Mensuration 3D — SPI Mathematics Formulas and Traps

Cuboids and Cubes

Surface area of a cuboid and a cube

Surface of a cuboid and a cube

S = 2(lb + bh + hl), \quad \text{walls} = 2(l + b)h, \quad S_{\text{cube}} = 6a^2, \quad \text{walls}_{\text{cube}} = 4a^2

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

Volume of a cuboid, litres and how many blocks fit

Volume and units

V = lbh, \quad V_{\text{cube}} = a^3, \quad 1\ \text{litre} = 1000\ \text{cm}^3, \quad 1\ \text{m}^3 = 1000\ \text{litres}, \quad \text{rise} = \frac{\text{volume}}{\text{area}}

The longest rod: the diagonal of a cuboid

Diagonal of a cuboid

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

Common traps

Total surface or four walls

Painting the sides only, or the walls of a room, leaves out the top and bottom:

2(l + b)h

, or

4a^2

for a cube. The whole outside is

6a^2

. Read which one is asked before you multiply.

Converting an area by the length factor

If 1 inch is 2.5 cm, then 1 square inch is

2.5^2 = 6.25

square cm, not 2.5. Convert each edge to inches first, then find the area.

Counting the hidden faces

Three separate cubes of edge

a

show

18a^2

. Joined in a row, two joints hide four faces, so the cuboid shows

14a^2

.

Mixing units inside one product

A tank 4 m by 150 cm by 1 m must be written as 400 cm by 150 cm by 100 cm (or all in metres) before multiplying. And 1 cubic m is 1000 litres, not 100.

Leaving the pit in the area

Earth dug from a pit is spread over the rest of the field, so subtract the pit's floor area from the field's area before dividing.

Forgetting the mortar

If one eighth of a wall is mortar, the bricks fill seven eighths of its volume. Take that share first, then divide by one brick's volume.

Dividing volumes when the blocks cannot fit

Cubes of edge 4 cm from a block 12 cm by 10 cm by 8 cm: the volumes divide to 15, but only

3 \times 2 \times 2 = 12

whole cubes can be cut, because 10 is not a multiple of 4. Check each edge when the question says cut.

Stopping at the floor diagonal

\sqrt{l^2 + b^2}

is the longest rod that lies flat on the floor. The longest rod in the room also uses the height:

\sqrt{l^2 + b^2 + h^2}

.

Leaving the root unsimplified

Options print roots in simplest form:

\sqrt{200} = 10\sqrt{2}

. Take out the largest square factor before matching.

Cylinders and Cones

Learn this subtopic in the notes

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)

$r$ radius of the base
$h$ height

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)

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

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}

Solids made by turning or cutting a shape

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.

Common traps

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.

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

.

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.

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.

Spheres and Hemispheres

Learn this subtopic in the notes

Surface area and volume of a sphere

Sphere

S = 4\pi r^2, \qquad V = \frac{4}{3}\pi r^3

Surface area and volume of a hemisphere

Hemisphere

\text{CSA} = 2\pi r^2, \quad \text{TSA} = 3\pi r^2, \quad V = \frac{2}{3}\pi r^3

Ratios and percent changes: square for area, cube for volume

Scaling rule

\frac{S_1}{S_2} = \left(\frac{r_1}{r_2}\right)^2, \qquad \frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^3

Common traps

Using the diameter as the radius

A ball of diameter 14 cm has radius 7 cm. Putting 14 into

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

makes the volume 8 times too big.

Answering with the radius when the diameter is asked

After the cube root gives

r

, check the question. If it asks for the diameter, double it; the radius is usually printed as a wrong option.

Bowl or solid

A hollow bowl is

2\pi r^2

. A solid hemisphere also has its flat circle, so its total is

3\pi r^2

. Read whether the question says solid.

A hemisphere is not half the sphere's surface

Half of

4\pi r^2

is

2\pi r^2

, the curved part only. A solid half also has its new flat face, so each half shows

3\pi r^2

, three quarters of the whole sphere's surface.

