# A regular hexagon can be divided into six equilateral triangles. If the length of a side of an equilateral triangle is a, the height is sqrt(3)/2a. What is the area of each base in terms of a? What is the surface in terms of a?

Jan 18, 2018

Area of each hexagonal base A_(base) = color(blue)(((3sqrt3)/2)*a^2

Total Surface Area ${A}_{t s a} = \textcolor{p u r p \le}{\left(\frac{9 \sqrt{3}}{2}\right) \cdot {a}^{2}}$

#### Explanation: Area of an equilateral triangle base ${A}_{t} = \left(\frac{1}{2}\right) a \cdot h = \left(\frac{1}{2}\right) \cdot a \cdot \left(\frac{\sqrt{3}}{2}\right) \cdot a = \left(\frac{\sqrt{3}}{4}\right) {a}^{2}$

Base of a hexagon consists of 6 equilateral triangles with side measuring ‘a’

Area of hexagonal base ${A}_{b a s e} = 6 \cdot \left(\frac{\sqrt{3}}{4}\right) \cdot {a}^{2} = \left(\frac{3 \sqrt{3}}{2}\right) {a}^{2}$

Lateral Surface area of the hexagonal prism ${A}_{l s a} = 6 \cdot a \cdot \left(\frac{\sqrt{3}}{2}\right) \cdot a = \left(\frac{3 \sqrt{3}}{2}\right) {a}^{2}$

Total Surface Area of the prism ${A}_{t s a} = \left(2 \cdot {A}_{b a s e}\right) + {A}_{l s a}$

${A}_{t s a} = \left(2 \cdot \left(\frac{3 \sqrt{3}}{2}\right) \cdot {a}^{2}\right) + \left(\frac{3 \sqrt{3}}{2}\right) {a}^{2} = \textcolor{p u r p \le}{\left(\frac{9 \sqrt{3}}{2}\right) \cdot {a}^{2}}$