Question

ABCDE is a square-based pyramid.

its base has side length 8cm and centre M.
Angle ECM = 72 degrees

Work out the volume of the pyramid.

Answer 1

371.41 cm^3.

Explanation:

The volume = 1/3 * area of the square base * height.

The area of the square base = 8^2 = 64 cm^2.

To find the height we need to consider the triangle ECM in which CM is the line between center M and C on the Base, and EM is the height of the triangle ( which also = height of the pyramid).

We use Pythagoras to find length of CM.

CM^2 = 4^2 + 4^2 ( because 4 is half of the side length of 8 cm).

CM = √32 cm.

Now tan 72 = height / CM = EM / CM

tan 72 = EM / √32

EM = √32 tan 72

So the volume of the pyramid is 1/3 * 64 * √32 tan 72

= 371.41 cm^3.

Answer 2

371 cm³

Explanation:

Assuming the base is ABCD and tip is at E

CM is half the diagonal

Length of the diagonal 'd'

d² = 8² + 8²

d² = 128

d = 8sqrt(3)

CM = 4sqrt(2)

EMC is a right angle triangle, where EM is the height of the pyramid.

tan(72) = EM/CM

EM = tan(72) × 4sqrt(2) = 17.41

Volume of pyramid;

V = ⅓(base area)×height

V = ⅓(8²)×17.41

V = 371.4134868 cm³

V = 371 cm³ (3sf)