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. s2+s2=d2{\displaystyle s^{2}+s^{2}=d^{2}}
s2+s2=d2{\displaystyle s^{2}+s^{2}=d^{2}} Simplify: 2s2=d2{\displaystyle 2s^{2}=d^{2}} Divide both sides by two: s2=d22{\displaystyle s^{2}={\frac {d^{2}}{2}}} Area = s2=d22{\displaystyle s^{2}={\frac {d^{2}}{2}}} Area = d22{\displaystyle {\frac {d^{2}}{2}}}
For example, let’s say a square has a diagonal that measures 10 cm. Area = 1022{\displaystyle {\frac {10^{2}}{2}}}= 1002{\displaystyle {\frac {100}{2}}}= 50 square centimeters.
2s2=d2{\displaystyle 2s^{2}=d^{2}}2s2=d2{\displaystyle {\sqrt {2s^{2}}}={\sqrt {d^{2}}}}s2=d{\displaystyle s{\sqrt {2}}=d} 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.
For example, a square with a diagonal of 10cm has sides with length 102=7. 071{\displaystyle {\frac {10}{\sqrt {2}}}=7. 071} 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 =s2=7. 0712=50{\displaystyle =s^{2}=7. 071^{2}=50} square centimeters. This is a bit less accurate, since 2{\displaystyle {\sqrt {2}}} is an irrational number that can lead to rounding errors.
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.
