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SAT Math : How to find the volume of a cone

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Example Question #181 : Solid Geometry

Cone

The above is a right circular cone. Give its volume.

Possible Answers:

$\frac{245}{3} \pi \sqrt{11}$

$175 \pi$

$294 \pi$

$49 \pi+35 \pi\sqrt{11}$

$\frac{1,925}{3} \pi$

Correct answer:

$294 \pi$

Explanation:

The volume of a right circular cone can be calculated from its height and the radius of its base using the formula

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

r = 7 and h = 18 , so substitute and evaluate:

$V = \frac{1}{3} \pi \cdot 7^{2} \cdot 18$

$= \frac{1}{3} \pi \cdot 49 \cdot 18$

$=\left ( \frac{1}{3} \cdot 49 \cdot 18 \right ) \cdot \pi$

$= 294 \pi$

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Example Question #182 : Solid Geometry

Cone

In terms of , express the volume of the above right circular cone.

Possible Answers:

$V = \frac{1}{3} \pi r^{2} \sqrt{ 576 - r^{2} }$

$V =8 \pi r^{2}$

$V = \pi r^{2} + 24 \pi r$

$V = \frac{1}{3} \pi r^{2} \sqrt{ r^{2} - 576 }$

$V = \pi r^{2} + \pi r\sqrt{ 576 - r^{2} }$

Correct answer:

$V = \frac{1}{3} \pi r^{2} \sqrt{ 576 - r^{2} }$

Explanation:

The volume of a cone can be calculated from its height and the radius of its base using the formula

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

The slant height is shown in the diagram to be 24. By the Pythagorean Theorem,

$h^{2}+ r ^{2} = l^{2}$

Setting l = 24 and solving for :

$h^{2}+ r ^{2} = 24^{2}$

$h^{2}+ r ^{2} = 576$

$h^{2}+ r ^{2} - r^{2} = 576 -r^{2}$

$h^{2} = 576 - r^{2}$

$h =\sqrt{ 576 - r^{2} }$

Substituting in the volume formula for :

$V = \frac{1}{3} \pi r^{2} \sqrt{ 576 - r^{2} }$ .

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Example Question #183 : Solid Geometry

Cone

In terms of , express the volume of the provided right circular cone.

Possible Answers:

$V = \frac{20}{3} \pi r^{2}$

$V = \pi r^{2}+ \pi r \sqrt{r^{2} + 400}$

$V = \frac{1}{3} \pi r^{2} \sqrt{r^{2} - 400}$

$V = \pi r^{2}+ 20 \pi r$

$V = \frac{1}{3} \pi r^{2} \sqrt{r^{2} + 400}$

Correct answer:

$V = \frac{20}{3} \pi r^{2}$

Explanation:

The volume of a cone can be calculated from its height and the radius of its base using the formula

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

The height of the cone is shown to be equal to 20, so substituting accordingly:

$V = \frac{1}{3} \pi r^{2} \cdot 20$

$V = \frac{20}{3} \pi r^{2}$

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SAT Math : How to find the volume of a cone

Study concepts, example questions & explanations for SAT Math

All SAT Math Resources

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Example Question #181 : Solid Geometry

Example Question #182 : Solid Geometry

Example Question #183 : Solid Geometry

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