What area and volume formulas are given on the SAT reference sheet?

The reference sheet provides the area and circumference of a circle, the area of a rectangle and triangle, the Pythagorean theorem, and the volumes of a rectangular prism, cylinder, cone, sphere, and pyramid. Memorizing them anyway saves time and prevents look-up errors under pressure.

How do you find the area of a composite or shaded figure?

Break the figure into standard shapes whose formulas you know, compute each one, then combine them — add stacked pieces and subtract cutouts. A shaded region between a rectangle and a circle punched out of it is simply the rectangle area minus the circle area.

Why does doubling the radius multiply a volume by 8?

Volume is three-dimensional, so it scales with the cube of any change in a linear dimension. Doubling the radius multiplies by $2$ in all three directions, and $2 \times 2 \times 2 = 8$. Area, being two-dimensional, would scale by $2^2 = 4$.

Do I use the radius or the diameter in circle formulas?

Always the radius. If a problem gives the diameter, halve it first. Using the diameter in $A = \pi r^2$ makes the area four times too large, which is exactly the mistake several answer choices are designed to catch.

CramClub • Area and Volume — SAT® Guide & Practice

$A = \tfrac{1}{2}bh$

$A = \pi r^2, \;\; C = 2\pi r$

$V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h, \;\; V_{\text{sphere}} = \tfrac{4}{3}\pi r^3$

length $\times k \Rightarrow$ area $\times k^2$ , volume $\times k^3$

Measuring Flat Shapes and Solid Figures

The SAT expects you to move fluently between two-dimensional measurements (perimeter and area) and three-dimensional measurements (surface area and volume). A reference sheet at the start of every Math module supplies the circle, rectangle, triangle, and common-solid formulas, but the test rewards students who already know them cold and — more importantly — who know when to use each one. The trap is almost never the formula itself; it is deciding which quantity the question actually wants.

Perimeter and Area in Two Dimensions

Perimeter is the total distance around a shape, measured in linear units. Area is the space enclosed, measured in square units. For a rectangle, $\text{Area} = \ell w$ and $\text{Perimeter} = 2\ell + 2w$ . For a triangle, $\text{Area} = \tfrac{1}{2}bh$ , where $h$ is the height measured perpendicular to the chosen base $b$ — not a slanted side. For a circle of radius $r$ , the circumference is $C = 2\pi r$ and the area is $A = \pi r^2$ .

A single mix-up sinks many circle problems: the formulas want the radius, but questions often give the diameter. Always halve a diameter before substituting. If a circle has diameter $10$ , its area is $\pi(5)^2 = 25\pi$ , not $\pi(10)^2 = 100\pi$ .

Surface Area and Volume in Three Dimensions

Volume measures the capacity of a solid in cubic units; surface area measures the total area of every face, in square units. The volume formulas you need:

Solid	Volume	Key idea
Rectangular prism	$V = \ell w h$	multiply all three dimensions
Cylinder	$V = \pi r^2 h$	circle area times height
Cone	$V = \tfrac{1}{3}\pi r^2 h$	one-third of the matching cylinder
Sphere	$V = \tfrac{4}{3}\pi r^3$	grows with the cube of the radius

Notice the pattern: a cone holds exactly one-third of the cylinder that shares its base and height. That relationship is a favorite SAT shortcut, and forgetting the $\tfrac{1}{3}$ is one of the most common errors on the whole geometry–trig domain.

Composite Figures: Break, Compute, Combine

Many "unusual shape" questions are just familiar shapes glued together or carved out of one another. The strategy is always the same three moves: break the figure into standard pieces, compute each piece with a formula you know, and combine them — adding for stacked pieces, subtracting for holes or cutouts.

