For example, you might have a hexagon with a side length of 8 cm. The base of each equilateral triangle, then, is also 8 cm.
For example, if the base of the equilateral triangle is 8 cm, when you divide the triangle into two right triangles, each right triangle now has a base of 4 cm.
For example, if a right triangle had a hypotenuse of 2{\displaystyle 2} inches, one leg of 1{\displaystyle 1} inch, and another leg of about 1. 732{\displaystyle 1. 732} inches (3{\displaystyle {\sqrt {3}}}), the Pythagorean Theorem would state that 12+32=22{\displaystyle 1^{2}+{\sqrt {3}}^{2}=2^{2}}, which is true when you complete the calculations: 1+3=4{\displaystyle 1+3=4}.
For example, if the length of the base is 4 cm, your formula will look like this: a2+42=c2{\displaystyle a^{2}+4^{2}=c^{2}}.
For example, if the side length of the hexagon is 8 cm, then the length of the right triangle’s hypotenuse is also 8 cm. So your formula will look like this: a2+42=82{\displaystyle a^{2}+4^{2}=8^{2}}.
For example, squaring the known values, your formula will look like this: a2+16=64{\displaystyle a^{2}+16=64}.
For example:a2+16−16=64−16{\displaystyle a^{2}+16-16=64-16}a2=48{\displaystyle a^{2}=48}
For example, using a calculator, you can calculate 48=6. 93{\displaystyle {\sqrt {48}}=6. 93}. Thus, the missing length of the right triangle, and the length of the hexagon’s apothem, equals 6. 93 cm.
For example, for a hexagon with a side length of 8 cm, the formula will look like this: 82tan(180n){\displaystyle {\frac {8}{2\tan({\frac {180}{n}})}}}.
For example: 82tan(1806){\displaystyle {\frac {8}{2\tan({\frac {180}{6}})}}}.
For example, 1806=30{\displaystyle {\frac {180}{6}}=30}, so the formula now look like this: 82tan(30){\displaystyle {\frac {8}{2\tan(30)}}}.
For example, the tangent of 30 is about . 577, so the formula will now look like this: 82(. 577){\displaystyle {\frac {8}{2(. 577)}}}.
For example:apothem=82(. 577){\displaystyle {\text{apothem}}={\frac {8}{2(. 577)}}}apothem=81. 154{\displaystyle {\text{apothem}}={\frac {8}{1. 154}}}apothem=6. 93{\displaystyle {\text{apothem}}=6. 93}So, the apothem of a regular hexagon with 8-cm sides is about 6. 93 cm.
