Area and Volume

This skill covers the area of flat shapes, the volume and surface area of solids, and how these measures change when a figure is scaled. The SAT gives you a reference sheet with most formulas, so the challenge is choosing the right one, finding missing dimensions, and handling composite figures and unit changes.

• Area is measured in square units; volume in cubic units. Keep track of which one the question wants.
• Composite figures: split into familiar pieces (rectangles, triangles, semicircles), then add or subtract.
• Cones and pyramids hold exactly $\frac{1}{3}$ of the matching cylinder or prism.
• Surface area is the total area of all faces. A cube with edge $s$ has $6$ square faces, so its surface area is $6s^2$.
• If every length is multiplied by $k$, areas are multiplied by $k^2$ and volumes by $k^3$.
• When only some dimensions change (for example, the radius doubles and the height is cut in half), substitute into the formula rather than using one scale factor.

• Similar solids with surface areas in ratio $4 : 9$ have lengths in ratio $2 : 3$ and volumes in ratio $8 : 27$.
• $1$ liter $= 1{,}000$ cm$^3$.

A closed box is $8$ cm long, $5$ cm wide, and $3$ cm tall. Find its volume and surface area.

1. Volume: $8 \times 5 \times 3 = 120$ cm$^3$.
2. Surface area: three pairs of faces, $2(8 \cdot 5 + 8 \cdot 3 + 5 \cdot 3) = 2(40 + 24 + 15) = 158$ cm$^2$.

Check: the faces are $40$, $40$, $24$, $24$, $15$, and $15$, which add to $158$. Correct.

A solid metal cylinder with radius $4$ and height $9$ is melted and recast as a cone with radius $4$. What is the height of the cone?

1. The volume stays the same. Cylinder: $\pi (4)^2 (9) = 144\pi$.
2. Cone: $\frac{1}{3}\pi (4)^2 h = \frac{16}{3}\pi h$.
3. Set them equal: $\frac{16}{3}\pi h = 144\pi$, so $h = 144 \cdot \frac{3}{16} = 27$.

Check: with the same base, a cone needs $3$ times the cylinder's height to hold the same volume, and $3 \times 9 = 27$. Correct.

• Using the diameter as the radius. A circle with diameter $10$ has area $25\pi$, not $100\pi$.
• Forgetting the $\frac{1}{3}$ for cones and pyramids.
• Scaling volume by $k$ instead of $k^3$. Doubling every edge of a box multiplies its volume by $8$.
• Using slant height as height. In a cone, the height is perpendicular to the base. If you are given the slant height, use the Pythagorean theorem with the radius first.
• Counting a shared side twice in a composite figure, or forgetting that a semicircle is half a circle's area.