$V_{\text{total}} = V_{\text{piece 1}} + V_{\text{piece 2}} \quad\text{or}\quad A_{\text{shaded}} = A_{\text{outer}} - A_{\text{cutout}}.$

For example, a "shaded region" between a rectangle and a circle punched out of it is simply $\ell w - \pi r^2$ . A grain silo shaped like a cylinder capped by a cone is $\pi r^2 h + \tfrac{1}{3}\pi r^2 h_{\text{cone}}$ . Once you see the pieces, the problem collapses into arithmetic you already trust.

The Scaling Rule That Trips Everyone Up

When every linear dimension of a figure is multiplied by a factor $k$ , its area scales by $k^2$ and its volume by $k^3$ . This is the single most counterintuitive idea in the topic. Doubling the radius of a sphere does not double its volume — it multiplies the volume by $2^3 = 8$ . Tripling the side of a cube multiplies its volume by $27$ . The SAT loves this because students instinctively expect volume to grow in step with length.

$\text{length} \times k \;\Rightarrow\; \text{area} \times k^2 \;\Rightarrow\; \text{volume} \times k^3.$

Watch the Units

Square and cubic units convert differently from linear ones. Because $1$ yard $= 3$ feet, $1$ square yard $= 9$ square feet and $1$ cubic yard $= 27$ cubic feet. If a tank is described in feet but the answer must be in cubic yards, convert consistently and remember the cube. A question that "changes units" mid-problem is testing exactly this awareness.

A Reliable Game Plan

Read the question twice and ask which quantity it wants — perimeter, area, surface area, or volume — because a figure often supplies enough to compute several. Convert every diameter to a radius before substituting. For odd shapes, break-compute-combine. And whenever a dimension changes by a factor, reach for $k^2$ (area) or $k^3$ (volume) instead of scaling the answer linearly. Those four habits handle essentially every area-and-volume question the SAT can ask.

A triangle has a base of $10$ and a height of $6$ . What is its area?

A circle has a diameter of $10$ . What are its area and circumference, in terms of $\pi$ ?

A cylindrical can has a radius of $4$ and a height of $9$ . What is its volume, in terms of $\pi$ ?

A circle of radius $3$ is cut out of the center of a rectangle that is $10$ wide and $6$ tall. What is the area of the remaining (shaded) region, in terms of $\pi$ ?

A sphere has a volume of $36\pi$ . What is its radius, and what is its surface area? (Surface area of a sphere is $4\pi r^2$ .)

A solid metal cylinder with radius $6$ and height $15$ is melted down and recast, with no waste, into small spheres of radius $3$ . How many spheres are produced?

Substituting the diameter where the formula wants the radius. This makes a circle area four times too large and a volume even further off.

Halve any diameter first: if $d = 10$ , then $r = 5$ and $A = \pi(5)^2 = 25\pi$ , never $\pi(10)^2$ .

Dropping the $\tfrac{1}{3}$ on cone volume, reporting the full cylinder value instead.

A cone holds one-third of its matching cylinder: $V = \tfrac{1}{3}\pi r^2 h$ . Write the fraction first so you cannot forget it.

Assuming that doubling a length doubles the area or volume.

Area scales by $k^2$ and volume by $k^3$ . Doubling a radius multiplies area by $4$ and volume by $8$ ; tripling multiplies volume by $27$ .

Squaring the height in a cylinder, or mixing linear and cubic units in a conversion.

In $V = \pi r^2 h$ only the radius is squared. And remember $1$ cubic yard $= 27$ cubic feet (not $3$ ) — cube the linear conversion factor.

A rectangular storage box has a length of $8$ inches, a width of $5$ inches, and a height of $3$ inches. What is its volume, in cubic inches?

A triangle has a base of $12$ and a height of $5$ . What is its area?

The volume of the larger of two spheres is how many times the volume of the smaller if the larger radius is twice the smaller radius?

A cylinder has a radius of $3$ and a height of $8$ . What is its volume, in terms of $\pi$ ?

A circle of radius $2$ is removed from the center of a $10$ -by- $6$ rectangle. What is the area of the remaining region, in terms of $\pi$ ?

A cylinder's radius is doubled and its height is halved. The new volume is how many times the original volume?