Area and volume
Geometry and Trigonometry is about 15% of SAT Math (5–7 of 44 questions). Area and volume questions appear as multiple choice and as typed answers, often with a figure, a composite shape or a short real-life story such as a tank, a can or a garden.
The formulas are given to you
In every SAT Math module you can open a reference sheet with the most common formulas: the area of a rectangle, triangle and circle, and the volume of a box, cylinder, sphere, cone and pyramid. You do not have to memorize them, but you should know them well enough to use them quickly. The table shows the ones you need in this lesson.
| Shape | Formula |
|---|---|
| Rectangle | A = ℓw |
| Triangle | A = ½bh |
| Circle | A = πr², circumference C = 2πr |
| Rectangular prism | V = ℓwh |
| Cylinder | V = πr²h |
| Sphere | V = (4/3)πr³ |
| Cone | V = (1/3)πr²h |
| Pyramid | V = (1/3)ℓwh |
Flat shapes and composite figures
In a triangle, the height must make a right angle with the base: it is not the slanted side. A composite figure is made of simple pieces. Find each piece separately, then add the pieces (a shape built from two parts) or subtract the missing piece (a shape with a hole or a corner cut out). Write down every piece before you calculate.
Solids
A prism or cylinder has the same cross-section from bottom to top, so its volume is base area × height. A cone or pyramid fits inside a prism or cylinder with the same base and height, and takes exactly one third of its volume. Be careful with the radius: if a problem gives the diameter, halve it first.
What happens when you scale a shape
If every length is multiplied by k, then every area is multiplied by k² and every volume by k³. Doubling all the edges of a cube makes each face 4 times larger and the volume 8 times larger. The SAT likes this idea because you can answer without knowing the original size.
| Lengths × | Areas × | Volumes × |
|---|---|---|
| 2 | 4 | 8 |
| 3 | 9 | 27 |
| 1/2 | 1/4 | 1/8 |
| k | k² | k³ |
Units
Area is in square units and volume in cubic units. Convert lengths first, then calculate; do not convert squares or cubes by the same number as lengths. For example, 1 m = 100 cm, so 1 m² = 100 × 100 = 10,000 cm². The SAT usually states the conversion you need, but you must apply it to the right power. Also check whether the answer should be in terms of π (such as 36π) or a number.
- 60 − 16π
- 60 − 4π
- 60 − 2π
- 56
- Rectangle: 10 × 6 = 60.
- Hole: πr² = π × 2² = 4π.
- The hole is removed, so subtract: 60 − 4π.
- The choice 60 − 2π comes from using πr instead of πr².
- 30π
- 60π
- 90π
- 300π
- Use V = πr²h.
- V = π × 3² × 10 = π × 9 × 10 = 90π.
- 24
- 72
- 216
- 512
- Every length is multiplied by k = 3.
- The volume is multiplied by k³ = 27.
- New volume = 8 × 27 = 216.
- Check: the original edge is 2, the new edge is 6, and 6³ = 216 ✓
- Convert to centimeters: 4 m = 400 cm and 5 m = 500 cm.
- Floor area: 400 × 500 = 200,000 cm².
- Tile area: 50 × 50 = 2,500 cm².
- Number of tiles: 200,000 ÷ 2,500 = 80.
- Check in meters: each tile is 0.5 m by 0.5 m = 0.25 m², and 20 ÷ 0.25 = 80 ✓
- Using the diameter as the radiusRead the figure and the text twice. If you see “diameter” or a segment through the center, halve it before using πr².
- Forgetting the ½ in the triangleA = ½bh. A triangle is half of a rectangle with the same base and height.
- Using a slanted side as the heightThe height of a triangle is perpendicular to the base. In a right triangle, the legs are base and height.
- Scaling area and volume like lengthsLengths × k, areas × k², volumes × k³. Doubling every edge multiplies the volume by 8, not 2.
- Mixing unitsConvert all lengths to one unit before multiplying. And remember: 1 m² = 10,000 cm², not 100 cm².
- Forgetting the 1/3 in cones and pyramidsA cone is one third of the cylinder with the same base and height.
Desmos has no geometry mode, but its calculator is useful for the arithmetic: type π to get its exact value, and type expressions such as 60 − 4π, (4/3)π(3)^3 or (1/3)π(6)^2(8) to get decimals for comparing answer choices. You can also use it to solve an equation, such as πr² = 50.27 for r, by graphing y = πx² and y = 50.27.
- Type 60 − 4π: Desmos shows 47.4336…
- Type (4/3)π(3)^3 for the volume of a sphere with radius 3
- To find a radius from an area, graph y = πx² and y = the area, and click the crossing point
Use Desmos to turn answers with π into decimals and compare them with the choices. If the choices contain π, you can often leave π in your answer and not use Desmos at all.
Open Desmos ↗