Volume and Surface Area | SAT Math Practice

The key idea

Prisms and cylinders: base area × height. Cones and pyramids:

\frac{1}{3}

of that. Spheres:

\frac{4}{3} π r^{3}

. When dimensions change by percents, multiply the factors, squaring any that apply to the radius.

The trap

Forgetting the $\frac{1}{3}$ for cones, or not squaring the radius factor.

Can Desmos do it?

The formulas are on the reference sheet; the calculator only multiplies.

College Board skill: Area and volume · Geometry and Trigonometry

Try one at each level

One question from each level. The drill has 12 at every level, and moves you up after four right in a row.

Level 1 · Warm-up

A right circular cone has a radius of 8 and a height of 12. Its volume is

k π

. What is the value of

k

Show the answer

Answer: 256

V = \frac{1}{3} π r^{2} h = \frac{1}{3} π (8)^{2} (12) = 256 π

, so

k = 256

. Leaving out the

\frac{1}{3}

gives the volume of a cylinder.

Level 2 · Test level

The radius of a right circular cylinder is increased by 20%, and its height is decreased by 40%. By what percent does the volume decrease?

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Answer: 68/5

V = π r^{2} h

, so the volume is multiplied by

(1.2)^{2} \times 0.6 = 0.864

, a 13.6% decrease. The radius factor is squared.

Level 3 · Beyond the test

A cylindrical container with an inside radius of 3 centimeters holds some water. A solid metal sphere with a radius of 2 centimeters is dropped in and is completely covered by the water, and no water spills. By how many centimeters does the water level rise?

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Answer: 32/27The water rises by the sphere's volume spread over the cylinder's base:

\frac{\frac{4}{3} π ( 2 ) ^{3}}{π ( 3 ) ^{2}} = \frac{32}{27} = \frac{32}{27}

centimeters.

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