Mensuration 3D — SSC CGL Quantitative Aptitude Formulas and Traps

Cubes and Cuboids

Volume and surface area of a cuboid and a cube

Cuboid and cube

V = lbh \qquad S = 2(lb + bh + hl) \qquad V_{\text{cube}} = a^3 \qquad S_{\text{cube}} = 6a^2

$l, b, h$ length, breadth and height of the cuboid
$a$ edge of the cube

Walls of a room and the capacity of a tank

Four walls and capacity

A_{\text{walls}} = 2(l + b)h \qquad 1\ \text{m}^3 = 1000\ \text{L} \qquad 1\ \text{L} = 1000\ \text{cm}^3

Cutting a cuboid into cubes and joining cubes

Cutting and joining

\text{Count} = \frac{V_{\text{big}}}{V_{\text{small}}} \qquad S_{n \text{ cubes in a row}} = (4n + 2)a^2

The longest diagonal of a cuboid and a cube

Body diagonal

d = \sqrt{l^2 + b^2 + h^2} \qquad d_{\text{cube}} = a\sqrt{3} \qquad d_{\text{face}} = a\sqrt{2}

Common traps

Leaving out the 2 in the surface formula

The total surface of a cuboid is

2(lb + bh + hl)

. For a cuboid 2 by 3 by 4 the bracket is 26 but the surface is 52; the half value is often a printed option.

Edge from the perimeter of one face

A face of a cube is a square, so its perimeter is

4a

. A face perimeter of 36 cm gives an edge of 9 cm. The total of all twelve edges,

12a

, is a different measure.

Taking the thickness off once

A wall 1 cm thick sits at both ends of every edge, so each inner edge is 2 cm shorter than the outer edge, not 1 cm shorter.

Mixing metres and centimetres

A depth of 80 cm is 0.8 m. Change every length to one unit before multiplying, then use

1\ \text{m}^3 = 1000

litres or

1000\ \text{cm}^3 = 1

litre.

Adding the floor to the walls by habit

Whitewashing walls and ceiling leaves out the floor; painting only the walls leaves out both. Add

lb

only for each surface the question names.

Adding the surfaces of joined cubes

Three separate cubes show 18 faces, but joined in a row they show only 14: the two faces at each join are hidden. The surface of

n

cubes in a row is

(4n + 2)a^2

, not

6na^2

.

One cut for every part

Dividing one edge into 4 parts needs 3 cuts, not 4: the last part needs no cut of its own.

Using the face diagonal for the longest rod

a\sqrt{2}

runs across one face. The rod through the inside of a cube is longer:

a\sqrt{3}

.

Prisms and Pyramids

Learn this subtopic in the notes

Volume of a prism

Prism and two base areas

V = \text{base area} \times h \qquad A_{\triangle} = \frac{\sqrt{3}}{4}a^2 \qquad A_{\text{hexagon}} = \frac{3\sqrt{3}}{2}a^2

Lateral surface area of a prism

Lateral and total surface of a prism

\text{LSA} = \text{perimeter of base} \times h \qquad \text{TSA} = \text{LSA} + 2 \times \text{base area}

A prism built in layers whose heights form an AP

Sum of the layer heights

H = \frac{n}{2}(a + l) \qquad l = a + (n - 1)d \qquad V = \text{base area} \times H

Volume and lateral surface of a pyramid

Pyramid

V = \frac{1}{3} \times \text{base area} \times h \qquad \text{LSA} = \frac{1}{2} \times \text{perimeter} \times l

$h$ vertical height, apex to base
$l$ slant height of a face

Common traps

A hexagon is six triangles

A regular hexagon of side

a

splits into six equilateral triangles of side

a

, so its area is

6 \times \frac{\sqrt{3}}{4}a^2 = \frac{3\sqrt{3}}{2}a^2

. Writing

\frac{\sqrt{3}}{4}a^2

alone gives one triangle.

Square centimetres times metres

A base of

60\ \text{cm}^2

and a height of 1.2 m give

60 \times 120 = 7200\ \text{cm}^3

, not

72

. Change the height to centimetres first.

