Which is greater in volume, a right
cone having a base radius of 25 centimeters and a height of 56 centimeters or a
right square-based pyramid having a base with perimeter of 176 centimeters and a
height of 48 centimeters?
Before starting this question, it
is worth recalling that the volume of a cone and the volume of a pyramid have very
similar formulas. The volume of both of these shapes
is equal to a third of the base area multiplied by the height. As we are dealing with a right cone
and a right pyramid, the height will be the perpendicular distance from the apex to
the center of the base. When dealing with a cone, the
volume is equal to one-third 𝜋𝑟 squared multiplied by ℎ. This is because the base area of a
cone is circular, and the area of a circle is equal to 𝜋𝑟 squared.
When dealing with a square-based
pyramid, the volume is equal to one-third of 𝑙 squared multiplied by ℎ, where 𝑙 is
the length of each side of the square base. In this question, the cone has a
base radius of 25 centimeters and a height of 56 centimeters. The volume 𝑉 is therefore equal to
one-third multiplied by 𝜋 multiplied by 25 squared multiplied by 56. Typing this into the calculator
gives us 36651.91 and so on. To the nearest cubic centimeter,
the volume of the cone is 36652 cubic centimeters.
Before we can calculate the volume
of the pyramid, we need to work out the length of each side of the square base. We know that the perimeter of the
base is equal to 176 centimeters. The side length 𝑙 is therefore
equal to 176 divided by four. This is equal to 44
centimeters. The volume of the pyramid is
therefore equal to one-third multiplied by 44 squared multiplied by 48, as 48 is the
height of the square-based pyramid. Clearing some space and typing this
into the calculator gives us 30976. The volume of the square-based
pyramids is 30976 cubic centimeters. As this value is less than the
volume of the cone, 36652 cubic centimeters, the shape that has the greater volume
is the cone.