ACT Math › Cones
The slant height of a cone is ; the diameter of its base is one-fifth its slant height. Give the surface area of the cone in terms of
.
The formula for the surface area of a cone with base of radius and slant height
is
.
The diameter of the base is ; the radius is half this, so
Substitute in the surface area formula:
The slant height of a cone is ; the diameter of its base is one-fifth its slant height. Give the surface area of the cone in terms of
.
The formula for the surface area of a cone with base of radius and slant height
is
.
The diameter of the base is ; the radius is half this, so
Substitute in the surface area formula:
You have two cones, one with a diameter of and a height of
and another with a diameter of
and a height of
. If you fill the smaller cone with sand and dump that sand into the larger cone, how much empty space will be left in the larger cone?
1. Find the volume of each cone:
Cone 1:
Since the diameter is , the radius is
.
Cone 2:
Since the diameter is , the radius is
.
2. Subtract the smaller volume from the larger volume:
You have an empty cone and a cylinder filled with water. The cone has a diameter of and a height of
. The cylinder has a diameter of
and a height of
. If you dump the water from the cylinder into the cone until it is filled, what volume of water will remain in the cylinder?
1. Find the volumes of the cone and cylinder:
Cone:
Since the diameter is , the radius is
.
Cylinder:
Since the diameter is , the radius is
.
2. Subtract the cone's volume from the cylinder's volume:
You have two cones, one with a diameter of and a height of
and another with a diameter of
and a height of
. If you fill the smaller cone with sand and dump that sand into the larger cone, how much empty space will be left in the larger cone?
1. Find the volume of each cone:
Cone 1:
Since the diameter is , the radius is
.
Cone 2:
Since the diameter is , the radius is
.
2. Subtract the smaller volume from the larger volume:
You have an empty cone and a cylinder filled with water. The cone has a diameter of and a height of
. The cylinder has a diameter of
and a height of
. If you dump the water from the cylinder into the cone until it is filled, what volume of water will remain in the cylinder?
1. Find the volumes of the cone and cylinder:
Cone:
Since the diameter is , the radius is
.
Cylinder:
Since the diameter is , the radius is
.
2. Subtract the cone's volume from the cylinder's volume:
Use the following formula to answer the question.
The slant height of a right circular cone is . The radius is
, and the height is
. Determine the surface area of the cone.
Notice that the height of the cone is not needed to answer this question and is simply extraneous information. We are told that the radius is , and the slant height is
.
First plug these numbers into the equation provided.
Then simplify by combining like terms.
Use the following formula to answer the question.
The slant height of a right circular cone is . The radius is
, and the height is
. Determine the surface area of the cone.
Notice that the height of the cone is not needed to answer this question and is simply extraneous information. We are told that the radius is , and the slant height is
.
First plug these numbers into the equation provided.
Then simplify by combining like terms.
What is the volume of a cone with a radius of and a height of
? Leave your answer in terms of
, reduce all fractions.
To find the volume of a cone with radius , and height
use the formula:
.
We plug in our given radius and height to find:
What is the volume of a cone with a radius of 3 mm and a height of 6 mm?
The formula for the volume of a cone is given by the equation:
.
Pluggin in our values we get: