Math 6

Cylinders and cones

A soup can and a traffic cone both stand on a circle. What are their parts called, and why are they not counted with boxes and pyramids?

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What a circle isMath 6

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Geometry11

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Cylinders and cones

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Why this exists

Boxes and pyramids are easy to describe: count the flat faces, measure the straight edges. A soup can, a hockey puck and a traffic cone have no straight edges round their sides. They are round, and each is built on a circle.

To describe these shapes, compare them, and later work out how much they hold, we need names for their parts and a clear sense of what each measurement means. Two solids cover many of them: the cylinder and the cone.

Definition

A cylinder is a solid with two bases that are circles of the same size, lying in parallel planes, joined by a curved surface. The curved surface is made of straight segments, all parallel to each other, each joining a point on the edge of one base to a point on the edge of the other.

Figure 1A cylinder has two equal circular bases, one curved surface, and an axis joining the centers.

The radius r of the cylinder is the radius of either base. Its diameter is d = 2r.
The axis is the segment joining the centers of the two bases.
The height h is the distance between the two bases, measured at right angles to both.

A cone is a solid with one circular base and one point, the apex, that is not in the plane of the base. Its curved surface is made of all the straight segments from the apex to the edge of the base.

Figure 2The height h drops straight down inside the cone. The slant height \ell runs along the outside.

The height h is the distance from the apex straight down to the plane of the base.
The slant height \ell is the distance from the apex to the edge of the base, measured along the curved surface.

Right and oblique. A cylinder is right when its axis is at right angles to its bases, so one base sits straight above the other. Otherwise it is oblique. A cone is right when its apex is straight above the center of its base, and oblique when it is not. This is the same idea as for prisms and pyramids, with a circle where the polygon was.

Figure 3Sliding the top base sideways leaves the height the same. Only the axis leans.

Figure 4In an oblique cone the apex is not above the center, and the height can land outside the base.

Intuition

Stack identical hockey pucks straight up and the stack is a taller cylinder. Every puck is the same circle, so a cylinder looks the same all the way up. Now push the stack so it leans. Each puck still lies flat, and the top puck is still the same distance above the table, so the height has not changed. Only the side slants: the stack is very nearly an oblique cylinder (its side has tiny steps, one per puck, which a true cylinder smooths away).

A cone works the other way. Move up from the base and the circles shrink steadily until, at the apex, they are a single point. That is why a traffic cone is wide at the bottom and comes to a tip.

A cylinder is to a prism what a cone is to a pyramid. Replace the polygon base with a circle and the flat side faces melt into one curved surface.

Results

Slices. Every slice of a cylinder parallel to its bases is a circle the same size as the bases.

Height and slant height. In a right cone, the slant height is longer than the height: \ell > h. In an oblique cylinder, each straight segment of the curved surface is longer than the height.

Not polyhedra. Neither a cylinder nor a cone is a polyhedron.

Views. Look straight at a solid from above, from the front and from the side, and draw only its outline. For a right cylinder standing on a base, the top view is a circle of diameter d and the front and side views are rectangles d wide and h tall. For a right cone standing on its base, the top view is a circle of diameter d with the apex seen as a point at its center, and the front and side views are triangles with a base of d and a height of h.

Figure 5Flat views of a standing cylinder and a standing cone, each seen straight on.

Why it is true

Slices of a cylinder. A cylinder is what a circle sweeps out when it slides from one base to the other without turning or tilting. At every level the slice is that same circle, moved along.

Why the slanted lengths are longer. The height of a cone is the shortest way from the apex to the flat plane of the base, because straight down is the shortest way to any floor. The slant height makes the same trip at a slant, so \ell > h. In the same way, a segment on the side of an oblique cylinder crosses from base to base at a slant, so it is longer than the height. In a right cylinder those segments run straight across, and each has length exactly h.

Not polyhedra. A polyhedron is a solid whose surface is made entirely of flat polygons. A circle is not a polygon, since it has no straight sides, and the curved surface is not flat. So neither solid qualifies, however large or small.

Views. From straight above, a standing cylinder’s two bases land on the same circle. From the front, it is a diameter wide, its sides rise straight up and its ends are level, so the outline is a rectangle. A cone’s sides meet at the apex, so from the front its outline is a triangle.

Worked examples

Example 1

Naming the parts of a can

Zarina’s soup can is 10 cm tall and 7 cm across the top. Name the solid, say how many bases it has, and give its height, diameter and radius.

Solution

The can has a flat circular top and bottom of the same size, one directly above the other, joined by a round side. That is a right cylinder, with 2 bases.

The height is the distance between the bases, h = 10 cm. “Across the top” is the diameter of a base, so d = 7 cm, and the radius is half the diameter:

r = 7 \div 2 = 3.5 \text{ cm}

Example 2

A cylinder does not have to be tall

Aleksy’s hockey puck is 7.6 cm across and 2.5 cm thick. Is it a cylinder? Compare it with Zarina’s can: which is wider, which is taller, and by how much?

