Math 10C · Surface area and volume
The outside and the inside
Surface area is how much material wraps a solid; volume is how much fits inside. Nearly every formula in this unit is one of two ideas — base times height, or a third of base times height — with a circle substituted in.
- 1. The words, first
- 2. The formulas, grouped by idea
- 3. Worked examples
- 4. What happens when you scale a solid
- 5. What costs marks
The words, first
The idea: The two words that get confused, and the three lengths that get confused.
| Word | What it means |
|---|---|
| Surface area | The total area of all the faces or surfaces. Measured in square units, always. |
| Volume | The space inside. Measured in cubic units, always. If your units do not match the question, the formula was wrong. |
| Prism | A solid with two identical parallel bases joined by flat faces. A cylinder is the circular version. |
| Pyramid | A solid with one base and faces meeting at a point. A cone is the circular version. |
| Slant height (s) | The distance from the apex down the sloping surface to the edge of the base. Used in the surface area of a cone or pyramid. |
| Height (h) | The perpendicular distance from the base to the apex. Used in volume. It is not the slant height, and questions often supply one and want the other. |
| Composite solid | A solid made of two or more simpler ones joined together. |
The formulas, grouped by idea
The idea: Learn them as two families and a sphere, rather than as eight unrelated lines.
Volume: base area × height.
Prism: V = (base area) × h · Cylinder: V = πr²h
Volume: a third of that, for anything that comes to a point.
Pyramid: V = ⅓ × (base area) × h · Cone: V = ⅓πr²h
Sphere is its own case: V = (4/3)πr³ and SA = 4πr².
Surface area is always the sum of the pieces:
Cylinder: 2πr² + 2πrh · Cone: πr² + πrs · Cube: 6s²
The cylinder is two circles plus a rectangle that has been rolled up — its length is the circumference, which is where the 2πr in the second term comes from. Seeing that once means never forgetting whether to use 2πr or πr².
Worked examples
The idea: Write the formula, substitute, then evaluate. Doing all three in one step is where the slips happen.
Cylinder, r = 3 cm and h = 10 cm.
SA = 2π(3)² + 2π(3)(10) = 56.5 + 188.5 ≈ 245 cm²
Forgetting the two circular ends gives 188.5, which is the most common wrong answer in this unit.
Cone, r = 4 cm and s = 9 cm.
SA = π(4)² + π(4)(9) = 50.3 + 113.1 ≈ 163 cm²
If a question gives the vertical height instead, find the slant first with Pythagoras: s² = r² + h².
Composite: a cylinder r = 2, h = 8 with a hemisphere on top.
V = π(2)²(8) + ½ × (4/3)π(2)³ = 100.5 + 16.8 ≈ 117 cm³
Volumes always add. Surface areas do not: the circle where the two parts meet is now inside the solid, so it is not part of the surface and must be left out.
What happens when you scale a solid
The idea: Lengths scale by k, areas by k², volumes by k³. One fact, and it answers a whole family of questions without any formula.
Double the edge of a cube and its surface area is multiplied by 4 while its volume is multiplied by 8. The solid has eight times as much inside and only four times as much skin.
Why it matters beyond maths. This is the surface-area-to-volume relationship: as things get bigger, their surface falls behind their bulk. It is why a cell cannot grow indefinitely, why small animals lose heat faster than large ones, and why a crushed tablet dissolves faster than a whole one.
What costs marks
The idea: Units and lengths.
- Cubic units for an area, or square units for a volume. Check before you write the answer.
- Using the slant height in a volume, or the vertical height in a cone's surface area.
- Forgetting the ends of a cylinder's surface area.
- Adding surface areas of joined solids without removing the hidden faces.