# Common Core: 8th Grade Math : Know and Use the Formulas for the Volumes of Cones, Cylinders, and Spheres: CCSS.Math.Content.8.G.C.9

## Example Questions

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### Example Question #1 : How To Find The Volume Of A Cone

Chestnut wood has a density of about . A right circular cone made out of chestnut wood has a height of three meters, and a base with a radius of two meters. What is its mass in kilograms (nearest whole kilogram)?

Explanation:

First, convert the dimensions to cubic centimeters by multiplying by : the cone has height , and its base has radius .

Its volume is found by using the formula and the converted height and radius.

Now multiply this by  to get the mass.

Finally, convert the answer to kilograms.

### Example Question #2 : How To Find The Volume Of A Cone

A cone has the height of 4 meters and the circular base area of 4 square meters. If we want to fill out the cone with water (density = ), what is the mass of required water (nearest whole kilogram)?

6333

Explanation:

The volume of a cone is:

where is the radius of the circular base, and is the height (the perpendicular distance from the base to the vertex).

As the circular base area is , so we can rewrite the volume formula as follows:

where  is the circular base area and known in this problem. So we can write:

We know that density is defined as mass per unit volume or:

Where is the density; is the mass and is the volume. So we get:

### Example Question #3 : How To Find The Volume Of A Cone

The vertical height (or altitude) of a right cone is . The radius of the circular base of the cone is . Find the volume of the cone in terms of .

Explanation:

The volume of a cone is:

where is the radius of the circular base, and is the height (the perpendicular distance from the base to the vertex).

### Example Question #1 : Know And Use The Formulas For The Volumes Of Cones, Cylinders, And Spheres: Ccss.Math.Content.8.G.C.9

A right cone has a volume of , a height of and a radius of the circular base of . Find .

Explanation:

The volume of a cone is given by:

where is the radius of the circular base, and is the height; the perpendicular distance from the base to the vertex. Substitute the known values in the formula:

### Example Question #5 : How To Find The Volume Of A Cone

A cone has a diameter of  and a height of . In cubic meters, what is the volume of this cone?

Explanation:

First, divide the diameter in half to find the radius.

Now, use the formula to find the volume of the cone.

### Example Question #2 : Know And Use The Formulas For The Volumes Of Cones, Cylinders, And Spheres: Ccss.Math.Content.8.G.C.9

The height of a cylinder is 3 inches and the radius of the circular end of the cylinder is 3 inches. Give the volume and surface area of the cylinder.

Explanation:

The volume of a cylinder is found by multiplying the area of one end of the cylinder (base) by its height or:

where is the radius of the circular end of the cylinder and is the height of the cylinder. So we can write:

The surface area of the cylinder is given by:

where is the surface area of the cylinder, is the radius of the cylinder and is the height of the cylinder. So we can write:

### Example Question #3 : Know And Use The Formulas For The Volumes Of Cones, Cylinders, And Spheres: Ccss.Math.Content.8.G.C.9

The height of a cylinder is two times the length of the radius of the circular end of a cylinder. If the volume of the cylinder is , what is the height of the cylinder?

Explanation:

The volume of a cylinder is:

where is the radius of the circular end of the cylinder and is the height of the cylinder.

Since , we can substitute that into the volume formula. So we can write:

So we get:

### Example Question #4 : Know And Use The Formulas For The Volumes Of Cones, Cylinders, And Spheres: Ccss.Math.Content.8.G.C.9

The end (base) of a cylinder has an area of square inches. If the height of the cylinder is half of the radius of the base of the cylinder, give the volume of the cylinder.

Explanation:

The area of the end (base) of a cylinder is , so we can write:

The height of the cylinder is half of the radius of the base of the cylinder, that means:

The volume of a cylinder is found by multiplying the area of one end of the cylinder (base) by its height:

or

### Example Question #2 : Know And Use The Formulas For The Volumes Of Cones, Cylinders, And Spheres: Ccss.Math.Content.8.G.C.9

We have two right cylinders. The radius of the base Cylinder 1 is  times more than that of Cylinder 2, and the height of Cylinder 2 is 4 times more than the height of Cylinder 1. The volume of Cylinder 1 is what fraction of the volume of Cylinder 2?

Explanation:

The volume of a cylinder is:

where is the volume of the cylinder,   is the radius of the circular end of the cylinder, and is the height of the cylinder.

So we can write:

and

Now we can summarize the given information:

Now substitute them in the formula:

### Example Question #6 : Know And Use The Formulas For The Volumes Of Cones, Cylinders, And Spheres: Ccss.Math.Content.8.G.C.9

Two right cylinders have the same height. The radius of the base of the first cylinder is two times more than that of the second cylinder. Compare the volume of the two cylinders.

Explanation:

The volume of a cylinder is:

where is the radius of the circular end of the cylinder and is the height of the cylinder. So we can write:

We know that

and

.

So we can write:

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