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Teacher Guide: Volume of Cones

Learn the cone volume formula V = (1/3)πr²h, why a cone holds one third of a cylinder, and how to use the perpendicular height.

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All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Volume & Capacity. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Calculate the volume of a cone using V=13πr2hV = \frac{1}{3}\pi r^2 h, in terms of π\pi and with π≈3.14\pi \approx 3.14
  • Explain why a cone holds one third of the cylinder with the same base and height
  • Tell the perpendicular height apart from the slant height and find hh with the Pythagorean theorem
  • Find a missing height or radius when the volume is known
  • Solve real-world problems about ice cream cones, funnels, cups and piles of material
Prerequisites
  • • Volume of cylinders (V=πr2hV = \pi r^2 h)
  • • Area of a circle (A=πr2A = \pi r^2) and the link between radius and diameter
  • • Squares and square roots
  • • The Pythagorean theorem (for the slant height problems)
Discussion Starters
  • 1. Before the pour test: how many cones do you think it takes to fill the matching cylinder? Why?
  • 2. Why is the slant height of a cone always longer than its height?
  • 3. If you double the height of a cone, what happens to its volume? What if you double the radius?
  • 4. An ice cream shop sells a cone and a cup with the same radius and height for the same price. Which is the better deal?
  • 5. Where else have you seen the fraction one third in a volume formula?
Common Misconceptions

A cone should be half of a cylinder, because it looks like it is cut in half

Remediation: Do the pour test. Then point out that the cone is narrow near the top, so it misses much more than half of the cylinder's space.

The slant height and the height are the same thing

Remediation: Draw the right triangle inside the cone and label rr, hh and ll. Show that ll is the hypotenuse, so it must be the longest side.

Doubling the radius doubles the volume

Remediation: Compute r=2r = 2 and r=4r = 4 with the same height: the volume goes from 4h3π\frac{4h}{3}\pi to 16h3π\frac{16h}{3}\pi, which is 4 times as much, because the radius is squared.

Differentiation Ideas

For Struggling Students:

  • • Give the radius directly and use whole numbers that make r2hr^2 h divisible by 3
  • • First compute the matching cylinder volume, then divide by 3
  • • Provide a labeled cone diagram showing rr, hh and ll

For On-Level Students:

  • • Mix problems that give the diameter or the radius
  • • Give answers both in terms of π\pi and with π≈3.14\pi \approx 3.14, with stated rounding
  • • Find a missing height from the volume and the radius

For Advanced Students:

  • • Find the height from the slant height with the Pythagorean theorem
  • • Find a missing radius from the volume (square roots)
  • • Explore how the volume changes when the radius or height is scaled by 2 or 3
Standards Alignment
  • 8.G.C.9 (CCSS.MATH.CONTENT.8.G.C.9)

    Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems

  • 8.G.B.7 (CCSS.MATH.CONTENT.8.G.B.7)

    Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions

  • HSG.GMD.A.3 (CCSS.MATH.CONTENT.HSG.GMD.A.3)

    Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems

Lesson Resources
  • visual3D Cone and Cylinder Viewer

    Change the radius and height of a cone and a cylinder side by side and watch the volume of the cone stay at one third

  • activityPour Test

    Students fill a paper cone with rice or water and count how many cones it takes to fill a cylinder with the same base and height

  • worksheetCones in Real Life

    Volume problems about waffle cones, funnels, paper cups and sand piles, including slant height and missing dimension questions

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A cone is a 3D shape with a flat circular base that narrows to a single point called the apex (or vertex). Think of an ice cream cone, a party hat or a traffic cone.

The Volume Formula

V=13πr2hV = \frac{1}{3}\pi r^2 h
Where:
  • rr = radius of the circular base
  • hh = height, the perpendicular distance from the apex straight down to the center of the base
  • πr2\pi r^2 = the area of the base

Why One Third?

