Area and Volume: Free SAT Math Lesson
Area and volume questions on the Digital SAT test whether you can apply the right geometric formula to a described or pictured figure, often inside a real-world word problem. This free, complete lesson covers the area formulas for triangles, trapezoids, parallelograms, and circles, the volume formulas for rectangular prisms, cylinders, cones, pyramids, and spheres, surface area, and composite figures that combine more than one shape. Each idea is followed by a fully worked example, and then a set of original practice problems with instant feedback. Everything here is free, and you can keep practicing afterward with the full SAT Math Question Bank linked below.
1. Area Formulas for 2D Shapes
| Shape | Area formula |
|---|---|
| Triangle | A = 12bh |
| Rectangle | A = lw |
| Parallelogram | A = bh |
| Trapezoid | A = 12(b1 + b2)h |
| Circle | A = πr² |
A trapezoid, with parallel sides b₁ and b₂ and height h
Worked Example: A trapezoid has parallel sides of length 8 and 14, and a height of 5. What is its area?
A = 12(8 + 14)(5) = 12(22)(5) = 55
1. A triangle has base 12 and height 7. What is its area?
Explanation: A = 12(12)(7) = 42.
2. A trapezoid has parallel sides of length 8 and 14, and height 5. What is its area?
Explanation: This is the worked example above: A = 12(8 + 14)(5) = 55.
3. A circle has radius 6. What is its area, in terms of π?
Explanation: A = πr² = π(6)² = 36π.
4. A parallelogram has base 9 and height 4. What is its area?
Explanation: A = bh = (9)(4) = 36.
5. A rectangle has a perimeter of 34 and a length of 10. What is its area?
Explanation: Perimeter = 2(l + w) = 34, so l + w = 17. Since l = 10, w = 7. Area = lw = (10)(7) = 70.
2. Volume Formulas for 3D Solids
| Solid | Volume formula |
|---|---|
| Rectangular prism | V = lwh |
| Cylinder | V = πr²h |
| Cone | V = 13πr²h |
| Pyramid | V = 13Bh (B = area of the base) |
| Sphere | V = 43πr³ |
A cylinder, with radius r and height h
Worked Example: A cone has radius 6 and height 9. What is its volume, in terms of π?
V = 13π(6)²(9) = 13π(36)(9) = 13(324π) = 108π
1. A rectangular prism has length 5, width 4, and height 3. What is its volume?
Explanation: V = lwh = (5)(4)(3) = 60.
2. A cylinder has radius 3 and height 10. What is its volume, in terms of π?
Explanation: V = πr²h = π(3)²(10) = 90π.
3. A cone has radius 6 and height 9. What is its volume, in terms of π?
Explanation: This is the worked example above: V = 13π(6)²(9) = 108π.
4. A sphere has radius 3. What is its volume, in terms of π?
Explanation: V = 43π(3)³ = 43π(27) = 36π.
5. A pyramid has a square base with side length 6 and a height of 10. What is its volume?
Explanation: The base area is 6² = 36. V = 13(36)(10) = 120.
3. Surface Area
Surface area is the total area covering the outside of a 3D solid. For many solids, it's the sum of the areas of each face; for curved solids like cylinders and spheres, specific formulas apply.
| Solid | Surface area formula |
|---|---|
| Rectangular prism | SA = 2(lw + lh + wh) |
| Cylinder | SA = 2πr² + 2πrh |
| Sphere | SA = 4πr² |
Worked Example: A rectangular prism has length 4, width 3, and height 5. What is its surface area?
SA = 2(lw + lh + wh) = 2[(4)(3) + (4)(5) + (3)(5)] = 2(12 + 20 + 15) = 2(47) = 94
1. A rectangular prism has length 4, width 3, and height 5. What is its surface area?
Explanation: This is the worked example above: SA = 2(12 + 20 + 15) = 94.
2. A cylinder has radius 4 and height 6. What is its surface area, in terms of π?
Explanation: SA = 2π(4)² + 2π(4)(6) = 32π + 48π = 80π.
3. A sphere has radius 5. What is its surface area, in terms of π?
Explanation: SA = 4π(5)² = 100π.
