SHSAT: Volume of cylinders, cones and spheres
Find volumes of cylinders, cones and spheres in terms of π or with a given value of π, and predict how changing the radius or height changes the volume.
How do you find the volume of a curved solid, and how do you avoid the radius and diameter mix-up?
Short answer
Find the radius first. Then use πr²h for a cylinder, one third of that for a cone, and (4/3)πr³ for a sphere.
Know first: Squaring and cubing numbers, Multiplying fractions
Step 1 of 5
Cylinders, cones and spheres all start from the radius
All three curved solids use the radius. If a figure shows the diameter, halve it before you do anything else.
The SHSAT guide says formulas and definitions won't be supplied, so learn these three and what each part means. Answers may be written as a multiple of π, such as 60π, so you can often leave π alone.
| Solid | Volume | Remember |
|---|---|---|
| Cylinder | πr²h | base area πr² times height |
| Cone | (1/3)πr²h | one third of the cylinder with the same radius and height |
| Sphere | (4/3)πr³ | the radius is cubed, not squared |
If a question says to use π = 3 or π = 3.14, substitute that value. Otherwise keep π as a symbol.
Step 2 of 5
Square the radius, then apply the fraction
Worked example: Volume of a cone
A cone has radius 3 centimeters and height 8 centimeters. Its volume is kπ cubic centimeters. What is k?
- A8
- B24
- C72
- D96
- 1
Square the radius
3² = 9.
- 2
Multiply by the height
9 × 8 = 72. That would be the cylinder's volume coefficient.
- 3
Take one third
72 ÷ 3 = 24, so the volume is 24π and k = 24.
- 4
See why the other choices are there
8 forgets to square the radius. 72 is the cylinder with the same radius and height. 96 uses the diameter, 6, as the radius.
Answer
k = 24
Find the wrong step
A ball has a diameter of 4 inches. Find its volume in terms of π.
- 1
Use V = (4/3)πr³.
- 2
Substitute r = 4: (4/3)π × 64 = (256/3)π.
What went wrong
4 is the diameter. The radius is half of it, 2.
- 3
The volume is (256/3)π cubic inches.
The fix
With r = 2: (4/3)π × 8 = (32/3)π cubic inches.
Step 3 of 5
Radius first, every time
Check yourself · Question 1
A cylinder has radius 4 inches and height 9 inches. Its volume is kπ cubic inches. What is k?
Answer: C
V = πr²h = π × 16 × 9 = 144π, so k = 144.
Check yourself · Question 2
A cylinder and a cone have the same radius and the same height. The cone's volume is 40 cubic units. How many cubic units greater is the cylinder's volume than the cone's?
Answer: B
The cone is one third of the matching cylinder, so the cylinder holds 3 × 40 = 120. The difference is 120 − 40 = 80.
Trap answer: 120
Answering with the cylinder's volume
- Why it looks right
- Finding the cylinder's volume is the main step, so it feels like the answer.
- Why it's wrong
- The question asks how much greater the cylinder is, which is a difference.
- The right answer
- 80, from 120 − 40.
Check yourself · Question 3
The radius of a sphere is doubled. Its new volume is how many times its old volume?
Answer: D
Volume depends on r³. Doubling r multiplies the volume by 2³ = 8.
Common mistake
I plug in the diameter because that's the number on the figure.
- Why it's tempting
- Figures often label the full width across the top, and it's the only length shown.
- Do this instead
- Before writing any formula, write r = and halve the diameter. Every formula here uses the radius.
Step 4 of 5
What to remember
Remember
- Cylinder πr²h, cone (1/3)πr²h, sphere (4/3)πr³. Halve a diameter first.
- A cone is one third of the cylinder with the same radius and height.
- Volume scales with r²h for cylinders and cones, and with r³ for spheres.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
In your own words
In the practice question, what was the radius, and which wrong choice would you get by using the diameter instead?