ACT Math: Volume and Surface Area Formulas for Solids

Rectangular solid (box): V=lwh (length × width × height). Cylinder: V=πr^2h (area of base times height), Surface Area = 2πr^2+2πrh (two circles plus side). Sphere: V=4/3 πr^3, Surface Area=4πr^2. These three shapes cover 90% of ACT solid geometry questions. Write them on a card and memorize before test day. The key to success is knowing which formula applies, not deriving formulas from scratch.

Example: A cylinder has radius 3 and height 5. Volume = π(3^2)(5)=45π. Surface area = 2π(3^2)+2π(3)(5)=18π+30π=48π. Both formulas apply to the same object, but you must know which to use based on the question.

Error 1: Confusing radius and diameter. If the problem gives diameter, divide by 2 for radius. Many students forget this and use the diameter directly in the formula. Error 2: Forgetting π in calculations. V=πr^2h means you must include π; don't drop it. Error 3: Mixing up surface area and volume. Surface area is the "wrapper" (outside measurement); volume is the "fill" (inside capacity). Before you solve, whisper aloud: "This question asks for [VOLUME/SURFACE AREA]."

Catch these errors upfront and you eliminate 90% of mistakes on solid geometry questions.

Problem 1: A rectangular solid has dimensions 2, 3, and 5. Find volume. Problem 2: A cylinder has radius 2 and height 6. Find volume. Problem 3: Same cylinder. Find surface area. Problem 4: A sphere has radius 4. Find volume. Write your work and answers: (1) V=2×3×5=30, (2) V=π(2^2)(6)=24π, (3) SA=2π(4)+2π(2)(6)=8π+24π=32π, (4) V=4/3 π(4^3)=256/3 π.

If you missed any, rework it and identify where your formula or arithmetic broke. Repeat until all four are correct.

Solid geometry questions appear on roughly one-third of ACT Math tests. They require you to know and apply three main formulas. Once you memorize these formulas, you can solve solid geometry questions in 1-2 minutes, gaining quick points.

Spend 20 minutes today drilling these three formulas and the practice problems. By test day, you'll solve any solid geometry question with confidence.