Base area in place of perimeter

The lateral surface multiplies the perimeter of the base by the height. Base area times height is the volume, a different quantity.

Using one layer's height

The prism is as tall as all its layers together. Base area times the tallest layer, or times the average layer without multiplying by the number of layers, gives only part of the volume.

Forgetting the one third

A pyramid is one third of the prism on the same base and height. Base area times height alone gives three times the answer.

Slant height in the volume

A square pyramid of base side 6 cm can have vertical height 4 cm and slant height 5 cm. Use 4 for the volume and 5 for the faces.

Cylinders

Learn this subtopic in the notes

Volume and surface area of a cylinder

Cylinder

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

$r$ radius of the base
$h$ height

Finding the radius from the curved surface or the circumference

Radius from the circumference, volume from the curved area

r = \frac{C}{2\pi} \qquad V = \frac{r}{2} \times \text{CSA}

Hollow cylinders, pipes and drilled holes

Pipe and drilled share

V_{\text{metal}} = \pi(R^2 - r^2)h \qquad \text{share removed} = \frac{r^2}{R^2}

Common traps

Using the diameter as the radius

Many cylinder questions give the diameter. Halve it first: a flask 10 cm across has radius 5 cm. Putting 10 into

\pi r^2 h

gives four times the volume.

Two ends on an open vessel

A tank or a glass open at the top has one circular end, so its outer surface is

2\pi rh + \pi r^2

, not

2\pi r(r + h)

.

Rolling the sheet the wrong way

A sheet rolled along its breadth has the breadth as the circumference. Swap the two edges and you get a different cylinder with a different volume, so check which edge goes round.

Litres against cubic metres

1\ \text{m}^3

is 1000 litres, but 1 litre is

1000\ \text{cm}^3

. A level drop worked in centimetres with a volume in litres is off by 1000 unless you convert first.

Subtracting the radii before squaring

\pi(R - r)^2 h

is not the metal in a pipe. Square each radius first:

R^2 - r^2 = (R + r)(R - r)

. For radii 4 and 3 that is 7, not 1.

Cones

Learn this subtopic in the notes

Radius, height and slant height of a cone

Slant height

l = \sqrt{r^2 + h^2}

$r$ radius of the base
$h$ vertical height
$l$ slant height

Volume of a cone

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

Curved and total surface area of a cone

Surface of a cone

\text{CSA} = \pi rl \qquad \text{TSA} = \pi r(l + r)

Common traps

Slant height used as the height

The slant height is the sloping edge; the height runs from the tip straight down to the centre of the base. With

r = 6

and

l = 10

the height is 8, not 10.

Turning about the wrong side

A 6, 8, 10 triangle turned about the 8 cm side gives a cone of height 8 cm and radius 6 cm. The side it turns about is always the height.

Leaving out the one third

A cone holds one third of the matching cylinder.

\pi r^2 h

alone gives three times the cone's volume, and that value is often among the options.

Slant height in the volume formula

The volume needs the vertical height. When a question gives the slant height, find

h = \sqrt{l^2 - r^2}

before you use

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

.

Adding a base to a tent

A tent or a cap has no base. Adding

\pi r^2

asks for canvas that is not there; use

\pi rl

alone.

Height in the curved surface

\pi rh

is not a cone formula. The curved surface uses the slant height:

\pi rl

.

Spheres, Hemispheres and Shells

Learn this subtopic in the notes

Volume and surface area of a sphere and a hemisphere

Sphere and hemisphere

V_{\text{sphere}} = \frac{4}{3}\pi r^3 \qquad S_{\text{sphere}} = 4\pi r^2 \qquad V_{\text{hemi}} = \frac{2}{3}\pi r^3 \qquad \text{CSA}_{\text{hemi}} = 2\pi r^2 \qquad \text{TSA}_{\text{hemi}} = 3\pi r^2

Hollow spheres and hemispherical shells

Metal in a shell

V_{\text{shell}} = \frac{4}{3}\pi(R^3 - r^3) \qquad V_{\text{hemi shell}} = \frac{2}{3}\pi(R^3 - r^3)