4. A cube has an edge length of 6. What is its total surface area?
Explanation: A cube has 6 identical square faces, each with area 6² = 36. Total surface area: 6(36) = 216.
4. Composite Figures and Word Problems
A composite figure combines two or more basic shapes. Break it into simpler pieces, find the area or volume of each piece, then add or subtract as the situation requires.
Worked Example: A rectangular garden measures 15 feet by 8 feet. A walkway of uniform width 2 feet surrounds it. What is the total area of the garden and walkway combined?
The outer dimensions include the 2-foot walkway on every side: (15 + 2 + 2) by (8 + 2 + 2), or 19 by 12.
Total area = 19 × 12 = 228 square feet
1. A cylindrical water tank has radius 4 feet and height 10 feet. How many cubic feet of water can it hold, rounded to the nearest whole number? (Use π ≈ 3.14)
Explanation: V = πr²h = π(4)²(10) = 160π ≈ 160(3.14) = 502.4, which rounds to 502.
2. A rectangular garden measures 15 feet by 8 feet. A walkway of uniform width 2 feet surrounds it. What is the total area of the garden and walkway combined?
Explanation: This is the worked example above: the outer dimensions are 19 by 12, giving a total area of 228 square feet.
3. A cone-shaped paper cup has radius 3 cm and height 8 cm. What is its volume, in terms of π?
Explanation: V = 13π(3)²(8) = 13π(9)(8) = 13(72π) = 24π.
4. Two similar rectangular prisms have a scale factor of 2 : 1 (larger to smaller). If the smaller prism has a volume of 15 cubic units, what is the volume of the larger prism?
Explanation: When linear dimensions scale by a factor of k, volume scales by k³. Here k = 2, so volume scales by 2³ = 8. The larger volume is 15 × 8 = 120.
5. A sphere is inscribed in a cube with edge length 10, touching all six faces. What is the radius of the sphere?
Explanation: An inscribed sphere's diameter equals the cube's edge length, so the diameter is 10, and the radius is 5.
Common Mistakes to Avoid
- Mixing up which dimension is the height in a trapezoid or triangle. The height must be measured perpendicular to the base, not along a slanted side. A slanted side length is not the same as the height unless the figure happens to be a right triangle.
- Forgetting the 13 factor for cones and pyramids. A cone or pyramid has exactly one-third the volume of a cylinder or prism with the same base and height, it's easy to forget this factor and overestimate the volume by 3 times.
- Confusing radius and diameter. Every area and volume formula involving a circle or sphere uses the radius, not the diameter. If a problem gives you a diameter, divide by 2 before substituting into any formula.
- Using the wrong scaling power for similar figures. When similar figures scale by a factor k, lengths scale by k, areas scale by k², and volumes scale by k³. Applying the wrong power is a common error on similarity-based area and volume questions.
- Forgetting to account for every side of a composite figure. When a walkway, border, or frame surrounds a shape on all sides, remember it adds to both dimensions on both ends, not just once.
Frequently Asked Questions
Do I need to memorize every area and volume formula for the Digital SAT?
Many common ones, including several used in this lesson, appear on the reference sheet provided at the start of the Math section. Still, knowing them well enough to apply them quickly and confidently, without pausing to look them up, will save valuable time on test day.
How do I know whether a question wants area, surface area, or volume?
Area applies to flat, 2D shapes or to one face of a 3D solid. Surface area applies to the entire outside covering of a 3D solid. Volume applies to how much space is inside a 3D solid. Watch for units too, area and surface area are in square units, while volume is in cubic units.
What's the key idea for solving composite figure problems?
Break the figure into shapes you already have formulas for, calculate each piece separately, and then combine them with addition or subtraction based on whether the pieces are being added together or one is being removed from another, like a hole cut out of a larger shape.
Where can I practice more problems like these?
The Area and Volume quizzes in the SAT Math Question Bank include additional original problems on this topic, along with quizzes covering every other Algebra, Advanced Math, Problem-Solving and Data Analysis, and Geometry and Trigonometry skill on the Digital SAT.