In the cylinder lesson you learned V=πr2hV = \pi r^2 h. Take a cylinder and a cone with the same base and the same height. If you fill the cone with water and pour it into the cylinder, it takes exactly 3 full cones to fill the cylinder. So the cone holds one third of the cylinder:
Vcone=13×Vcylinder=13πr2hV_{\text{cone}} = \frac{1}{3} \times V_{\text{cylinder}} = \frac{1}{3}\pi r^2 h
Careful: the height hh is not the slant height ll (the length along the sloping side). The radius, the height and the slant height form a right triangle, so l2=r2+h2l^2 = r^2 + h^2.

Worked Examples

A waffle cone has a radius of 3 cm and a height of 10 cm. Find its volume in terms of π\pi, then with π≈3.14\pi \approx 3.14, rounded to the nearest tenth.
1

Write the formula

V=13πr2hV = \frac{1}{3}\pi r^2 h → Formula ready
2

Square the radius

r2=32=9r^2 = 3^2 = 9 → r2=9r^2 = 9
3

Multiply by the height and divide by 3

13×9×10=903=30\frac{1}{3} \times 9 \times 10 = \frac{90}{3} = 30 → V=30πV = 30\pi cm³
4

Use π≈3.14\pi \approx 3.14

30×3.14=94.230 \times 3.14 = 94.2 → V≈94.2V \approx 94.2 cm³

Common Mistakes

Forgetting the 13\frac{1}{3} and using V=πr2hV = \pi r^2 h

Why it's wrong: That is the cylinder formula. A cone with the same base and height holds only one third of the cylinder, so the answer comes out 3 times too big.

Correct: Always write V=13πr2hV = \frac{1}{3}\pi r^2 h and divide by 3. For r=3r = 3, h=10h = 10: 13π×9×10=30π\frac{1}{3}\pi \times 9 \times 10 = 30\pi, not 90π90\pi.

Using the diameter instead of the radius

Why it's wrong: The radius is half the diameter. Squaring the diameter makes the answer 4 times too big.

Correct: If the problem gives the diameter, halve it first: a diameter of 8 cm means r=4r = 4 cm and r2=16r^2 = 16, not 6464.

Using the slant height as the height

Why it's wrong: The slant height runs along the sloping side, so it is always longer than the perpendicular height. Using it makes the volume too big.

Correct: The height goes straight from the apex down to the center of the base. If you only know the slant height ll, find hh with h2=l2−r2h^2 = l^2 - r^2.

Writing the answer in square units, like cm²

Why it's wrong: Volume is three-dimensional: r×r×hr \times r \times h multiplies three lengths.

Correct: Volume is always in cubic units: cm³, m³, in³. Remember that 11 cm³ =1= 1 mL.

Why It Matters

Cones are all around you, and knowing their volume answers real questions:
  • Food: How much ice cream fits in a waffle cone? How much popcorn does a paper cone hold?
  • Kitchen and garage: How much liquid sits in a funnel before it drains?
  • Construction: How many cubic meters of sand, gravel or road salt are in a cone-shaped pile?
  • Industry: Grain silos and hoppers often have a cone-shaped bottom, so engineers need cone volumes to know their capacity.
  • Nature: Volcanoes and anthills are roughly cone-shaped, so scientists estimate their volume with this formula.
The one third rule also comes back in the pyramid lesson, so learning it here gives you two formulas for the price of one!

Real World Applications

Ice Cream Shops

Ice cream shops choose cone sizes and prices by how much ice cream fits inside each cone.

Example:

A small cone with radius 2 cm and height 9 cm holds 13π×4×9=12π≈37.68\frac{1}{3}\pi \times 4 \times 9 = 12\pi \approx 37.68 cm³.

1Try It Yourself

A large waffle cone has a radius of 3 cm and a height of 12 cm. The shop fills it right up to the rim.

How many cubic centimeters of ice cream fit inside? Use 3.14 for π\pi.

Step 1: Write the mathematical expression

Calculate 13×3.14×32×12\frac{1}{3} \times 3.14 \times 3^2 \times 12:

Paper Cups at a Water Dispenser

Many office water dispensers use cone-shaped paper cups. They are cheap to make and easy to stack.

Example:

A paper cone cup with radius 3 cm and height 10 cm holds 30π≈94.230\pi \approx 94.2 cm³, which is about 9494 mL of water.

