PLAIN AND SOLID GEOMETRY

Question

A right circular cone has altitude 18 cm and radius of base 5 cm. inscribe in it is a square pyramid.

a. Compare the volume of the cone with the volume of the pyramid.

b. Compare the lateral area of the cone with that of pyramid

Answer 1

Since square pyramid is inscribed in a cone, we can say that the cone is larger in every way.

(a) First, calculate the volume of cone.

Formula is: V = (1/3)Bh

V_c = (1/3)•(π•5²)•18

V_c = 150π cm³ ≈ 471.24 cm³

second, the volume of square pyramid:

Formula is: V = (1/3)Bh

Using the 45°-45°-90° triangle to compute the side of a square if A_square = s²

s =5•2^1/2

∴A_square = (5•2^1/2)² =50 cm²

∴ V_sp = (1/3)(50)(18) = 300 cm³

(b)The formula for Lateral Area of a cone is:

LA_cone = π•r•(r²+h²)^1/2

Where r = radius fo the base

h = height fo the prism

LA_cone = π•5•(5²+18²)^1/2

∴LA_cone ≈ 93.41π ≈ 293.45 cm²

The formula for Lateral Area of square pyramid is:

LA_sp = 2s(a²+h²)^1/2

s = measure of one side of the square

a = apothem of the square base (It is the distance from the center to one of its side)

h = height of the prism.

LA_sp = 2(5•2^1/2)((2.5•2^1/2)²+18²)^1/2

LA_sp ≈ 259.42 cm²