did I find the length of side of each square?LyKaxandra CaimoyCommunity AnswerSuppose that the diagonal is 8. Square 8, so you will get 64. Now divide 64 by 2, you will get 32. Get the square root of 32. That is the length of the side of a square. The square root of 32 is equal to 4 square root of 2.

QuestionOne side of the square is (2x+5) meters long. How do I find the area of the square in terms of x?Community AnswerThis problem is trying to get you used to working with algebraic terms. Although it looks strange at first, you can substitute (2x + 5) for the side length in any formula. In this case, start with the formula A = s x s. Substitute (2x + 5) for each s, then simplify as much as you can.

QuestionHow do I find the diagonal of a square with different side lengths?Community AnswerA square always has equal lengths. If your shape has four right angles, it is a rectangle. You can use Pythagoras' theorem to find the diagonal from the length and width of the rectangle: d^2 = l^2 + w^2.

QuestionWhat is the length of a square if one side of it is 12 feet?Community AnswerThe length is 12 ft if one of the sides is 12 ft. The length of the diagonal can be found using the Pythagorean Theorem (a^2+b^2=c^2). In this formula, a and b are the sides of the right triangle, and c is the long side or the hypotenuse. The diagonal (c) would be found with the equation 12^2+12^2=c^2. 144+144=c^2, 288=c^2, c=sqrt(288)=approximately 16.97. Therefore, the diagonal of a square that is 12 feet would be around 17 feet.

QuestionHow do I find the area of a circle inside a square?Community AnswerIf the circle's diameter fits perfectly across the square, it must be equal to the side's square length. Divide this by two to get the circle's radius, then use the standard area formula for circles: Area = pi * r^2.

QuestionHow do I determine the length of a side of a square given the area?Community AnswerStart with the area formula A = s^2 and solve for s. (Hint: a square root will cancel out the exponent.)

QuestionWhat is the area of a floor that is 10 1/2 feet long on each side?DanoyachtcaptTop AnswererArea = length X width 10.5 X 10.5 = 110.25 sq ft.

QuestionHow do I find the surface area of a square?Community AnswerA square is a 2D shape, so therefore does not have a surface area. It only has an area, which is calculated by multiplying two sides together, or squaring one. as a square has equal edges.
Video
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.Thanks!
 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 Category:Microsoft Excel Imagery, Category:Mathematics, Category:Spreadsheets or Category:Graphics.Thanks!
 If you don't have a calculator and you need a more precise estimate for the square root of 2, there are ways to estimate it by hand. The NewtonRaphson method is one example.^{[5] X Research source }Thanks!
References
About This Article
To find the area of a square using the length of its diagonal, use the formula area = d^2 divided by 2, where d is the length of the diagonal. Just square the length of the diagonal and then divide that number by 2 to find the square's area. To learn how to find the length of a square's sides using the diagonal, scroll down!