Volume and Surface Area — SAT Math

Introduction

Volume and surface area problems appear under the Geometry and Trigonometry domain, which makes up about 15% of the SAT Math section. You will be asked to select the right formula, substitute given measurements, and calculate a result — sometimes in a real-world context (a water tank, a shipping box, an ice cream cone). The key skill is formula selection: knowing which formula applies and which values plug into which variable.

Core Concept

Every 3-D solid question falls into one of two categories:

Volume — how much space the solid occupies (cubic units)
Surface area — the total area of all faces or curved surfaces (square units)

Which formulas are on the SAT reference sheet?

The reference sheet provides:

V_{\text{rectangular prism}} = \ell w h, \quad V_{\text{cylinder}} = \pi r^2 h, \quad V_{\text{sphere}} = \tfrac{4}{3}\pi r^3

V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h, \quad V_{\text{pyramid}} = \tfrac{1}{3}\ell w h

Everything else must be memorized. That means all surface area formulas are on you.

Surface area: sum of the areas of all faces

Think of "unfolding" a solid into flat pieces and summing their areas.

Solid	Surface area formula (memorize)
Rectangular prism	$2(\ell w + \ell h + wh)$
Cylinder	$2\pi r^2 + 2\pi r h$
Cone	$\pi r^2 + \pi r \ell$
Sphere	$4\pi r^2$

For a cone, $\ell$ is the slant height (not the vertical height $h$ ). These are related by $\ell = \sqrt{r^2 + h^2}$ .

For a pyramid, the surface area is the base area plus the area of each triangular face, computed individually — there is no single memorizable formula because pyramid shapes vary.

Quick illustrative calculation

A cylinder has radius 2 m and height 5 m. Its volume (from the reference sheet):

V = \pi(2)^2(5) = 20\pi \approx 62.8 \text{ m}^3

Its total surface area (memorized formula):

SA = 2\pi(2)^2 + 2\pi(2)(5) = 8\pi + 20\pi = 28\pi \approx 87.96 \text{ m}^2

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