What is the difference between slant height and height in a cone?

Height (h) is the perpendicular distance from the center of the circular base straight up to the apex (point) of the cone. Slant height is the distance measured along the side of the cone from the edge of the base to the apex. Slant height is always longer than height. The volume formula uses only the perpendicular height, not the slant height. You need slant height for surface area, not volume.

Why does a cone have a 1/3 in its volume formula but a cylinder doesn't?

A cone with the same radius and height as a cylinder holds exactly one-third as much volume. Imagine filling a cone-shaped cup with water and pouring it into a cylinder-shaped container with the same base and height—you would need to fill and pour the cone three times to fill the cylinder completely. This 1/3 relationship comes from the cone tapering to a point, so less material fits inside compared to a cylinder with the same base dimensions.

Can I use the sphere volume formula if I only know the diameter?

Yes, but you must first convert diameter to radius. Diameter is twice the radius, so r = d/2. For example, if a sphere has a diameter of 10 cm, then r = 5 cm. Then use V = 4/3 r^3 = 4/3 (5)^3. Many students accidentally use the full diameter in the formula, which gives a volume that's 8 times too large.

Do I always have to leave my answer in terms of π?

Not necessarily. If the problem asks you to find an exact answer, leave it in terms of π (like 48 cubic cm). If it asks for a decimal approximation or the answer to a word problem about a real container, multiply by π and round to a reasonable number of decimal places (often two or three). Always check what the problem is asking for and whether your answer makes sense in context.

M8MATH-6.4

Volume of Cylinders, Cones & Spheres

Learn to find the volume of cylinders, cones, and spheres using formulas and solve real-world problems by working backwards from volume to radius or height.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Volume of Cylinders, Cones & Spheres, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

You already know how to find the volume of a rectangular prism using

V = Bh

, where

B

is the area of the base and

h

is the height. This same principle works for curved shapes too. A cylinder is like a prism with a circular base, so you can use the same formula—just replace the base area with the area of a circle. Cones and spheres follow their own formulas, but they're all built on the same idea: volume measures how much space a solid object takes up. In this lesson, you'll learn all three formulas, see why they work, and solve problems where you know the volume and have to find the radius or height.

Volume of a Cylinder

A cylinder has two parallel circular bases connected by a curved side. To find its volume, use the formula

V = Bh

, where

B

is the area of the circular base and

h

is the height.

Since the base is a circle,

B = \pi r^2

. So the cylinder volume formula is:

V = \pi r^2 h

Here,

r

is the radius of the base and

h

is the perpendicular distance between the two circular bases. Think of stacking circular disks on top of each other—the volume tells you how many cubic units fit inside.

When solving problems, make sure the radius and height are in the same units. If a cylinder has radius 3 cm and height 10 cm, then

V = \pi (3)^2 (10) = 90\pi

cubic cm, or about 282.7 cubic cm. Many students forget to square the radius before multiplying by

\pi

and height—that's a key place where mistakes happen.

Volume of a Cone

A cone looks like a cylinder, but it tapers to a point at the top instead of having a flat circular top. The volume of a cone is always one-third the volume of a cylinder with the same base and height:

V = \frac{1}{3}\pi r^2 h

The

\frac{1}{3}

factor is the key difference. If you filled a cone and a cylinder that have the same radius and height with water, you could pour the cone's contents into the cylinder exactly three times.

Common errors: students sometimes forget the

\frac{1}{3}

and use

V = \pi r^2 h

instead, which gives an answer three times too large. Also, make sure you're using the height of the cone (measured perpendicular from the base to the apex), not the slant height, which is the distance along the side. The slant height is longer and should not be used in the volume formula.

Volume of a Sphere

A sphere is a perfectly round solid, like a basketball or a globe. Its volume formula is:

V = \frac{4}{3}\pi r^3

Here,

r

is the radius—the distance from the center to any point on the surface. Notice that the exponent on

r

is 3, not 2, because volume always involves three dimensions.

The

\frac{4}{3}

coefficient comes from calculus, but the important thing to remember is that it appears in every sphere volume problem. A common mistake is to use

\frac{1}{3}

(like in the cone formula) or to forget the coefficient entirely. Also, many students confuse the formula for the surface area of a sphere,

A = 4\pi r^2

, with the volume formula. Surface area has

r^2

; volume has

\frac{4}{3}\pi r^3

Solving Backwards: Finding Radius or Height

Sometimes you're given the volume and need to find the radius or height. This is an algebra problem—you substitute the volume into the formula and solve for the unknown.

For example: A cylinder has volume 150 cubic inches and height 6 inches. Find the radius.

Substitute into

V = \pi r^2 h

150 = \pi r^2 (6)

150 = 6\pi r^2

\frac{150}{6\pi} = r^2

r^2 = \frac{25}{\pi}

r = \sqrt{\frac{25}{\pi}} = \frac{5}{\sqrt{\pi}} \approx 2.82 \text{ inches}

The key steps are: substitute the known values, isolate the variable, and solve. With cones and spheres, the algebra is similar but you must be careful with exponents. For a sphere, you'll need to take the cube root; for a cylinder or cone, you may need to take a square root.

Comparing the Three Formulas

All three formulas measure volume, but they're used for different shapes. A helpful way to see the pattern:

Shape	Formula	Key Feature
Cylinder	$V = \pi r^2 h$	Straight sides, circular base and top
Cone	$V = \frac{1}{3}\pi r^2 h$	Tapers to a point; one-third of a cylinder
Sphere	$V = \frac{4}{3}\pi r^3$	Perfectly round in all directions; $r^3$ instead of $r^2 h$

Notice that the cylinder and cone both use

\pi r^2 h

in structure, but the cone has the

\frac{1}{3}

factor. The sphere is different because it has no separate height—it's defined entirely by its radius, so the formula uses

r^3

.

