Surface Area and Volume — AFCAT Numerical Ability Formulas and Traps

Cube, Cuboid, Prism and Pyramid

Cube: volume, surface and diagonal

Cube of edge a

V = a^3,\quad S = 6a^2,\quad \text{LSA} = 4a^2,\quad d = a\sqrt{3}

Cuboid: volume, surface, diagonal and the open tank

Cuboid l × b × h

V = lbh,\quad S = 2(lb + bh + hl),\quad d = \sqrt{l^2 + b^2 + h^2}

Scaling a cube, and comparing a cube with a cuboid

Two cubes of edges a₁ and a₂

\dfrac{S_1}{S_2} = \left(\dfrac{a_1}{a_2}\right)^2,\quad \dfrac{V_1}{V_2} = \left(\dfrac{a_1}{a_2}\right)^3

Prisms and pyramids: base area times height

Base area A, height h

V_{\text{prism}} = Ah,\quad V_{\text{pyramid}} = \tfrac13 Ah

Common traps

Lateral surface is four faces, not six

A cube of edge 7 cm has total surface 294 cm² but lateral surface 196 cm². Lateral means the walls only.

Divide by 6 before the square root

A total surface of 600 cm² gives

a^2 = 100

and

a = 10

cm. Taking

\sqrt{600}

skips the six faces.

Lateral surface leaves out top and bottom

A box 12 cm × 9 cm × 8 cm has total surface 552 cm² but lateral surface 336 cm². Read which one the stem asks for.

An open tank has no lid, and wet walls stop at the water

A tank 5 m × 3 m with water 2 m deep has wetted surface

15 + 2 \times 2 \times 8 = 47

m², not the closed box's 62 m².

Square for area, cube for volume

Cubes with volumes

1 : 1000

have edges

1 : 10

and surfaces

1 : 100

. The volume ratio is not the surface ratio.

A 10% longer edge is not 30% more volume

1.1^3 = 1.331

, so the volume grows by 33.1%. Adding the percentages three times leaves out the cross terms.

Do not forget the third

A pyramid on a base of 30 cm² and 12 cm tall holds 120 cm³, not 360 cm³. Only the prism is base times height.

Slant height for faces, vertical height for volume

A square pyramid's triangular faces use the slant height; its volume uses the height straight up from the base.

Cylinder

Learn this subtopic in the notes

Volume of a cylinder

Cylinder of radius r and height h

V = \pi r^2 h

Curved and total surface, and the stories that use them

Cylinder surfaces

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

Hollow cylinders and pipes

Hollow cylinder

V = \pi h (R^2 - r^2)

Comparing two cylinders, and changing their size

Two cylinders

\dfrac{V_1}{V_2} = \dfrac{r_1^2 h_1}{r_2^2 h_2},\quad \dfrac{V}{\text{CSA}} = \dfrac{r}{2}

Common traps

Which side it turns about

A rectangle 10 cm × 3 cm turned about the 10 cm side gives radius 3 and height 10, volume

90\pi

. Turned about the 3 cm side it gives

300\pi

. The side it turns on is the height.

Halve the diameter

A cylinder 10 cm across and 6 cm tall holds

150\pi

cm³. Using 10 as the radius gives

600\pi

, four times too much.

A wetted surface has one base, not two

A tank of radius 7 m with water 5 m deep wets

220 + 154 = 374

m²: the wall up to the water and the floor. The lid and the dry wall are not wet.

Turn centimetres into metres before a rate per m²

A pillar 56 cm across has radius 0.28 m. Multiplying 28 by a height in metres gives an area a hundred times too big.

R² − r² is not (R − r)²

With radii 9 and 7,

81 - 49 = 32

, but

(9 - 7)^2 = 4

. Square each radius first, then subtract.

A pipe's total surface has four parts

Outside, inside and the two end rings. Leaving out the inner wall or the rings is the usual way to land on a wrong option.

The radius counts twice in volume, once in surface

Cylinders of equal height with radii

1 : 4

have curved surfaces

1 : 4

but volumes

1 : 16

.

