Volume of 3D Solids — ACT Math Guide

Q: What You Need to Know

• **Rectangular prism/cube**: V = length × width × height • **Cylinder**: V = πr²h (circular base area times height) • **Sphere**: V = (4/3)πr³ (four-thirds pi r-cubed) • **Cone**: V = (1/3)πr²h (one-third the volume of a cylinder) • **Pyramid**: V = (1/3)Bh (one-third base area times height) • Alwa

Volume of 3D solids ACT questions test your ability to calculate the space inside three-dimensional shapes like cubes, spheres, cylinders, and pyramids. These problems require you to identify the correct formula and substitute given measurements accurately. Expect 2-3 volume questions on your ACT math section out of the 60 questions you'll tackle in 60 minutes. With solid formula knowledge and practice, these points are completely within your reach.

What You Need to Know

Rectangular prism/cube: V = length × width × height

Cylinder: V = πr²h (circular base area times height)

Sphere: V = (4/3)πr³ (four-thirds pi r-cubed)

Cone: V = (1/3)πr²h (one-third the volume of a cylinder)

Pyramid: V = (1/3)Bh (one-third base area times height)

Always check units — ACT loves mixing feet, inches, and centimeters

Your calculator is allowed for all computations including π calculations

📐 KEY FORMULA: Most volume formulas involve multiplying a base area by height

⏱️ ACT TIME TIP: Memorize the basic formulas cold — don't waste precious seconds deriving them during the 60-minute time limit

How to Solve Volume of 3D Solids on the ACT

Example Question 1 — Easy/Medium Difficulty

A cylindrical water tank has a radius of 4 feet and a height of 10 feet. What is the volume of the tank in cubic feet?

A) 40π

B) 80π

C) 160π

D) 320π

E) 640π

Solution:

Step 1: Identify the shape and formula — cylinder, so V = πr²h

Step 2: Substitute the given values — V = π(4)²(10)

Step 3: Calculate — V = π(16)(10) = 160π cubic feet

✅Answer: C — The volume equals π times radius squared times height.

Example Question 2 — Hard Difficulty

A cone has the same base radius and height as a cylinder. If the cylinder's volume is 150π cubic inches, what is the volume of the cone?

A) 25π

B) 50π

C) 75π

D) 100π

E) 450π

Solution:

Step 1: Recall that cone volume is 1/3 of cylinder volume when they share dimensions

Step 2: Set up the relationship — V_cone = (1/3) × V_cylinder

Step 3: Calculate — V_cone = (1/3) × 150π = 50π cubic inches

✅Answer: B — A cone's volume is exactly one-third that of a cylinder with identical base and height.

Common ACT Math Mistakes to Avoid

❌Mistake: Forgetting to cube the radius in sphere volume calculations

✅Fix: Write out V = (4/3)πr³ and carefully cube the radius value

❌Mistake: Using diameter instead of radius in circular-based solids

✅Fix: Always identify whether you're given radius or diameter, then convert if needed

❌Mistake: Mixing up cone and cylinder formulas

✅Fix: Remember cones and pyramids always have the 1/3 factor

❌Mistake: Forgetting to include π in final answers when working with circles

✅Fix: Double-check that your answer choice format matches your calculation

Practice Question — Try It Yourself

A rectangular swimming pool is 20 feet long, 15 feet wide, and 6 feet deep. How many cubic feet of water does it hold when completely filled?

A) 41

B) 180

C) 900

D) 1,800

E) 3,600

Show Answer

Answer: D — Volume = 20 × 15 × 6 = 1,800 cubic feet

Key Takeaways for the ACT

Memorize the five core volume formulas before test day — you can't afford to derive them during the ACT math section

Always identify the 3D shape first, then select the appropriate formula

Pay close attention to units and whether you need radius or diameter

Use your calculator freely for π calculations and large number multiplication

Remember the 1/3 relationship: cones are 1/3 of cylinders, pyramids are 1/3 of prisms