Area measures the flat space inside a shape, in square units. Volume measures the space inside a solid, in cubic units. Surface area is the total area of all the faces of a solid, in square units. Getting the units right is half the battle, so state them in every answer.
Area of flat shapes
| Shape | Area | Notes |
|---|---|---|
| Rectangle | A = lw | |
| Square | A = s² | |
| Triangle | A = ½bh | h is perpendicular to the base |
| Parallelogram | A = bh | h is the perpendicular height, not the slanted side |
| Trapezoid | A = ½(b₁ + b₂)h | b₁ and b₂ are the parallel sides |
| Circle | A = πr² | Circumference C = 2πr |
Example Find the area of a triangle with base 10 and height 6
A = ½(10)(6) = 30 square units.
Example Find the area of a trapezoid with parallel sides 6 and 10 and height 4
A = ½(6 + 10)(4) = ½(16)(4) = 32 square units.
Example A circle has radius 5. Find its area and circumference.
A = π(5²) = 25π ≈ 78.54 square units. C = 2π(5) = 10π ≈ 31.42 units.
Radius or diameter?
The circle formulas use the radius. If a problem gives a diameter of 12, the radius is 6, and the area is 36π. Using 12 gives 144π, four times too big.
Composite shapes
Break the shape into pieces you know, find each area, and add. If there’s a hole, subtract it.
Example A window is a 10-by-6 rectangle with a semicircle on top of the 6-unit side. Find its area.
- Rectangle: 10 × 6 = 60.
- The semicircle’s diameter is 6, so its radius is 3. Half a circle: ½π(3²) = 4.5π ≈ 14.14.
- Total: 60 + 4.5π ≈ 74.14 square units.
Volume and surface area of solids
| Solid | Volume | Surface area |
|---|---|---|
| Rectangular prism | V = lwh | SA = 2(lw + lh + wh) |
| Cube | V = s³ | SA = 6s² |
| Cylinder | V = πr²h | SA = 2πr² + 2πrh |
| Cone | V = ⅓πr²h | SA = πr² + πrl, l = slant height |
| Sphere | V = (4/3)πr³ | SA = 4πr² |
| Pyramid | V = ⅓Bh, B = base area | Base plus the triangular faces |
A pattern: a prism or cylinder is “base area × height.” A pyramid or cone with the same base and height holds exactly one third as much.
Example A box is 4 by 3 by 2. Find its volume and surface area.
V = 4 × 3 × 2 = 24 cubic units. SA = 2(12 + 8 + 6) = 52 square units.
Example A cylinder has radius 3 and height 10. Find its volume and surface area.
- V = π(3²)(10) = 90π ≈ 282.74 cubic units.
- SA = 2π(3²) + 2π(3)(10) = 18π + 60π = 78π ≈ 245.04 square units.
Example A cone has radius 3 and height 4. Find its volume and surface area.
- V = ⅓π(9)(4) = 12π ≈ 37.70 cubic units.
- The slant height comes from the Pythagorean theorem: l = √(3² + 4²) = 5.
- SA = π(9) + π(3)(5) = 9π + 15π = 24π ≈ 75.40 square units.
Example A sphere has radius 6. Find its volume and surface area.
V = (4/3)π(216) = 288π ≈ 904.78 cubic units. SA = 4π(36) = 144π ≈ 452.39 square units.
Working backwards
Sometimes you know the area or volume and need a length. Set up the formula and solve.
Example A sphere has volume 36π. Find its radius.
(4/3)πr³ = 36π. Divide by π and multiply by ¾: r³ = 27, so r = 3.
Scaling
If every length of a shape is multiplied by k:
- Perimeter is multiplied by k.
- Area and surface area are multiplied by k².
- Volume is multiplied by k³.
Double the side of a cube and its volume becomes 2³ = 8 times bigger. This explains why a pizza with twice the diameter has four times the pizza.
Units
Square and cubic units convert by squaring or cubing the length conversion.
- 1 ft = 12 in, so 1 ft² = 144 in² and 1 ft³ = 1,728 in³.
- 1 yd = 3 ft, so 1 yd² = 9 ft² and 1 yd³ = 27 ft³.
- 1 m³ = 1,000 liters.
Example How much mulch covers a 12 ft by 4 ft garden bed 3 inches deep?
- Convert the depth to feet: 3 in = 0.25 ft.
- Volume: 12 × 4 × 0.25 = 12 ft³.
- Mulch is sold by the cubic yard: 12 ÷ 27 ≈ 0.44 yd³.
Example A pool is 20 m long, 10 m wide and 1.5 m deep. How many liters of water does it hold?
20 × 10 × 1.5 = 300 m³, which is 300,000 liters.
Check yourself
Find the area of a triangle with base 14 and height 9.
Show answer
63 square units.
Find the area of a circle with diameter 12.
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36π ≈ 113.10 square units. The radius is 6.
Find the volume of a cylinder with radius 2 and height 7.
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28π ≈ 87.96 cubic units.
Find the surface area and volume of a cube with side 5.
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SA = 150 square units, V = 125 cubic units.
A sphere has a volume of 36π. What is its radius?
Show answer
3.
A prism has a volume of 10 cm³. Every length is tripled. What is the new volume?
Show answer
270 cm³. Volume scales by 3³ = 27.
Practice
Questions students ask
Do I need to memorize these for the SAT?
The SAT gives you a reference sheet with the main area and volume formulas, including the sphere, cone and pyramid. Knowing them already is still faster, and many school tests don’t give a sheet.
What’s the difference between surface area and volume?
Surface area is the wrapping paper. Volume is what fits inside. A box’s surface area is how much cardboard it takes; its volume is how much it holds.
Why use π instead of 3.14?
An answer in terms of π, like 25π, is exact. Use a decimal only when the question asks for one, and then use your calculator’s π key for accuracy.