$R$ outer radius
$r$ inner radius

When the volume is a multiple of the surface area

Volume over surface for a sphere

\frac{V}{S} = \frac{r}{3}

The largest sphere inside a cube, a box or a cone

Radius of the largest ball

\rho_{\text{cube}} = \frac{a}{2} \qquad \rho_{\text{cuboid}} = \frac{\text{smallest edge}}{2} \qquad \rho_{\text{cone}} = \frac{rh}{r + l}

$\rho$ radius of the ball
$r, h, l$ base radius, height and slant height of the cone

Common traps

Curved or total surface of a hemisphere

A solid hemisphere has a flat face too, so its total surface is

3\pi r^2

. The curved part alone is

2\pi r^2

. Read which one the question names.

Diameter put in as the radius

A ball 7 cm across has radius 3.5 cm. Putting 7 into

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

gives eight times the true volume.

Cubing the thickness

The metal is

\frac{4}{3}\pi(R^3 - r^3)

, not

\frac{4}{3}\pi(R - r)^3

. For radii 3 and 2 the bracket is 19 in the first and only 1 in the second.

The wrong surface for a hemisphere

The

\frac{r}{3}

shortcut for a hemisphere uses its curved surface

2\pi r^2

. With the total surface

3\pi r^2

the ratio is

\frac{2r}{9}

, so check which surface the question means.

The longest edge limits the ball

A ball is stopped by the smallest edge of the box. In a box 9 cm by 8 cm by 4 cm the largest ball has diameter 4 cm, not 9 cm.

Edge of the cube used as the radius

A ball that just fits a cube of edge 14 cm has radius 7 cm. Using 14 as the radius gives eight times the true volume.

Scaling and Comparing Solids

Learn this subtopic in the notes

How a change in length changes area and volume

Scale factors

\text{area} \times k^2 \qquad \text{volume} \times k^3 \qquad V_{\text{cone, cylinder}} \propto r^2 h

Ratios of two cylinders or two cones

Two cylinders or two cones

\frac{V_1}{V_2} = \frac{r_1^2 h_1}{r_2^2 h_2} \qquad \frac{\text{CSA}_1}{\text{CSA}_2} = \frac{r_1 h_1}{r_2 h_2}

Slicing a cone by a plane parallel to the base

Cut cone and frustum

\frac{V_{\text{top}}}{V_{\text{whole}}} = \left(\frac{h}{H}\right)^3 \qquad V_{\text{frustum}} = \frac{1}{3}\pi h(R^2 + Rr + r^2)

$h, H$ height of the top cone and of the whole cone, both from the tip
$R, r$ radii of the two circular faces of the frustum

Volume and surface formulas of the standard solids

Solid	Volume	Curved or side surface	Total surface
Cube, edge $a$	$a^3$	$4a^2$	$6a^2$
Cuboid	$lbh$	$2(l + b)h$	$2(lb + bh + hl)$
Prism	base area $\times h$	base perimeter $\times h$	side surface $+ 2 \times$ base area
Pyramid	$\frac{1}{3} \times$ base area $\times h$	$\frac{1}{2} \times$ base perimeter $\times$ slant height	side surface $+$ base area
Cylinder	$\pi r^2 h$	$2\pi rh$	$2\pi r(r + h)$
Cone	$\frac{1}{3}\pi r^2 h$	$\pi rl$	$\pi r(l + r)$
Sphere	$\frac{4}{3}\pi r^3$	$4\pi r^2$	$4\pi r^2$
Solid hemisphere	$\frac{2}{3}\pi r^3$	$2\pi r^2$	$3\pi r^2$ The total surface includes the flat circle; a bowl has no lid, so it has only the curved part.

In the cone row

l

is the slant height; in the cuboid row

l

is the length.

Common traps

Adding the percentages

Radius up 10% and height down 10% does not leave a cone's volume unchanged. The radius counts twice, as

r^2

:

1.1 \times 1.1 \times 0.9 = 1.089

, a rise of 8.9%.

