E. Geometry and Measurement · Geometry and Measurement: Circles, Triangles, Measurement Systems, Pythagoras and Volume

The Volume of Cylinders and Cones

E1.6 E1.5 E1.3 A1 A2About 70 minutes

Learning goals

I can find the volume of a cylinder with V = πr²h.
I can explain why the volume of a cone is one-third of the volume of a cylinder with the same radius and height.
I can find the volume of a cone, using the Pythagorean theorem when I know the slant height.
I can find a missing height or radius and solve problems with mixed units and composite objects.

Words to know

cylinder
A 3D object with two parallel, identical circular bases joined by a curved surface.

cone
A 3D object with a circular base and a curved surface that narrows to a point (the apex).

radius (r)
The distance from the centre of a circle to its edge: half the diameter.

height (h)
For a cylinder, the distance between the bases; for a cone, the perpendicular distance from the base to the apex.

slant height (s)
For a cone, the distance from the apex down the side to the edge of the base: s² = r² + h².

composite object
An object made of two or more simpler solids, such as a cylinder with a cone on top.

A hook: the road salt pile

In winter, many Ontario towns store road salt in huge piles. When salt is poured from above, it forms a cone. The works department needs to know how much salt is in the pile to decide whether to order more before the next snowstorm.

You can't count grains of salt, but you can measure the pile's width and height and use geometry.

Cylinders: base area × height

A cylinder is like a prism with a circular base. B = πr², so

V = πr²h

Cones: one-third again

A cone is to a cylinder what a pyramid is to a prism. A cone with the same radius and height as a cylinder holds exactly one-third as much:

V = ⅓πr²h

You can test it by filling a paper cone with rice and pouring it into a cylinder with the same radius and height: it takes 3 cones.

A cylinder and a cone with the same radius and height. The cone holds one-third as much.

Three cones with the same radius and height fill one cylinder.

Worked example

Example 1: A can

A can has a diameter of 6.6 cm and a height of 12 cm. Find its volume in mL, to one decimal.

r = 6.6 ÷ 2 = 3.3 cm.
V = π × 3.3² × 12 = π × 10.89 × 12 = 130.68π ≈ 410.5 cm³.
1 cm³ = 1 mL.

About 410.5 mL

Worked example

Example 2: The cone with the same size

Find the volume of a cone with the same radius and height as the can.

V = ⅓ × 130.68π = 43.56π ≈ 136.8 cm³.

About 136.8 cm³ (one-third of 410.5)

A road salt pile shaped like a cone, drawn to scale. Measure the diameter across the bottom and the height to the top.

Worked example

Example 3: The salt pile

A cone-shaped salt pile is 12 m across and 4 m high. Find its volume.

r = 12 ÷ 2 = 6 m.
V = ⅓ × π × 6² × 4 = ⅓ × π × 144 = 48π.
48π ≈ 150.8 m³.

About 150.8 m³

Worked example

Example 4: A paper water cup

A cone-shaped paper cup has a radius of 4 cm and a slant height of 10 cm. How many mL does it hold, to the nearest mL?

Find the height: h = √(10² − 4²) = √84 ≈ 9.165 cm.
V = ⅓ × π × 4² × 9.165 = ⅓ × π × 146.64 ≈ 153.6 cm³.
153.6 cm³ ≈ 154 mL.

About 154 mL

Worked example

Example 5: Missing height

A cylindrical jug with a radius of 6 cm must hold 2 L. How tall must it be (inside)?

2 L = 2000 cm³.
2000 = π × 6² × h = 36πh.
h = 2000 ÷ (36π) ≈ 17.7 cm.

About 17.7 cm

Worked example

Example 6: A grain bin

A grain bin is a cylinder with radius 3 m and height 6 m, with a cone-shaped roof 2 m high. Find the total volume.

Cylinder: π × 3² × 6 = 54π m³.
Cone: ⅓ × π × 3² × 2 = 6π m³.
Total: 60π ≈ 188.5 m³.

About 188.5 m³

Watch out!

