Question

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`?

Answer 1

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

Total Surface Area `A_(tsa) = color(purple)(( (9sqrt3)/2)*a^2)`

Explanation:

Area of an equilateral triangle base `A_t = (1/2) a * h = (1/2) * a * (sqrt3/2) * a = (sqrt3 / 4) a^2`

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

Area of hexagonal base `A_(base) = 6 * (sqrt3/4) * a^2 = ((3sqrt3)/2) a^2`

Lateral Surface area of the hexagonal prism `A_(lsa) = 6 * a* (sqrt3/2) * a = ((3sqrt3)/2) a^2`

Total Surface Area of the prism `A_(tsa) = (2 * A_(base)) + A_(lsa)`

`A_(tsa) = (2 * ((3sqrt3)/2) * a^2) + ((3sqrt3)/2) a^2 = color(purple)(( (9sqrt3)/2)*a^2)`