Solution

The puck has two equal flat circular faces, one straight above the other, joined by a curved side. So it is a right cylinder lying on a base, with d = 7.6 cm and h = 2.5 cm. Nothing in the definition says the height must be bigger than the diameter.

Diameters tell which is wider: 7.6 - 7 = 0.6, so the puck is wider by 0.6 cm. Heights tell which is taller: 10 - 2.5 = 7.5, so the can is taller by 7.5 cm. The puck’s radius is 7.6 \div 2 = 3.8 cm.

Example 3

Height or slant height?

Danika measures a traffic cone, ignoring its square foot. Its base is 36 cm across. Her notebook says 80 cm and 82 cm, but she forgot which was the height and which the slant height. Sort them out, and give the radius.

Solution

In a right cone the slant height is longer than the height, because the height runs straight down to the base plane, the shortest way there. So the shorter measurement is the height:

h = 80 \text{ cm} \qquad \ell = 82 \text{ cm}

The radius is half the diameter: r = 36 \div 2 = 18 cm.

Example 4

A leaning stack of pucks

Frida stacks 6 of Aleksy’s pucks straight up, then gently pushes the stack so it leans. What solid is each stack very nearly, and how tall is each?

Solution

The straight stack has circular bases one directly above the other: a right cylinder. Each puck is 2.5 cm thick, so

h = 6 \times 2.5 = 15 \text{ cm}

In the leaning stack each puck still lies flat on the one below, so the top is still 6 pucks above the table. It is very nearly an oblique cylinder, and its height is still 15 cm. Only the side changed: it slants, so a straight segment on the side of that oblique cylinder is longer than 15 cm.

Example 5

Views of a can on its side

Larissa lays a soup can on its side, with its axis running from left to right as she faces it. The can is 12 cm long and 8 cm across. Describe the top, front and side views.

Solution

From the front she sees the curved side face on: as long as the can, 12 cm, and as tall as the can is wide, 8 cm, with straight ends where the bases are seen edge on. The front view is a rectangle 12 cm by 8 cm. From above she sees the same: another rectangle 12 cm by 8 cm.

From the side she looks straight at one base, so the side view is a circle 8 cm across. For a standing can the circle was the top view: which view shows the circle depends on where the axis points.

Example 6

Naming a solid from its views

A solid’s top view is a circle 30 cm across with a dot at its center. Its front and side views are triangles with a base of 30 cm and a height of 45 cm. Name the solid and give its radius and height.

Solution

A circle from above and a triangle from the front and side is the outline of a cone standing on its base. The dot is the apex seen from above, and it sits at the center, so the apex is straight above the center: a right cone. A square-based pyramid would show a square from above.

The width of each triangle is the diameter, so r = 30 \div 2 = 15 cm. The height of the triangle is the height of the cone, h = 45 cm.

Boundaries

The bases must be circles. A solid with two equal parallel polygon bases is a prism, not a cylinder, and a solid with an oval base is not a cone in the sense of this lesson.

A cylinder’s bases must be the same size and parallel. A bucket has a circular top and bottom of different sizes; it is a cone with its tip sliced off, not a cylinder.

Slant height belongs to the right cone. In an oblique cone the distance from the apex to the edge of the base changes as you go round, so there is no single slant height. The height makes sense for every cone and every cylinder.

Real objects are close, not exact. Cans have rims and traffic cones have a square foot. The solids here are the ideal shapes such objects approximate.

Where this goes next

Once the parts have names, they can be measured. The circle that forms a base has a Circumference and an area, and the volume of a cylinder is the base’s area times the height, which works for oblique cylinders too, for the same reason the leaning stack kept its height. Unrolling the curved surface flat is the idea behind nets of solids, and slicing solids in other directions is taken up in cross sections. The round solid with no base at all is the sphere.

Exercises

Parts and measurements

A soup can is 11 cm tall and its base has radius 4.5 cm. Find its diameter. Which is greater, the height or the diameter, and by how much?
A small game puck has radius 2.9 cm and is 1.8 cm thick. Find its diameter, and say how much wider it is than it is thick.
The base of a traffic cone is 34 cm across. What is its radius?
Can A has base radius 3.9 cm and height 10.2 cm. Can B has base diameter 7.4 cm and height 11.3 cm. Which can is wider, and by how much? Which is taller, and by how much?
How many bases and how many curved surfaces does a cylinder have? A cone? How many apexes does each have?

Height, slant height and leaning

A right cone has a height and a slant height of 60 cm and 61 cm, in some order. Which is the height? Its base radius is 11 cm; what is its diameter?
Could a right cone have a height of 30 cm and a slant height of 25 cm? Explain.
In a cylinder, the straight segments of the curved surface are 26 cm long and the bases are 24 cm apart. What is the height? Is the cylinder right or oblique?
A straight line drawn on the curved surface of a right cylinder, from one base to the other, is 14.5 cm long. How tall is the cylinder?
9 pucks, each 2.4 cm thick, are stacked straight, and then a second stack of 9 is pushed so it leans. How tall is each stack, and what solid is each very nearly?

