Area and volume

What the exam asks

• Direct formulas. Find the area of a rectangle, triangle or circle, or the volume of a prism, cylinder, cone, pyramid or sphere.
• Working backward. Given an area or volume, find a radius, a height or a side.
• Composite figures. Add pieces together, or subtract a hole or a cutout. Many shaded-region questions are this type.
• Surface area. Add up the faces of a box, cube or cylinder. These formulas are not on the reference sheet.
• Scale and change. What happens to an area or volume when lengths are multiplied by $k$ or changed by a percent.
• Inscribed figures. A square in a circle, a circle in a square or a sphere in a cube.
• Units. Liters vs. cubic centimeters, square feet vs. square inches.

Answers often stay in terms of $\pi$ (“the volume is $k\pi$; what is $k$?”), so don’t convert to decimals unless the choices are decimals.

Core ideas

The reference sheet, and what it leaves out

Any pointed solid (a cone or pyramid) has exactly $\frac{1}{3}$ the volume of the cylinder or prism with the same base and height. If a question needs something unusual, such as the surface area of a sphere, it gives you the formula.

Surface area is a sum of faces

A closed rectangular box has three pairs of congruent faces, so its surface area is $2(\ell w + \ell h + wh)$. An open-top box loses one $\ell w$ face. A closed cylinder is two circles plus a rolled-up rectangle whose width is the circumference: $2\pi r^2 + 2\pi rh$.

Composite figures: add or subtract

Break the figure into standard pieces. Add pieces that are joined, such as a rectangle with a semicircle attached. Subtract holes and cutouts. For a shaded region, use outer area − inner area.

If a figure is made only of horizontal and vertical segments, and a rectangle has had a corner cut out, its perimeter equals the perimeter of the full rectangle. The two edges of the notch replace the two pieces of the sides that were removed. A notch cut into the middle of a side is different: its two depth edges add extra length.

Working backward

Write the formula, substitute everything you know, then solve. A cone with height 6 and volume $50\pi$:

Take roots last, and keep only positive lengths: $r^3 = 64$ gives $r = 4$.

Scale factors: $k$, $k^2$, $k^3$

If every length of a figure is multiplied by $k$ (so the figures are similar):

To go the other way, find $k$ from whatever ratio you’re given. If two similar solids have surface areas in the ratio $9 : 49$, their lengths are in the ratio $3 : 7$ and their volumes in the ratio $27 : 343$. A percent change is a scale factor too: increasing every edge by 10% multiplies the volume by , a 33.1% increase.

When dimensions change by different factors, multiply the factor for each length in the formula. Doubling a cylinder’s radius and halving its height multiplies the volume by $2^2 \times \frac{1}{2} = 2$.

Units: square and cube the conversion

1 ft = 12 in, so 1 ft² = 144 in² and 1 ft³ = 1,728 in³. Likewise 1 yd³ = 27 ft³, 1 m² = 10,000 cm² and 1 L = 1,000 cm³. When possible, convert lengths before you compute.

Inscribed and circumscribed figures

A shared length links the two figures.

• Circle inscribed in a square: the circle’s diameter equals the square’s side.
• Square inscribed in a circle: the square’s diagonal equals the circle’s diameter, so the square’s area is $\frac{d^2}{2}$.
• Sphere inscribed in a cube: the sphere’s diameter equals the cube’s edge.

Worked examples

Exam technique

• Write the formula before you substitute, especially for cones and pyramids, where the $\frac{1}{3}$ is easy to drop.
• Radius or diameter? Note which one the question gives before doing anything else.
• Keep $\pi$ as a symbol when the choices contain $\pi$, and compare coefficients.
• For scale items, find first from whatever ratio is given (lengths, areas or volumes), then raise it to the power you need.

Common mistakes

Quick recap

• Most volume formulas are on the reference sheet. Surface area, trapezoid, equilateral triangle and hemisphere formulas are not.
• A cone or pyramid is $\frac{1}{3}$ of the matching cylinder or prism.
• For composite figures, add joined pieces and subtract holes. Shaded area = outer − inner.
• To work backward, write the formula, substitute, and solve (Desmos intersections for cubic equations).
• For similar figures, lengths scale by $k$, areas by $k^2$ and volumes by .