A thinner wire grows by more than the cut

Cut the diameter by 20% and the cross-section becomes 0.64 of itself, so the length must grow to

\tfrac{1}{0.64} = 1.5625

times: up 56.25%, not 20%.

Cone and Frustum

Learn this subtopic in the notes

Slant height and the surface of a cone

Cone of radius r, height h, slant l

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

Volume of a cone

Cone

V = \tfrac13 \pi r^2 h

A cone cut parallel to its base

Small cone cut off the top

\dfrac{V_{\text{top}}}{V} = k^3,\quad \dfrac{\text{CSA}_{\text{top}}}{\text{CSA}} = k^2

The frustum: a bucket shape

Frustum of radii R and r, height h

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

Common traps

Curved surface uses the slant height

For radius 9 and height 12,

\pi rh = 108\pi

is not a surface of anything. The curved surface is

\pi rl = 135\pi

.

Cloth length is area divided by width

600 m² of canvas from cloth 3 m wide needs 200 m of cloth. Stopping at the area answers a different question.

Doubling the height is a 100% rise

Volume is proportional to height, so twice the height is twice the volume: an increase of 100%, not 200%.

The radius counts squared

Doubling only the radius makes the cone hold 4 times as much. Treating the radius like the height gives 2 times.

Measure from the right end

If the small cone is 9 cm of a 45 cm cone, the cut is 36 cm above the base. The question may ask for either number.

Halving the height does not halve the volume

A cut through the mid-point leaves a top cone with

\tfrac 18

of the volume and a frustum with

\tfrac 78

.

The slant uses R − r, not R + r

Radii 11 and 5 with height 8 give slant

\sqrt{64 + 36} = 10

. Using

R + r = 16

gives

\sqrt{320}

, which is wrong.

An open bucket has one circle

The metal in a bucket with no lid is its curved surface plus the small bottom circle only.

Sphere and Hemisphere

Learn this subtopic in the notes

Hemispheres: curved or total surface, solid or hollow

Hemisphere of radius r

\text{CSA} = 2\pi r^2,\quad \text{TSA} = 3\pi r^2,\quad V = \tfrac23 \pi r^3

Changing the radius of a sphere

Two spheres

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

Sphere and hemisphere formulas

Solid	Curved surface	Total surface	Volume
Sphere of radius r	$4\pi r^2$	$4\pi r^2$	$\tfrac43 \pi r^3$
Solid hemisphere of radius r	$2\pi r^2$	$3\pi r^2$	$\tfrac23 \pi r^3$
Hollow sphere, radii R outside and r inside	$4\pi R^2$ outside	$4\pi (R^2 + r^2)$	$\tfrac43 \pi (R^3 - r^3)$

Common traps

Four πr², not four thirds

The surface is

4\pi r^2

; the

\tfrac43

belongs to the volume. Mixing them is the commonest slip on this page.

Halve the diameter

A ball 14 cm across has radius 7 cm. Putting 14 into

4\pi r^2

gives a surface four times too big.

A solid hemisphere's total surface counts the flat face

For radius 7 cm the curved surface is 308 cm² but the total is 462 cm². Read whether the stem says curved or total.

R³ − r³ is not (R − r)³

A bowl with radii 6 and 3 holds

\tfrac23 \pi \times 189

of metal. Cubing the thickness,

3^3 = 27

, gives far too little.

Becomes 27 times means up by 26 times

Tripling the radius makes the volume 27 times as large, which is an increase of 2,600%. Read whether the stem asks what it becomes or how much it grows.

Square for surface, cube for volume

Spheres with surfaces

1 : 100

have radii

1 : 10

and volumes

1 : 1000

. Cubing the surface ratio, or squaring the volume ratio, is a different question.

Combined and Inscribed Solids

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Solids on the same base, or with equal volumes

Same base, same height

V_{\text{cone}} = \tfrac13 V_{\text{cylinder}},\quad V_{\text{cylinder}} : V_{\text{hemisphere}} : V_{\text{cone}} = 3 : 2 : 1 \;(h = r)

The largest solid inside another

Carving one solid out of another

V_{\text{left}} = V_{\text{outer}} - V_{\text{inner}}

Solids joined together or hollowed out

Two solids joined on a face

V = V_1 + V_2,\quad S = S_1 + S_2 - 2 \times (\text{area of the joined face})

Common traps

Square the ratio of radii

A cone with three times a cylinder's radius and the same height holds

\tfrac13 \times 9 = 3

times as much, not as much. The radius enters squared.

