Answer: The area of the hexagon is 261 square centimeters
To calculate the area of the of regular hexagon, we need to find the apothem first
Step 1: find the apothem
To find the apothem, we will be applying the pythagora theorem
The above triangle can be split into two
The hypotenus of the triangle is the same as the side of the hexagon
[tex]\begin{gathered} \text{Pythagora's theorem} \\ \text{Hypoyenus}^2=opposite^2+adjacent^2 \\ \text{Hypotenus = c, opposite = a, and adjacent = b} \\ 10^2=5^2+a^2 \\ 100=25+a^2 \\ \text{Collect the like terms} \\ a^2\text{ = 100 - 25} \\ a^2\text{ = 75} \\ \text{Take the square root of both sides} \\ \text{a = }\sqrt[]{75} \\ \text{a = }8.7cm \end{gathered}[/tex]
apothem = s / 2 tan (180/n)
let s = length of the side
n = number of sides
[tex]\begin{gathered} \text{apothem = }\frac{s}{2\text{ tan (}\frac{180}{n})} \\ s\text{ = 10, n = 6} \\ \text{apothem = }\frac{10}{2\text{ tan (180/6)}} \\ \text{apothem = }\frac{10}{2\tan (30)} \\ \text{Apothem = }\frac{10}{2\text{ x 0.577}} \\ \text{apothem = }\frac{10}{1.54} \\ \text{Apothem = }8.7\operatorname{cm} \end{gathered}[/tex]
Step 2: find the area using the calculated apothem
Area = 1/2 x perimeter x apothem
Perimeter = 6a
a = 10cm
Perimeter = 6 x 10
Perimeter = 60cm
Area = 1/2 x 60 x 8.7
Area = 30 x 8.7
Area = 261 square centimeters
Therefore, the area of the hexagon is 261 square cm
