SAT Tip: The Basics of Volume

Photograph by Fotografia Basica

This tip on improving your SAT score was provided by Vivian Kerr at Veritas Prep.

Geometry knowledge is significantly tested on the SAT. There are a number of need-to-know formulas you should have memorized as part of your test-prep plan: formulas for area, perimeter, volume, and surface area, to name a few. Though the front of each SAT math section gives you reference information, including these formulas, you’ll feel more confident on questions that involve solids if you have a firm foundation in the basics and don’t need to flip back to the front of each section every time. Before we get into volume, let’s quickly refresh on just what a “solid” is.

A “solid” is a three-dimension figure (meaning it has width, length, and height). The volume of a solid is essentially the amount of space enclosed by that solid. The volume of any solid is typically equal to the area of the base multiplied by the height.

The volume of a rectangular solid, therefore, is V = lwh (this is one of the formulas given to you for free by the SAT testmakers). The volume of a cube (with six equal sides) is V = s3. The volume of a cylinder (a solid whose cross-section is a circle) is found using the formula V = πr2h.

Here are some irregular formulas. You won’t likely see them on the SAT, and if you did, they would be provided to you with some explanation. But in case you are curious: The volume of a pyramid is V = (1/3)bh. The volume of a cone is V = (1/3)πr2, and the volume of a sphere is V = (4/3)πr2.

Let’s try out a challenging volume question together.

In a rectangular solid, the length, width, and height are all integers. If two adjacent faces have areas 10 and 12, and two opposite faces each have an area of 30, what is the volume of the solid?

(A) 40

(B) 60

(C) 90

(D) 120

(E) 170

First, we need to recall the formula: V = lwh. To find the volume, we need to find the length, width, and height. The question-stem doesn’t tell us these dimensions, but it does tell us the areas of all three different faces of the solid.

If l,w,h denote the length, width, and height, then lw = 10, lh = 12, and wh = 30.

Combining these dimensions,(lw)(lh)(wh) = (lwh)(lwh) = 10*12*30 = 3600

(lwh)2 = 3600

If lwh = V, then if we take the square root of both sides of (lwh)2 = 3600, we’ll have our volume. The square root of 3600 = 60. The correct answer is (B).

Remember that not all figures are drawn to scale on the SAT (in fact, most of them are not drawn to scale to trick you), and it can sometimes be difficult to visualize a solid on a 2D test booklet. Don’t be afraid to redraw figures as you practice so you can “see” them better.

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