How to find the volume of a cone - SSAT Upper Level Quantitative

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Question

A cone has a radius of inches and a height of inches. Find the volume of the cone.

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Answer

The volume of a cone is given by the formula:

$\text{Volume}$=\frac{1}{3}$\pi $r^2h$

Now, plug in the values of the radius and height to find the volume of the given cone.

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Question

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)?

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Answer

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

$3m$\frac{100cm}{1m}$=300cm\ \ \2m$\frac{100cm}{1m}$=200cm$

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

$V=\frac{1}{3}$ \pi $r^{2}$h$

$V=\frac{1}{3}$ \pi $(200cm)^{2}$ ( 300cm )\approx 12,566,371 ; $\textrm{cm}^{3}$$

Now multiply this by to get the mass.

$m=V\rho$

Finally, convert the answer to kilograms.

$5654867g $\frac{1kg}{1,000g}$ \approx 5,655kg$

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Question

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)?

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Answer

The volume of a cone is:

$Volume=\frac{1}{3}$\pi $r^2h$$

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 $\pi $r^2$$ , so we can rewrite the volume formula as follows:

$Volume =\frac{A\times h}{3}$$

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

$Volume =\frac{A\times h}{3}$=\frac{4\times 4}{3}$=\frac{16}{3}$\approx 5.333 $m^3$$

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

$\rho =\frac{m}{v}$$

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

$m=\rho v=1000\times 5.333=5333 Kg$

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Question

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 .

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Answer

The volume of a cone is:

$Volume=\frac{1}$3{}\pi $r^2h$$

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

$Volume=\frac{1}$3{}\pi $r^2h$ =\frac{1}$3{}\pi $t^2$\times 2t=\frac{2}{3}$\pi $t^3$$

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Question

A right cone has a volume of $8\pi$ , a height of and a radius of the circular base of . Find .

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Answer

The volume of a cone is given by:

$Volume=\frac{1}{3}$\pi $r^2h$$

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:

$Volume=\frac{1}{3}$\pi $r^2h$\Rightarrow 8\pi=\frac{1}{3}$\pi\times $(2t)^2$\times 3t$

$\Rightarrow 8\pi=4\pi $t^3$\Rightarrow $t^3$=2\Rightarrow t=\sqrt[3]{2}$

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Question

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

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Answer

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

$$\frac{d}{2}$=r$

$$\frac{12}{2}$=6:m$

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

$$\text{Volume}$=\frac{1}{3}$\pi $r^2$ h$

$\text{Volume}$=\frac{1}{3}$\pi \times $6^2$\times4=\frac{1}{3}$\pi \times $36\times4=\frac{144}{3}$\pi=48\pi:m^3$

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Question

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)?

Tap to reveal answer

Answer

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

$3m$\frac{100cm}{1m}$=300cm\ \ \2m$\frac{100cm}{1m}$=200cm$

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

$V=\frac{1}{3}$ \pi $r^{2}$h$

$V=\frac{1}{3}$ \pi $(200cm)^{2}$ ( 300cm )\approx 12,566,371 ; $\textrm{cm}^{3}$$

Now multiply this by to get the mass.

$m=V\rho$

Finally, convert the answer to kilograms.

$5654867g $\frac{1kg}{1,000g}$ \approx 5,655kg$

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Question

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)?

Tap to reveal answer

Answer

The volume of a cone is:

$Volume=\frac{1}{3}$\pi $r^2h$$

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 $\pi $r^2$$ , so we can rewrite the volume formula as follows:

$Volume =\frac{A\times h}{3}$$

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

$Volume =\frac{A\times h}{3}$=\frac{4\times 4}{3}$=\frac{16}{3}$\approx 5.333 $m^3$$

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

$\rho =\frac{m}{v}$$

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

$m=\rho v=1000\times 5.333=5333 Kg$

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Question

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 .

Tap to reveal answer

Answer

The volume of a cone is:

$Volume=\frac{1}$3{}\pi $r^2h$$

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

$Volume=\frac{1}$3{}\pi $r^2h$ =\frac{1}$3{}\pi $t^2$\times 2t=\frac{2}{3}$\pi $t^3$$

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Question

A right cone has a volume of $8\pi$ , a height of and a radius of the circular base of . Find .

Tap to reveal answer

Answer

The volume of a cone is given by:

$Volume=\frac{1}{3}$\pi $r^2h$$

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:

$Volume=\frac{1}{3}$\pi $r^2h$\Rightarrow 8\pi=\frac{1}{3}$\pi\times $(2t)^2$\times 3t$

$\Rightarrow 8\pi=4\pi $t^3$\Rightarrow $t^3$=2\Rightarrow t=\sqrt[3]{2}$

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Question

A cone has a radius of inches and a height of inches. Find the volume of the cone.