Use the radius, not the diameter. Halve the diameter first.
Square only the radius: πr²h means π × r × r × h.
For a cone, use the height, not the slant height. Find h with s² = r² + h² if needed.
Keep π (or many decimal places) until the last step, then round.

Careers

Civil engineers and public works staff estimate volumes of salt, gravel and sand piles. Farmers use bin volumes to plan storage. Chemists and pharmacists use cone-shaped and cylindrical containers and convert between cm³ and mL every day.

Hands-on activity

Three cones, one cup

You need: a straight-sided cylindrical cup or can (with no sharp edges), paper, tape, dry rice, a ruler, and scissors (a grown-up should help with cutting).

Measure the inside radius r and height h of the cup.
Make a paper cone with the same radius and height. Hint: the slant height must be s = √(r² + h²). Draw a circle with radius s, cut a wedge out, and tape it into a cone. Trim the top edge until the cone's opening matches the cup and its height matches h.
Fill the cone with rice (level), pour it into the cup, and repeat. How many cones fill the cup?
Calculate both volumes with the formulas and compare.

Model answer: a cup with r = 4 cm and h = 9 cm has V = π × 16 × 9 ≈ 452 cm³; the cone has s = √(16 + 81) ≈ 9.8 cm and V ≈ 151 cm³. About 3 cones fill the cup.

Just for fun

Archimedes was so proud of discovering that a sphere holds exactly ⅔ of the cylinder that fits around it that, according to the Roman writer Cicero, his tomb was marked with a sphere inside a cylinder. You will study spheres in later grades.

Your turn

Find the volume of a cylinder with r = 5 cm and h = 10 cm.

Find the volume of a cone with r = 5 cm and h = 12 cm.

A cylinder holds 900 mL. How much does a cone with the same radius and height hold?

A cone has a diameter of 10 cm and a height of 9 cm. Find its volume.

A cylinder with r = 5 cm holds 1 L. How tall is it, to two decimals?

A cone has r = 6 cm and slant height 10 cm. Find its volume.

A glass is a cylinder with r = 3.5 cm and h = 10 cm. How many mL does it hold?

More practice: warm-up

Volume formula for a cylinder?

A cone holds what fraction of a matching cylinder?

Diameter 10 cm. Radius?

1000 cm³ = ? L

π to two decimal places?

More practice: on your own

A funnel is a cone with r = 7.5 cm and h = 15 cm. Find its volume.

A round above-ground pool has a diameter of 4.6 m and is filled to a depth of 1.2 m. How many litres of water is that, to the nearest 10 L?

A cone has a volume of 50π cm³ and a radius of 5 cm. Find its height.

A pencil is a cylinder with r = 0.4 cm and length 15 cm, with a cone tip 1.5 cm long. Find its volume.

A cone-shaped cup with r = 3 cm holds 100 mL. How tall is it, to two decimals?

Challenge

A cone-shaped salt pile has a radius of 10 m and a height of 6 m. A truck carries 12 m³. How many truckloads are in the pile?

A cone and a cylinder have the same volume and the same radius. The cone is 24 cm tall. How tall is the cylinder?

A cone's radius is doubled and its height is halved. What happens to its volume?

Check your understanding

Quiz

A cylinder has r = 2 cm and h = 5 cm. Its volume is:

A cylinder holds 1.2 L. A cone with the same radius and height holds:

A cone has a diameter of 6 cm and a height of 10 cm. Its volume is:

A cone has r = 3 cm and slant height 5 cm. What height do you use in the volume formula?

Which change increases a cylinder's volume the most?

A cylindrical tank holds 3.14 m³. About how many litres is that?

For parents and teachers

Big idea: This lesson completes E1.6: a cone holds one-third of a cylinder with the same radius and height, just as a pyramid holds one-third of a prism. Students use various units (cm³, mL, L, m³) and the Pythagorean theorem for slant heights.

Common difficulties:

Using the diameter in πr²h.
Using the slant height as the height of a cone.
Rounding π too early. Encourage using the π key.

How to help: Measure cans and glasses at home and compare your calculations with the mL on the labels (labels show the amount of product, which is usually a little less than the full can). Supervise cutting in the activity.