Views

A puck 7.2 cm across and 2.2 cm thick lies flat on a table. Describe its top, front and side views, with their sizes.
A traffic cone 64 cm tall with a base 32 cm across stands on the ground. Describe its top and front views, with their sizes.
A solid’s top view is a circle 9 cm across. Its front and side views are rectangles 9 cm wide and 4 cm tall. Name the solid and give its radius and height.
A solid’s top view is a circle with a dot at its center. Its front and side views are triangles 20 cm wide and 33 cm tall. Name the solid, give its radius and height, and say whether it is right or oblique.
A soup can 9.6 cm long and 6.6 cm across lies on its side with its axis pointing straight at you. Describe its front view and its top view.
A solid’s front and side views are rectangles and its top view is a square. Could it be a cylinder standing on its base, or lying on its side with its axis pointing left to right or straight at you? Explain.

Putting it together

A cylinder’s diameter is 14 cm and its height is 3 times its radius. How tall is it?
A right cone standing on its base has a front view that is a triangle 18 cm wide and 25 cm tall. What is its radius? Could its slant height be 25 cm?
A cone’s apex is straight above a point 3 cm from the center of its base. Is it right or oblique? Does it have a single slant height?
A soup can 12.4 cm tall with base radius 3.1 cm is sliced level, 5 cm above its bottom. What shape is the slice, and how wide is it?

Reasoning

A cylinder and a prism both have two equal, parallel bases. Explain why a cylinder is not a prism.
Explain why no cone, of any size, is a polyhedron.
Explain why the straight segments on the curved surface of an oblique cylinder are longer than its height.
Can the top view and the front view of a right cylinder both be circles? Explain.

Answers

d = 9 cm, and the height is greater by 2 cm: 2 \times 4.5 = 9 and 11 - 9 = 2.
d = 5.8 cm, and it is 4 cm wider than thick: 2 \times 2.9 = 5.8 and 5.8 - 1.8 = 4.
17 cm, half the diameter: 34 \div 2 = 17.
Can A is wider by 0.4 cm and Can B is taller by 1.1 cm. Can A’s diameter is 2 \times 3.9 = 7.8, and 7.8 - 7.4 = 0.4; 11.3 - 10.2 = 1.1.
A cylinder has 2 bases, 1 curved surface and no apex. A cone has 1 base, 1 curved surface and 1 apex.
The height is 60 cm, since the slant height is the longer one, and the diameter is 22 cm: 2 \times 11 = 22.
No. A right cone’s slant height is always longer than its height, and 25 is less than 30.
The height is 24 cm, and the cylinder is oblique. In a right cylinder the side segments would also be 24 cm; longer segments mean the side slants.
14.5 cm. In a right cylinder each side segment runs straight across, so its length is the height.
Both are 21.6 cm tall: 9 \times 2.4 = 21.6. The straight stack is very nearly a right cylinder, the leaning one an oblique cylinder; leaning does not change the height.
Top view: a circle 7.2 cm across. Front and side views: rectangles 7.2 cm wide and 2.2 cm tall.
Top view: a circle 32 cm across with the tip as a dot at its center. Front view: a triangle with a base of 32 cm and a height of 64 cm.
A cylinder standing on its base, with radius 4.5 cm and height 4 cm: 9 \div 2 = 4.5.
A right cone with radius 10 cm and height 33 cm: 20 \div 2 = 10, and the center dot puts the apex straight above the center.
Front view: a circle 6.6 cm across, since you look straight at a base. Top view: a rectangle 6.6 cm wide and 9.6 cm from front to back.
No. Standing, its top view is a circle; lying left to right, its side view is a circle; pointing at you, its front view is a circle. Here no view is a circle. A box with a square top would fit.
21 cm. The radius is half the diameter, 14 \div 2 = 7, and 3 \times 7 = 21.
The radius is 9 cm: the triangle’s width is the diameter, and 18 \div 2 = 9. No: the triangle’s height is the cone’s height, 25 cm, and the slant height is longer than that.
Oblique, since the apex is not straight above the center. It has no single slant height: the distance from the apex to the edge of the base changes as you go round.
A circle 6.2 cm across, the same size as the bases: 2 \times 3.1 = 6.2. Every level slice of a cylinder matches its bases.
A prism’s bases are polygons and its sides are flat faces. A cylinder’s bases are circles, which have no straight sides, and its side is one curved surface.
A polyhedron’s surface is made entirely of flat polygons. A cone’s base is a circle, not a polygon, and its curved surface is not flat. Neither fact depends on size.
The height is the shortest way from the top base to the plane of the bottom base, since straight down is the shortest way to a floor. Each side segment makes that trip at a slant, so it is longer.
No. The top view is a circle only when the axis points straight up, and then the front view is a rectangle. If the axis points straight at you the front view is a circle, but the top view is a rectangle.