Question

A pyramid has a regular hexagonal base with the side lengths of 4 and a slant height of 6. Find the total area of the pyramid..

Answer 1

for get the length of apothem write sin60 = a/4 so sqrt3 /2 = a/4 so a = 4sqrt3 /2

a = 2sqrt3

so area of base = 6(4*2sqrt3)/2 = 6*4sqrt3 = 24sqrt3 unit squared

the lateral area = 4*4*6/2 = 4*2*6 = 48 unit squared

total area = 24sqrt3 +48 = 24(sqrt3 +2) unit squared

hope this will help you

Answer 2

113.57 unit²

Explanation:

Surface area of a pyramid with regular hexagonal base

= Area of slant sides + area of Hexagonal base

Area of one slant side =
`(1)/(2)` (side of base) × slant height

=
`(1)/(2)` × 6 × 4

= 3 × 4

= 12 unit²

Since hexagonal pyramid has 6 slant sides.

So area of sic slant sides = 6 × 12 = 72 unit²

Now for the area of hexagonal base we will take triangle ABC.

∠BAC = 60° [angle formed at center =
`\frac{360}{\text{number of sides}}` ]

and ∠CAD = 30°

Now tan 30° =
`(DC)/(AD)` =
`(2)/(AD)`

`(1)/(√(3))=(2)/(AD)`

AD =
`2√(3)`

Now hexagonal base area = 6 × [
`(1)/(2)` (BC)(AD)]

6 × [
`(4)/(2)` ×
`2√(3)`] =
`24√(3)`

Therefore area of the pyramid = 72 + 41.57 = 113.57 unit²