SPI Mathematics · Mensuration 3D

Cylinders and Cones

A cylinder is a circle carried straight up through a height; a cone on the same base and height holds exactly one third of it.

All Mensuration 3D notes

Why this matters

Fifteen questions in the bank, the largest page of the chapter: ten EASY, four MODERATE and one HARD (the canvas for a conical tent). Most are one formula with π as 22/7, or the one-third link between a cone and a cylinder written as a ratio.

Concept 1 of 4: Volume and surface of a cylinder

A cylinder is a stack of equal circles. Its volume is one circle's area times the height. Unroll the side and it is a rectangle, the circumference wide and the height tall.

Definition

Volume $= \pi r^2 h$ . A stack of coins or plates is a cylinder whose height is the thicknesses added up.
Curved surface $= 2\pi rh$ ; total surface of a closed cylinder adds two circles: $2\pi r(r + h)$ .
Take $\pi = \frac{22}{7}$ when the radius is a multiple of 7, unless the question gives 3.14.
If $r + h$ is given with the total surface, put it straight into $2\pi r(r + h)$ and solve for $r$ .
With the height fixed, volume goes as $r^2$ : halve the radius and the volume becomes one quarter.

Cylinder

V = \pi r^2 h, \quad \text{CSA} = 2\pi rh, \quad \text{TSA} = 2\pi r(r + h)

rradius of the base
hheight

Worked example

A closed cylinder has radius 7 cm and height 10 cm. Find its volume and its total surface area.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Mensuration 3D · Cylinders and Cones

The sum of the radius of the base and the height of a solid cylinder is 37 cm. If the total surface area of the solid cylinder is 1628

\text{cm}^2

. Then the volume of the cylinder is

Curved or total surface

An open pipe or the label on a tin is the curved surface

2\pi rh

alone. A closed solid cylinder adds its two circles:

2\pi r(r + h)

Halving the radius does not halve the volume

The radius is squared in

\pi r^2 h

. Cutting the radius by 50% leaves

\left(\frac{1}{2}\right)^2 = \frac{1}{4}

of the volume, so the ratio is 1 : 4, not 1 : 2.

Drill 3 more on volume and surface of a cylinder

Concept 2 of 4: Volume and surface of a cone

Cut a cone straight down its middle and you see a right triangle: the radius along the bottom, the height up the middle, and the slant height as the sloping side. Every cone question uses that triangle.

Definition

Slant height: $l^2 = r^2 + h^2$ . The slant height $l$ is always the longest of the three.
Volume $= \frac{1}{3}\pi r^2 h$ uses the vertical height $h$ .
Curved surface $= \pi rl$ uses the slant height; total surface adds the base: $\pi r(l + r)$ .
A conical tent needs canvas for the curved surface only. Its floor is the circle $\pi r^2$ , so a given floor area gives $r$ . Length of canvas $=$ area $\div$ width of the canvas.

Cone

l^2 = r^2 + h^2, \quad V = \frac{1}{3}\pi r^2 h, \quad \text{CSA} = \pi rl, \quad \text{TSA} = \pi r(l + r)

rradius of the base
hvertical height
lslant height

Worked example

A cone has radius 6 cm and height 8 cm. Find its slant height, curved surface area and volume. (Use π = 3.14.)

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 2 · Mensuration 3D · Cylinders and Cones

The height of the conical tent is 14 m. and its floor area is 346.5

\text{m}^2

. How much canvas of width 1.1 m. will be required for it ?

Height where the slant height belongs

Volume uses the vertical height

h

; the curved surface

\pi rl

uses the slant height

l

. If only one is given, find the other from

l^2 = r^2 + h^2

first.

Adding a floor to the tent

A tent has no canvas floor. Its canvas is the curved surface

\pi rl

only, not

\pi r(l + r)

Keeping the negative root

If the total surface gives

r^2 + 7r - 60 = 0

, it factors as

(r + 12)(r - 5) = 0

. A radius cannot be negative, so

r = 5

Drill 2 more on volume and surface of a cone

Concept 3 of 4: One third: comparing cones and cylinders

Fill a cone with water and pour it into a cylinder with the same base and height: it takes three cones to fill it. Ratio questions are this fact, or the same formula written for two solids and divided.

Definition

Same base, same height: cylinder $=$ 3 cones. The largest cone cut from a cylinder leaves $\frac{2}{3}$ of the wood, so wood left : cone $= 2 : 1$ .
Same base, same volume: the cone must be 3 times as tall as the cylinder.
Two cones (or two cylinders): $V \propto r^2 h$ . Divide the two formulas and $\pi$ and $\frac{1}{3}$ cancel.
Radii are in the same ratio as diameters.