Using the volume ratio as the radius ratio

Volumes 1 : 8 do not mean radii 1 : 8. Take the cube root: radii 1 : 2.

Copying the percent of the length to the area

A diameter 20% shorter gives an area

0.8^2 = 0.64

of the old, so 36% less, not 20% and not 40%.

Combined Solids and the Frustum

Learn this subtopic in the notes

Solids joined together

Cone on a hemisphere

V = \frac{1}{3}\pi r^2 h + \frac{2}{3}\pi r^3, \qquad S = \pi rl + 2\pi r^2

The frustum: buckets and glasses

Frustum

l = \sqrt{h^2 + (R - r)^2}, \quad \text{CSA} = \pi(R + r)l, \quad V = \frac{1}{3}\pi h(R^2 + r^2 + Rr)

$R, r$ radii of the larger and the smaller circular ends
$h$ vertical height
$l$ slant height

A cone cut parallel to its base

Similar top cone

\frac{h}{H} = \frac{r}{R} = k, \qquad \frac{V_{\text{top}}}{V} = k^3

Common traps

Counting the glued faces

A cone on a hemisphere shows the cone's curved surface and the hemisphere's curved surface:

\pi rl + 2\pi r^2

. Writing

3\pi r^2

for the hemisphere counts a flat face that is glued out of sight.

A right vertical angle is the whole angle

The vertical angle is measured across the tip. If it is

90^\circ

, each side makes

45^\circ

with the axis, so the height equals the radius.

Adding the radii in the slant height

The slant height uses the difference:

\sqrt{h^2 + (R - r)^2}

. The sum

R + r

belongs in the curved surface, not here.

Closing the top of a bucket

A bucket is open at the top. Its metal is the curved surface plus the smaller bottom circle only, not both circles.

Leaving the capacity in cubic cm

If the options are in litres, divide by 1000: 45 600 cubic cm is 45.6 litres.

Measuring from the wrong end

The cube root gives the height of the small cone, measured from the tip. The height of the cut above the base is the whole height minus that.

Square root for a volume fraction

A volume fraction needs the cube root.

\frac{1}{8}

of the volume means half the height, not

\frac{1}{\sqrt{8}}

.

Melting, Recasting and Water Level

Learn this subtopic in the notes

Melting and recasting: how many pieces

Recasting

V_{\text{old}} = n \times V_{\text{piece}}, \qquad n = \frac{V_{\text{old}}}{V_{\text{piece}}}

Rise in water level

Water rise in a cylinder

\pi R^2 \times \text{rise} = V_{\text{solid}}

$R$ radius of the vessel (not of the solid)

Common traps

Using the coin's diameter as its radius

A coin of diameter 3 cm has radius 1.5 cm. Using 3 makes each coin four times too big and the count four times too small.

Same shape, smaller size: cube the ratio

Small spheres from a big sphere:

n = \left(\frac{R}{r}\right)^3

. A sphere of radius 4 cm makes 64 balls of radius 1 cm, not 4 and not 16.

Dividing by the wrong circle

The risen water spreads over the vessel's base, so divide by

\pi R^2

of the vessel, not by the area of the ball or cone dropped in.

Worrying about the starting depth

'Partly filled' only tells you the solid is covered. The rise depends on the solid's volume and the vessel's base, not on how much water was there.

Mensuration 3D formulas

Cuboids and Cubes

Surface area of a cuboid and a cube

Volume of a cuboid, litres and how many blocks fit

The longest rod: the diagonal of a cuboid

Common traps

Cylinders and Cones

Volume and surface of a cylinder

Volume and surface of a cone

One third: comparing cones and cylinders

Solids made by turning or cutting a shape

Common traps

Spheres and Hemispheres

Surface area and volume of a sphere

Surface area and volume of a hemisphere

Ratios and percent changes: square for area, cube for volume

Common traps

Combined Solids and the Frustum

Solids joined together

The frustum: buckets and glasses

A cone cut parallel to its base

Common traps

Melting, Recasting and Water Level

Melting and recasting: how many pieces

Rise in water level

Common traps

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