Equal volumes do not mean equal surfaces

A cone and a cylinder on the same base with equal volumes have heights

3h

and

h

. For curved surfaces you still need the cone's slant height,

\sqrt{r^2 + 9h^2}

.

The edge is the diameter, not the radius

A sphere in a cube of edge 10 cm has radius 5 cm. Taking the radius as 10 makes the sphere eight times too big, and bigger than the cube.

In a short cylinder the height decides

A cylinder of radius 5 and height 8 holds a sphere of radius 4, not 5. Check both the width and the height.

No floor under a tent

Canvas covers the cylinder's wall and the cone's curved surface. Adding the base circle, or the circle where the cone meets the cylinder, overcounts.

A cavity adds a surface

Drilling a cone out of a cylinder removes one flat end but opens up the cone's inner surface. Both changes must be made.

Melting, Recasting and Displacement

Learn this subtopic in the notes

One solid melted into another

Volume is kept

V_{\text{before}} = V_{\text{after}}

Many small solids from one, or many into one

Spheres recast

n = \left(\dfrac{R}{r}\right)^3,\quad R^3 = r_1^3 + r_2^3 + \dots

A body sunk in water raises the level

Rise in level

\text{rise} = \dfrac{V_{\text{sunk}}}{\text{base area of the vessel}}

Earth dug and spread, water flowing and stored

Flow in a pipe

\text{volume per hour} = \text{cross-section} \times \text{speed}

Common traps

Melt only what is left

If a sphere is first carved from a block and the rest is recast, the new solid holds the block minus the sphere, not the whole block.

Radius in, diameter out

Stems give diameters for one solid and radii for the other, and some ask for a diameter at the end. Halve going in and double coming out.

Radii do not add

Spheres of radii 3, 4 and 5 make one of radius 6, not 12. Add the cubes, then take the cube root.

Merging cuts the surface

64 drops merge into one drop of four times the radius, with 16 times one drop's surface. The 64 drops had 64 times, so the surface falls to a quarter.

Divide by the vessel's base, not the object's

A cube sunk in a tank raises the water by its volume over the tank's floor. The cube's own face area plays no part.

Match the units before dividing

0.08 m³ over a pool of 200 m² is a rise of 0.0004 m, which is 0.04 cm. Convert at the end, not halfway.

Spread over the field less the pit

With 192 m³ of earth on a 2,000 m² field holding an 80 m² pit, dividing by 2,000 gives 9.6 cm. The earth covers only 1,920 m², so the rise is 10 cm.

cm² to m² is ten thousand, not a hundred

A pipe of cross-section 25 cm² is 0.0025 m². Dividing by 100 gives 0.25 m², a pipe a hundred times too wide.

Surface Area and Volume formulas

Cube, Cuboid, Prism and Pyramid

Cube: volume, surface and diagonal

Cuboid: volume, surface, diagonal and the open tank

Scaling a cube, and comparing a cube with a cuboid

Prisms and pyramids: base area times height

Common traps

Cylinder

Volume of a cylinder

Curved and total surface, and the stories that use them

Hollow cylinders and pipes

Comparing two cylinders, and changing their size

Common traps

Cone and Frustum

Slant height and the surface of a cone

Volume of a cone

A cone cut parallel to its base

The frustum: a bucket shape

Common traps

Sphere and Hemisphere

Hemispheres: curved or total surface, solid or hollow

Changing the radius of a sphere

Sphere and hemisphere formulas

Common traps

Combined and Inscribed Solids

Solids on the same base, or with equal volumes

The largest solid inside another

Solids joined together or hollowed out

Common traps

Melting, Recasting and Displacement

One solid melted into another

Many small solids from one, or many into one

A body sunk in water raises the level

Earth dug and spread, water flowing and stored

Common traps

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