Cone against cylinder

V_{\text{cone}} = \frac{1}{3}V_{\text{cylinder}} \ (\text{same } r, h), \qquad \frac{V_1}{V_2} = \frac{r_1^2 h_1}{r_2^2 h_2}

Worked example

Two cones have radii in the ratio 3 : 2 and heights in the ratio 2 : 3. Find the ratio of their volumes.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Mensuration 3D · Cylinders and Cones

If the volumes of two cones are in the ratio 1 : 4 and their diameters are in the ratio 4 : 5, then the ratio of their heights, is

Ratio of radii is not ratio of volumes

With equal heights, radii 2 : 5 give volumes

4 : 25

, because the radius is squared. 2 : 5 is printed as a wrong option.

Turning the ratio the wrong way

Cylinder to cone with the same base and height is 3 : 1. With the same base and the same volume, cone height to cylinder height is also 3 : 1. Write which solid comes first before you answer.

Drill 5 more on one third: comparing cones and cylinders

Concept 4 of 4: Solids made by turning or cutting a shape

Spin a flat shape about a line and it sweeps out a solid. Slice a solid with a flat plane and the cut face is a flat shape. A few of these come up as one-mark questions.

Definition

Turning a shape about a line gives a solid of revolution.
A cross-section is the flat face where a plane cuts a solid. It is never a solid itself.

Shape and what is done	What you get
Rectangle turned about one side	Cylinder
Right triangle turned about a side that makes the right angle	Right circular cone
Right triangle turned about its hypotenuse	Two cones joined base to base
Semicircle turned about its diameter	Sphere
Cone cut by a plane parallel to its base	A circle The solid piece left below the cut is a frustum; the cut face itself is a circle.
Cone cut through its vertex, at right angles to the base	An isosceles triangle
Cylinder cut parallel to its base	A circle
Sphere cut by any plane	A circle

The side you turn about becomes the height; the other side becomes the radius.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 4 · Mensuration 3D · Cylinders and Cones

A solid formed on revolving a right angled triangle about its height is

Naming the piece instead of the section

A cone cut parallel to its base has a circular cross-section. A frustum is the solid piece below the cut, and a cross-section is never a solid.

Turning about the hypotenuse

A right triangle turned about its hypotenuse gives two cones joined at their bases, not one cone.

Drill 1 more on solids made by turning or cutting a shape

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Volume and surface of a cylinder
Cylinder
$V = \pi r^2 h, \quad \text{CSA} = 2\pi rh, \quad \text{TSA} = 2\pi r(r + h)$
Volume and surface of a cone
Cone
$l^2 = r^2 + h^2, \quad V = \frac{1}{3}\pi r^2 h, \quad \text{CSA} = \pi rl, \quad \text{TSA} = \pi r(l + r)$
One third: comparing cones and cylinders
Cone against cylinder
$V_{\text{cone}} = \frac{1}{3}V_{\text{cylinder}} \ (\text{same } r, h), \qquad \frac{V_1}{V_2} = \frac{r_1^2 h_1}{r_2^2 h_2}$

Reference tables (1)

Solids made by turning or cutting a shape8 rows

Shape and what is done	What you get
Rectangle turned about one side	Cylinder
Right triangle turned about a side that makes the right angle	Right circular cone
Right triangle turned about its hypotenuse	Two cones joined base to base
Semicircle turned about its diameter	Sphere
Cone cut by a plane parallel to its base	A circle The solid piece left below the cut is a frustum; the cut face itself is a circle.
Cone cut through its vertex, at right angles to the base	An isosceles triangle
Cylinder cut parallel to its base	A circle
Sphere cut by any plane	A circle

The side you turn about becomes the height; the other side becomes the radius.

Watch out for (9)

Curved or total surface
→ Volume and surface of a cylinder
Halving the radius does not halve the volume
→ Volume and surface of a cylinder
Height where the slant height belongs
→ Volume and surface of a cone
Adding a floor to the tent
→ Volume and surface of a cone
Keeping the negative root
→ Volume and surface of a cone
Ratio of radii is not ratio of volumes
→ One third: comparing cones and cylinders
Turning the ratio the wrong way
→ One third: comparing cones and cylinders
Naming the piece instead of the section
→ Solids made by turning or cutting a shape
Turning about the hypotenuse
→ Solids made by turning or cutting a shape

Test yourself on a real paper

Sit a past SPI paper, timed and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Sit an SPI past paper Drill all 15 questions