Increased by 200% means three times

An increase of 200% adds twice the length to itself, so the new length is 3 times the old one, not 2 times.

New size or increase

If a volume becomes 8 times as large, the increase is 7 times the original, that is 700%. Options often print both 8 and 7, or 800% and 700%.

Squaring the height too

Only the radius is squared in a volume ratio. With radii

1 : 2

and heights

1 : 2

the volumes are

1 \times 1 : 4 \times 2 = 1 : 8

, not

1 : 16

.

Measuring from the base

Similar-cone ratios use heights from the tip. A cut 6 cm above the base of a 24 cm cone leaves a top cone 18 cm tall, three-quarters of the height.

Height ratio used as the volume ratio

A top cone with half the height holds one-eighth of the volume, not one-half. Volumes go as the cube of the height ratio.

Cone against cylinder the wrong way round

A cone holds one third of the cylinder on the same base and height. For equal volumes the cone must be 3 times as tall as the cylinder, not one third as tall.

Melting, Recasting and Water

Learn this subtopic in the notes

Melting and recasting keeps the volume the same

Volume is conserved

V_{\text{before}} = V_{\text{after}} \qquad R^3 = r_1^3 + r_2^3 + r_3^3

How many smaller solids, or how many containers

Number of pieces

n = \frac{V_{\text{big}}}{V_{\text{small}}} \qquad n_{\text{same shape}} = \left(\frac{R}{r}\right)^3

Surface area after recasting

Total surface after splitting into n equal copies

S_{\text{after}} = \sqrt[3]{n} \times S_{\text{before}}

Water displaced and the rise in level

Rise in water level

\text{rise} = \frac{V_{\text{sunk}}}{\text{base area}}

Common traps

Adding the radii

Spheres of radii 1 cm, 6 cm and 8 cm melt into a sphere of radius 9 cm, not 15 cm. Volumes add, so it is the cubes of the radii that add:

1 + 216 + 512 = 729 = 9^3

.

Forgetting the one third in a cone

A cylinder melted into a cone on the same base gives a cone 3 times as tall as the cylinder, because the cone's formula carries

\frac{1}{3}

.

Dividing the radii instead of the volumes

A ball of radius 4 cm makes 64 balls of radius 1 cm, not 4. The count is the ratio of the volumes,

\left(\frac{4}{1}\right)^3

.

Assuming the surface stays the same

Only the volume is kept. Splitting a ball into 27 equal balls triples the total surface, so treating the two surfaces as equal is wrong.

Using the water already in the tank

The rise depends only on the volume sunk and the base area. The starting depth of the water never enters the sum.

Dividing by the tank's volume

Divide the volume sunk by the base area of the tank, not by its volume: the answer is a length.

Mensuration 3D formulas

Cubes and Cuboids

Volume and surface area of a cuboid and a cube

Walls of a room and the capacity of a tank

Cutting a cuboid into cubes and joining cubes

The longest diagonal of a cuboid and a cube

Common traps

Prisms and Pyramids

Volume of a prism

Lateral surface area of a prism

A prism built in layers whose heights form an AP

Volume and lateral surface of a pyramid

Common traps

Cylinders

Volume and surface area of a cylinder

Finding the radius from the curved surface or the circumference

Hollow cylinders, pipes and drilled holes

Common traps

Cones

Radius, height and slant height of a cone

Curved and total surface area of a cone

Common traps

Spheres, Hemispheres and Shells

Volume and surface area of a sphere and a hemisphere

Hollow spheres and hemispherical shells

When the volume is a multiple of the surface area

The largest sphere inside a cube, a box or a cone

Common traps

Scaling and Comparing Solids

How a change in length changes area and volume

Ratios of two cylinders or two cones

Slicing a cone by a plane parallel to the base

Volume and surface formulas of the standard solids

Common traps

Melting, Recasting and Water

Melting and recasting keeps the volume the same

How many smaller solids, or how many containers

Surface area after recasting

Water displaced and the rise in level

Common traps

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