Area and Volume Formulas

Measuring Flat and Solid Figures

The ACT does not provide a formula sheet, so the geometry questions on area, perimeter, surface area, and volume reward memorizing a compact set of formulas and knowing exactly when each applies. The good news: a dozen formulas cover almost everything the test asks.

Keep the dimensions of each measurement in mind. Perimeter and circumference are lengths (units). Area and surface area are two-dimensional (units²). Volume is three-dimensional (units³). Matching the dimension to the question is often half the battle.

Two-Dimensional Figures

Figure	Area	Perimeter
Rectangle	$lw$	$2l + 2w$
Triangle	$\tfrac{1}{2}bh$	sum of the three sides
Parallelogram	$bh$	$2(\text{base} + \text{side})$
Trapezoid	$\tfrac{1}{2}(b_1 + b_2)h$	sum of the four sides
Circle	$\pi r^2$	$2\pi r$ (circumference)

For a triangle, $h$ is the perpendicular height, not a slanted side. For a trapezoid, $b_1$ and $b_2$ are the two parallel sides and $h$ is the distance between them.

Circle caution. The single most common ACT slip is confusing the radius and the diameter. The radius $r$ is half the diameter. If a problem gives a diameter of $10$, use $r = 5$: the area is $\pi(5)^2 = 25\pi$, not $\pi(10)^2$.

Composite Areas

Many questions build a shape out of simpler pieces. To find the area of an L-shaped room, split it into two rectangles and add. To find the shaded region between a circle and an inscribed square, subtract the square's area from the circle's. Look for the underlying rectangles, triangles, and circles.

Three-Dimensional Figures — Volume

Volume measures how much a solid holds. The most common ACT solids:

$V_{\text{box}} = lwh, \qquad V_{\text{cylinder}} = \pi r^2 h$

$V_{\text{cone}} = \tfrac{1}{3}\pi r^2 h, \qquad V_{\text{sphere}} = \tfrac{4}{3}\pi r^3, \qquad V_{\text{pyramid}} = \tfrac{1}{3}(\text{base area})h$

Notice the pattern: a prism or cylinder is (base area) $\times$ height, while a cone or pyramid is one-third of that — they taper to a point. A can of radius $3$ and height $10$ holds $\pi(3)^2(10) = 90\pi$ cubic units.

Three-Dimensional Figures — Surface Area

Surface area adds up the areas of every face. For a rectangular box it is the sum of three pairs of faces:

$SA_{\text{box}} = 2(lw + lh + wh)$

For a cylinder, unroll it: two circular caps plus a rectangular wrap whose width equals the circumference:

$SA_{\text{cylinder}} = 2\pi r^2 + 2\pi r h$

Watch the Height You Use

Cones and pyramids have two "heights": the vertical height $h$ (straight up from the center of the base) and the slant height $\ell$ (up the sloped face). Volume always uses the vertical height. If a problem gives the slant height and the radius, recover the vertical height with the Pythagorean theorem: $\ell^2 = r^2 + h^2$.

A Reliable Game Plan

• Identify the figure and whether the question wants length, area, or volume.
• Convert diameter to radius the moment you see a circle, sphere, or cylinder.
• Cone or pyramid? Multiply the prism/cylinder volume by $\tfrac{1}{3}$, and use the vertical height.
• Composite shape? Break it into standard pieces and add or subtract.
• Match the units: length (units), area (units²), volume (units³).

Common Mistakes to Avoid

• Squaring or cubing the diameter instead of the radius.
• Forgetting the $\tfrac{1}{2}$ in triangle or trapezoid area, or the $\tfrac{1}{3}$ in cone or pyramid volume.
• Using slant height in a cone/pyramid volume instead of the vertical height.
• Mixing dimensions — reporting an area when the question asked for a perimeter, or vice versa.