Tap to reveal answer

Answer

The volume of a cone is given by the formula:

$\text{Volume}$=\frac{1}{3}$\pi $r^2h$

Now, plug in the values of the radius and height to find the volume of the given cone.

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Question

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

Tap to reveal answer

Answer

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

$$\frac{d}{2}$=r$

$$\frac{12}{2}$=6:m$

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

$$\text{Volume}$=\frac{1}{3}$\pi $r^2$ h$

$\text{Volume}$=\frac{1}{3}$\pi \times $6^2$\times4=\frac{1}{3}$\pi \times $36\times4=\frac{144}{3}$\pi=48\pi:m^3$

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Question

A cone has a radius of inches and a height of inches. Find the volume of the cone.

Tap to reveal answer

Answer

The volume of a cone is given by the formula:

$\text{Volume}$=\frac{1}{3}$\pi $r^2h$

Now, plug in the values of the radius and height to find the volume of the given cone.

← Didn't Know|Knew It →

Question

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)?

Tap to reveal answer

Answer

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

$3m$\frac{100cm}{1m}$=300cm\ \ \2m$\frac{100cm}{1m}$=200cm$

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

$V=\frac{1}{3}$ \pi $r^{2}$h$

$V=\frac{1}{3}$ \pi $(200cm)^{2}$ ( 300cm )\approx 12,566,371 ; $\textrm{cm}^{3}$$

Now multiply this by to get the mass.

$m=V\rho$

Finally, convert the answer to kilograms.

$5654867g $\frac{1kg}{1,000g}$ \approx 5,655kg$

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Question

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)?

Tap to reveal answer

Answer

The volume of a cone is:

$Volume=\frac{1}{3}$\pi $r^2h$$

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 $\pi $r^2$$ , so we can rewrite the volume formula as follows:

$Volume =\frac{A\times h}{3}$$

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

$Volume =\frac{A\times h}{3}$=\frac{4\times 4}{3}$=\frac{16}{3}$\approx 5.333 $m^3$$

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

$\rho =\frac{m}{v}$$

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

$m=\rho v=1000\times 5.333=5333 Kg$

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Question

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 .

Tap to reveal answer

Answer

The volume of a cone is:

$Volume=\frac{1}$3{}\pi $r^2h$$

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

$Volume=\frac{1}$3{}\pi $r^2h$ =\frac{1}$3{}\pi $t^2$\times 2t=\frac{2}{3}$\pi $t^3$$

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Question

A right cone has a volume of $8\pi$ , a height of and a radius of the circular base of . Find .

Tap to reveal answer

Answer

The volume of a cone is given by:

$Volume=\frac{1}{3}$\pi $r^2h$$

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:

$Volume=\frac{1}{3}$\pi $r^2h$\Rightarrow 8\pi=\frac{1}{3}$\pi\times $(2t)^2$\times 3t$

$\Rightarrow 8\pi=4\pi $t^3$\Rightarrow $t^3$=2\Rightarrow t=\sqrt[3]{2}$

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Question

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

Tap to reveal answer

Answer

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

$$\frac{d}{2}$=r$

$$\frac{12}{2}$=6:m$

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

$$\text{Volume}$=\frac{1}{3}$\pi $r^2$ h$

$\text{Volume}$=\frac{1}{3}$\pi \times $6^2$\times4=\frac{1}{3}$\pi \times $36\times4=\frac{144}{3}$\pi=48\pi:m^3$

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Question

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)?

Tap to reveal answer

Answer

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

$3m$\frac{100cm}{1m}$=300cm\ \ \2m$\frac{100cm}{1m}$=200cm$

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

$V=\frac{1}{3}$ \pi $r^{2}$h$

$V=\frac{1}{3}$ \pi $(200cm)^{2}$ ( 300cm )\approx 12,566,371 ; $\textrm{cm}^{3}$$

Now multiply this by to get the mass.

$m=V\rho$

Finally, convert the answer to kilograms.

$5654867g $\frac{1kg}{1,000g}$ \approx 5,655kg$

← Didn't Know|Knew It →

Question

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)?

Tap to reveal answer

Answer

The volume of a cone is:

$Volume=\frac{1}{3}$\pi $r^2h$$

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 $\pi $r^2$$ , so we can rewrite the volume formula as follows:

$Volume =\frac{A\times h}{3}$$

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

$Volume =\frac{A\times h}{3}$=\frac{4\times 4}{3}$=\frac{16}{3}$\approx 5.333 $m^3$$

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

$\rho =\frac{m}{v}$$

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

$m=\rho v=1000\times 5.333=5333 Kg$

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