When solving real-world problems, the first step is always to identify which shape you're dealing with, then use the correct formula.

Key terms

Cylinder.: A solid with two parallel, congruent circular bases connected by a curved surface.
Cone.: A solid with one circular base that tapers to a single point (apex) at the top.
Sphere.: A perfectly round solid where every point on the surface is the same distance (the radius) from the center.
Radius.: The distance from the center of a circle or sphere to any point on its edge or surface.
Height (h).: For a cylinder or cone, the perpendicular distance between the base and the top (or apex). Do not confuse with slant height.
Base (B).: The area of the bottom (or top) surface. For circles, $B = \pi r^2$ .
Volume.: The amount of three-dimensional space occupied by a solid, measured in cubic units.
Slant height.: The distance along the side of a cone from the base to the apex. This is not the same as the perpendicular height used in volume formulas.

Worked example

A cone-shaped water cup has a radius of 4 cm and a height of 9 cm. How much water can it hold? Then, if the cup holds exactly 100 cubic cm of water, what would its height need to be (keeping the radius at 4 cm)?

Part 1: Find the volume of the original cone.

Use the cone volume formula:

V = \frac{1}{3}\pi r^2 h

Substitute

r = 4

and

h = 9

V = \frac{1}{3}\pi (4)^2 (9)

V = \frac{1}{3}\pi (16)(9)

V = \frac{1}{3}\pi (144)

V = 48\pi \approx 150.8 \text{ cubic cm}

So the original cup holds about 150.8 cubic cm.

Part 2: Find the height if volume is 100 cubic cm and radius stays 4 cm.

Substitute

V = 100

and

r = 4

into

V = \frac{1}{3}\pi r^2 h

100 = \frac{1}{3}\pi (4)^2 h

100 = \frac{1}{3}\pi (16)h

100 = \frac{16\pi h}{3}

Multiply both sides by 3:

300 = 16\pi h

Divide by

16\pi

h = \frac{300}{16\pi} = \frac{75}{4\pi} \approx 5.97 \text{ cm}

So the cup would need to be about 5.97 cm tall to hold exactly 100 cubic cm of water.

Practice questions

A cylinder has a radius of 5 inches and a height of 12 inches. What is its volume in cubic inches?

60π
300π
2880π
Not enough information

Answer: 300π

Use the formula

V = \pi r^2 h

. Substitute

r = 5

and

h = 12

V = \pi (5)^2 (12) = \pi (25)(12) = 300\pi

cubic inches. A common wrong answer is 60π, which comes from forgetting to square the radius and computing

\pi \cdot 5 \cdot 12

instead. Another error is 2880π, which results from accidentally using the surface area formula or other confusion.

A sphere has a volume of 288π cubic meters. What is its radius?

6 meters
12 meters
36 meters
72 meters

Answer: 6 meters

Use the sphere volume formula

V = \frac{4}{3}\pi r^3

. Substitute

V = 288\pi

288\pi = \frac{4}{3}\pi r^3

. Divide both sides by

\pi

288 = \frac{4}{3}r^3

. Multiply both sides by

\frac{3}{4}

r^3 = 288 \cdot \frac{3}{4} = 216

. Take the cube root:

r = \sqrt[3]{216} = 6

meters. Students often make errors by dividing by 4 instead of multiplying by

\frac{3}{4}

, or by taking the square root instead of the cube root.

A cone has a volume of 63π cubic cm and a radius of 3 cm. What is its height?

Answer: 21 cm

Use

V = \frac{1}{3}\pi r^2 h

. Substitute

V = 63\pi

and

r = 3

63\pi = \frac{1}{3}\pi (3)^2 h

. Simplify:

63\pi = \frac{1}{3}\pi (9)h = 3\pi h

. Divide both sides by

3\pi

h = 21

cm. A common error is forgetting the

\frac{1}{3}

in the cone formula; that would give

h = 7

cm instead. Always check that you're using the right formula for the right shape.

FAQ

What is the difference between slant height and height in a cone?: Height (h) is the perpendicular distance from the center of the circular base straight up to the apex (point) of the cone. Slant height is the distance measured along the side of the cone from the edge of the base to the apex. Slant height is always longer than height. The volume formula uses only the perpendicular height, not the slant height. You need slant height for surface area, not volume.
Why does a cone have a 1/3 in its volume formula but a cylinder doesn't?: A cone with the same radius and height as a cylinder holds exactly one-third as much volume. Imagine filling a cone-shaped cup with water and pouring it into a cylinder-shaped container with the same base and height—you would need to fill and pour the cone three times to fill the cylinder completely. This 1/3 relationship comes from the cone tapering to a point, so less material fits inside compared to a cylinder with the same base dimensions.
Can I use the sphere volume formula if I only know the diameter?: Yes, but you must first convert diameter to radius. Diameter is twice the radius, so $r = \frac{d}{2}$ . For example, if a sphere has a diameter of 10 cm, then $r = 5$ cm. Then use $V = \frac{4}{3}\pi r^3 = \frac{4}{3}\pi (5)^3$ . Many students accidentally use the full diameter in the formula, which gives a volume that's 8 times too large.
Do I always have to leave my answer in terms of π?: Not necessarily. If the problem asks you to find an exact answer, leave it in terms of π (like $48\pi$ cubic cm). If it asks for a decimal approximation or the answer to a word problem about a real container, multiply by π and round to a reasonable number of decimal places (often two or three). Always check what the problem is asking for and whether your answer makes sense in context.

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