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Area & Volume

Master the essential formulas for two-dimensional area and three-dimensional volume that appear on every SAT Math section.

SAT Math

What is Area?

Area measures the amount of space inside a flat (2D) shape. It is always expressed in square units: \(\text{cm}^2\), \(\text{ft}^2\), \(\text{m}^2\), etc.

20 squares 5 units 4 units r \(\pi r^2\)

What is Volume?

Volume measures the amount of space inside a 3D solid. It is always expressed in cubic units: \(\text{cm}^3\), \(\text{ft}^3\), \(\text{m}^3\), etc.

l h w \(V = lwh\) r h \(V = \pi r^2 h\)

Why Area & Volume Matter on the SAT

Frequency
3 to 5 questions per test directly involve area or volume calculations.
Formula Sheet
The SAT gives you a reference sheet, but speed comes from knowing the formulas.
Word Problems
Many questions wrap geometry inside real-world contexts: tanks, rooms, containers.

Core 2D Area Formulas

Rectangle
\(A = lw\)
Triangle
\(A = \tfrac{1}{2}bh\)
Circle
\(A = \pi r^2\)
Trapezoid
\(A = \tfrac{1}{2}(b_1 + b_2)h\)
\(lw\) \(\tfrac{1}{2}bh\) \(\pi r^2\) \(\tfrac{1}{2}(b_1\!+\!b_2)h\)

Core 3D Volume Formulas

Rect. Prism
\(V = lwh\)
Cylinder
\(V = \pi r^2 h\)
Cone
\(V = \tfrac{1}{3}\pi r^2 h\)
Sphere
\(V = \tfrac{4}{3}\pi r^3\)

Area vs. Volume: Key Distinction

Area Volume
Dimension 2D (flat) 3D (solid)
Measures Surface covered Space enclosed
Units \(\text{units}^2\) \(\text{units}^3\)
SAT clue words "paint," "cover," "shade" "fill," "hold," "capacity"
Scale factor \(k\) Scales by \(k^2\) Scales by \(k^3\)

Surface Area: The 3D Wrapper

Surface area is the total area of all faces of a 3D solid. Think of it as the wrapping paper needed to cover the shape.

Rect. Prism
\(SA = 2(lw + lh + wh)\)
Cylinder
\(SA = 2\pi r^2 + 2\pi rh\)
Sphere
\(SA = 4\pi r^2\)

How to Solve Area & Volume Problems

1
Identify the shape. Is it 2D (area) or 3D (volume/surface area)?
2
Select the formula. Match the shape to the correct formula from your reference sheet or memory.
3
Extract the dimensions. Label each variable: \(r\), \(h\), \(l\), \(w\), \(b\). Watch for units.
4
Substitute and compute. Plug values in, simplify. Leave \(\pi\) symbolic unless told otherwise.
5
Check units and context. Is the answer in \(\text{units}^2\) or \(\text{units}^3\)? Does it make sense?

Example 1: Rectangle Area

A garden is 12 feet long and 8 feet wide. What is the area of the garden?

12 ft 8 ft \(A = lw\)

Example 2: Cylinder Volume

A cylindrical water tank has a radius of 5 cm and a height of 12 cm. What is its volume?

5 cm 12 cm

Example 3: Composite Shape

A shape consists of a rectangle (10 × 6) topped by a semicircle with diameter 10. Find the total area.

10 6 \(r = 5\)
1
Rectangle area: \(A_{\text{rect}} = 10 \times 6 = 60\)
2
Semicircle area: \(A_{\text{semi}} = \tfrac{1}{2}\pi r^2 = \tfrac{1}{2}\pi(5)^2 = \tfrac{25\pi}{2}\)
3
Total: \(60 + \tfrac{25\pi}{2} \approx 60 + 39.27 \approx\) 99.27

Example 4: Sphere Volume

A basketball has a diameter of 9.4 inches. Find the volume of the basketball to the nearest cubic inch.

4.7 in

Example 5: Cone Inside a Cylinder

A cone with radius 3 and height 7 sits inside a cylinder with the same radius and height. What is the volume of the space between them?

3 7
1
Cylinder: \(V_{\text{cyl}} = \pi(3)^2(7) = 63\pi\)
2
Cone: \(V_{\text{cone}} = \tfrac{1}{3}\pi(3)^2(7) = 21\pi\)
3
Difference: \(63\pi - 21\pi =\) \(42\pi \approx 131.9\)

Common Mistakes to Avoid

✕ Diameter vs. Radius
Using the diameter where the formula requires the radius. Always divide the diameter by 2 first.
✕ Forgetting the \(\frac{1}{3}\)
Computing a cone or pyramid volume using the full cylinder or prism formula instead of multiplying by \(\tfrac{1}{3}\).
✕ Wrong Units
Mixing inches and feet, or reporting area in cubic units and volume in square units.
✕ Slant vs. Height
Using the slant height of a cone or pyramid as the perpendicular height \(h\) in the volume formula.

Formula Reference Table

Shape Area / Surface Area Volume
Rectangle \(lw\) —
Triangle \(\tfrac{1}{2}bh\) —
Circle \(\pi r^2\) —
Rect. Prism \(2(lw+lh+wh)\) \(lwh\)
Cylinder \(2\pi r^2 + 2\pi rh\) \(\pi r^2 h\)
Cone \(\pi r^2 + \pi r \ell\)
(where \(\ell\) is slant height)		 \(\tfrac{1}{3}\pi r^2 h\)
Sphere \(4\pi r^2\) \(\tfrac{4}{3}\pi r^3\)

Question 1 of 5

A circle has a diameter of 10 cm. What is its area?

Question 2 of 5

A rectangular prism has dimensions 4 × 5 × 6. What is its volume?

Question 3 of 5

A cone has a radius of 6 and a height of 10. What is the volume?

Question 4 of 5

If every dimension of a cube is tripled, by what factor does its volume increase?

Question 5 of 5

A cylinder with radius 4 and height 9 is filled with water. The water is poured into a rectangular tank measuring 8 × 6 at the base. To the nearest tenth, how high does the water rise?