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Area and volume

Compute area, surface area, and volume for standard solids, including scaling and working backward to a dimension.

Geometry and TrigonometryMath
Practice area and volume free

What the SAT actually asks

You use area and volume formulas for rectangles, triangles, circles, prisms, cylinders, cones, pyramids, and spheres. The formulas for solids are printed on the reference sheet, so the work is choosing the right one and substituting carefully. Common tasks are finding a missing dimension when the volume is known, comparing two solids, finding the volume of a composite shape made of two solids joined together, and predicting how volume changes when a dimension is multiplied by a factor.

Area and volume questions usually account for about 1 to 2 of the 44 math questions.

Where students lose points

How to attack these questions

  1. 1Write the formula down from the reference sheet before you substitute anything, and check radius versus diameter.
  2. 2Break composite figures into standard solids, compute each separately, then add or subtract.
  3. 3For scaling questions, apply the factor to every length in the formula and see what the total factor becomes.

Example questions

Written for this skill, with the answer explained. 73 more are in the app.

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Area and volume
Real question
OAB3
Note: Figure not drawn to scale.

The figure shows a cross-section through the center of a sphere with radius 3 inches. The volume of the sphere is kπ cubic inches. What is the value of k?

Medium

A solid consists of a right circular cone of radius 3 inches and height 12 inches with a hemisphere of radius 3 inches attached to the circular base of the cone. The total volume of the solid is k times pi cubic inches. What is the value of k?

  • A) 36
  • B) 54 ✓
  • C) 72
  • D) 108

Why: The cone contributes (1/3) pi(9)(12) = 36 pi. A full sphere of radius 3 has volume (4/3) pi(27) = 36 pi, so the hemisphere is half of that, 18 pi. The total is 54 pi, so k = 54. Answering 36 counts only the cone. Answering 72 uses a full sphere instead of a hemisphere, and 108 adds the cone to a full sphere, 36 + 72.

Medium

A grain silo is a right circular cylinder of radius 6 feet and height 20 feet topped by a hemisphere of radius 6 feet. The total volume of the silo is k times pi cubic feet. What is the value of k?

  • A) 792
  • B) 864 ✓
  • C) 936
  • D) 1,008

Why: The cylinder contributes pi(36)(20) = 720 pi. A full sphere of radius 6 has volume (4/3) pi(216) = 288 pi, so the hemisphere adds half of that, 144 pi. The total is 864 pi, so k = 864. Answering 1,008 adds a full sphere rather than a hemisphere. Answering 936 adds 216, which is r³ rather than the hemisphere's volume, and 792 adds only 72, half of the hemisphere.

Harder

A right circular cylinder has a volume of 1,296 pi cubic centimeters and a radius of 12 centimeters. What is the height, in centimeters, of the cylinder?

  • A) 2.25
  • B) 9 ✓
  • C) 27
  • D) 108

Why: For a cylinder, V = pi r squared h, so 1,296 pi = pi (144) h and h = 9 centimeters. Dividing 1,296 by 12 instead of by 144 gives 108. Answering 2.25 means you used the diameter where the radius belongs. Answering 27 means you applied the cone formula, tripling the height.

Questions students ask

Are volume formulas given on the SAT?

Yes. The reference information at the start of each math module includes the volume formulas for rectangular solids, cylinders, spheres, cones, and pyramids, along with basic area formulas and the special right triangle ratios. You still need to know when to use each one.

What happens to volume when you double a dimension?

If you double only one dimension, the volume doubles. If you scale a solid so that every length doubles, the volume becomes eight times as large, because volume depends on three dimensions. Surface area in that case becomes four times as large.

How do I find a missing dimension from a known volume?

Write the volume formula, substitute everything you know including the volume itself, then solve the resulting equation for the unknown length. Take a square root or cube root at the end if the unknown appears to a power.

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Questions on area and volume alone, with explanations on every one, and the app tracks which ones you keep missing.

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