2Try It Yourself

A cone-shaped paper cup has a radius of 4 cm and a height of 9 cm.

How many milliliters of water does it hold? Use 3.14 for π\pi (1 cm³ = 1 mL).

Step 1: Write the mathematical expression

Calculate 13×3.14×42×9\frac{1}{3} \times 3.14 \times 4^2 \times 9:

Piles of Sand and Road Salt

When sand, gravel or road salt is poured from above, it forms a cone. Builders and road crews use the cone formula to estimate how much material is in a pile.

Example:

A sand pile with radius 3 m and height 2 m contains 6π≈18.846\pi \approx 18.84 m³ of sand.

3Try It Yourself

A pile of road salt for the winter forms a cone with a diameter of 10 m and a height of 3 m.

How many cubic meters of salt are in the pile? Use 3.14 for π\pi.

Step 1: Write the mathematical expression

Halve the diameter, then calculate 13×3.14×r2×3\frac{1}{3} \times 3.14 \times r^2 \times 3:

Key Takeaways

  • 1The volume of a cone is V=13πr2hV = \frac{1}{3}\pi r^2 h, one third of base area times height
  • 2A cone holds exactly one third of a cylinder with the same base and the same height
  • 3Use the radius, not the diameter: halve the diameter before squaring
  • 4The height hh is the perpendicular height, not the slant height; if you know the slant height ll, use h2=l2−r2h^2 = l^2 - r^2
  • 5Give exact answers in terms of π\pi (like 30π30\pi cm³) or approximate ones with π≈3.14\pi \approx 3.14, rounded as the problem asks
  • 6To find a missing height, use h=3Vπr2h = \frac{3V}{\pi r^2}; to find a missing radius, use r2=3Vπhr^2 = \frac{3V}{\pi h} and take the square root

Frequently Asked Questions

Why is it exactly one third, and not one half?

A cone gets narrower all the way up, so most of its volume sits near the base. You can test it with water or rice: three full cones fill a cylinder with the same base and height. Later, in calculus, you can prove that the factor is exactly 13\frac{1}{3}. Pyramids follow the same one third rule.

What if the cone is tilted (an oblique cone)?

The formula still works, as long as hh is the perpendicular height from the apex to the plane of the base. A tilted cone with the same base and the same perpendicular height has the same volume as a straight one.

When should I leave the answer in terms of π\pi?

When the problem asks for an exact answer or says "in terms of π\pi", write it like 48π48\pi cm³. When you need a number for a real situation, multiply by 3.143.14 (or use the π\pi key) and round to the place the problem asks for.

How do I find the height if I only know the slant height?

The radius, the height and the slant height make a right triangle with the slant height as the hypotenuse. So h=l2−r2h = \sqrt{l^2 - r^2}. For r=5r = 5 and l=13l = 13: h=169−25=144=12h = \sqrt{169 - 25} = \sqrt{144} = 12.

Glossary

Cone
A 3D shape with a circular base and curved sides that meet at one point, the apex
Apex
The pointed tip of a cone (also called the vertex)
Radius
The distance from the center of the circular base to its edge, half the diameter
Height (perpendicular height)
The straight-line distance from the apex to the center of the base, at a right angle to the base
Slant height
The distance from the apex to the edge of the base along the sloping side; l2=r2+h2l^2 = r^2 + h^2
In terms of π\pi
An exact answer that keeps the symbol π\pi, like 30π30\pi cm³, instead of a rounded decimal

Formula Card

Volume of a cone

V=13πr2hV = \frac{1}{3}\pi r^2 h

One third of base area ($\pi r^2$) times perpendicular height

Cone and cylinder

Vcone=13VcylinderV_{\text{cone}} = \frac{1}{3} V_{\text{cylinder}}

Same base and same height: three cones fill one cylinder

Missing height

h=3Vπr2h = \frac{3V}{\pi r^2}

Multiply the volume by 3, then divide by the base area

Height from slant height

h=l2−r2h = \sqrt{l^2 - r^2}

Pythagorean theorem in the right triangle formed by $r$, $h$ and $l$

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