AFCAT Numerical Ability · Surface Area and Volume

Cube, Cuboid, Prism and Pyramid

Volume, surface area and diagonal of a cube and a cuboid, the open tank and its wetted surface, how a cube's measures scale, and the base area rule for prisms and pyramids.

All Surface Area and Volume notes

Why this matters

Most questions here are one formula and one line of arithmetic, so the marks go on reading the stem: total or lateral surface, a closed box or an open tank, a face diagonal or the space diagonal. All six prism and pyramid questions are marked HARD, yet each is base area times height, with a third for the pyramid.

Concept 1 of 4: Cube: volume, surface and diagonal

A cube is six equal square faces. Its volume is the edge used three times, its surface is six squares, and the walls alone (no top, no bottom) are four squares.

Definition

Volume $a^3$ , total surface $6a^2$ , lateral surface (four walls) $4a^2$ .
Face diagonal $a\sqrt 2$ ; space diagonal (corner to opposite corner) $a\sqrt 3$ . So two corners of a cube are $a$ , $a\sqrt 2$ or $a\sqrt 3$ apart.
Working back: from the volume take the cube root; from the total surface divide by 6, then take the square root.
With total surface $S$ and volume $V$ : $S^3 = 216a^6 = 216V^2$ .

Cube of edge a

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

Worked example

A cube has edge 7 cm. Find its volume, total surface, lateral surface and space diagonal.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Surface Area and Volume · Cube, Cuboid, Prism and Pyramid

TSA of a cube with volume

125

cm³ is:

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.

Drill 8 more on cube: volume, surface and diagonal

Concept 2 of 4: Cuboid: volume, surface, diagonal and the open tank

A cuboid has three pairs of equal rectangles. Its volume is length times breadth times height, and its surface counts each pair twice. The space diagonal is Pythagoras used twice: across the floor, then up.

Definition

Volume $lbh$ ; total surface $2(lb + bh + hl)$ ; lateral surface (four walls) $2h(l + b)$ .
Space diagonal $\sqrt{l^2 + b^2 + h^2}$ : the longest rod that fits in the box.
An open tank or a wetted surface has the floor once and the walls only up to the water: $lb + 2d(l + b)$ for water $d$ deep.
Cubes of edge $a$ set in a row of $n$ make a cuboid $na \times a \times a$ . The faces that touch are lost.
Given the three face diagonals: they are $l^2 + b^2$ , $b^2 + h^2$ and $h^2 + l^2$ . Add them and halve to get $l^2 + b^2 + h^2$ , then subtract each to get one edge squared.

Cuboid l × b × h

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

Worked example

A box is 12 cm × 9 cm × 8 cm. Find its volume, total surface, lateral surface and the longest rod that fits inside.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Surface Area and Volume · Cube, Cuboid, Prism and Pyramid

A cistern 6 m long and 4 m wide contains water to a depth of 1.25 m. What is the area of wetted surface?

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.

Drill 9 more on cuboid: volume, surface, diagonal and the open tank

Concept 3 of 4: Scaling a cube, and comparing a cube with a cuboid

Make every length k times as long and every area becomes k² times as big, every volume k³ times. Two cubes are always the same shape, so their edges, surfaces and volumes follow k, k² and k³.

Definition

Edges in the ratio $k$ : surfaces in $k^2$ , volumes in $k^3$ .
Back from volumes, take the cube root to get the edges; back from surfaces, the square root.
An edge up by $r\%$ multiplies the volume by $\left(1 + \tfrac{r}{100}\right)^3$ and the surface by $\left(1 + \tfrac{r}{100}\right)^2$ .
A cuboid with edges in a given ratio: write them $pk, qk, rk$ , set its volume or surface equal to the cube's, and find the cube's edge in $k$ .

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

Worked example

Two cubes have surfaces in the ratio

25 : 49

. Find the ratio of their volumes.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionHard

Example 3 · Surface Area and Volume · Cube, Cuboid, Prism and Pyramid

Two cubes have volumes in ratio

8:27

. Ratio of their surface areas is:

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.

Drill 3 more on scaling a cube, and comparing a cube with a cuboid

Concept 4 of 4: Prisms and pyramids: base area times height

A prism has the same cross-section all the way along, so its volume is that cross-section times its length; a cuboid is a prism on a rectangle. A pyramid narrows to a point and holds exactly a third of the prism on the same base and height.

Definition

Prism: volume = base area × height; lateral surface = perimeter of the base × height.
Pyramid: volume = $\tfrac 13$ × base area × height. For a regular pyramid the lateral surface is $\tfrac 12$ × perimeter of the base × slant height.
An equilateral triangle of side $a$ has area $\dfrac{\sqrt 3}{4}a^2$ ; a right triangle has half the product of its legs.
A prism and a pyramid of equal base and equal volume: the pyramid is three times as tall.

Base area A, height h

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

Worked example

A prism 15 cm long has a right-triangle base with legs 6 cm and 8 cm. Find its volume, and the volume of a pyramid on the same base with the same height.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionHard

Example 4 · Surface Area and Volume · Cube, Cuboid, Prism and Pyramid

A triangular prism has equilateral triangle base with side

4

cm and height

10

cm. Volume is:

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.

Drill 5 more on prisms and pyramids: base area times height

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 (4)

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

Watch out for (8)

Lateral surface is four faces, not six
→ Cube: volume, surface and diagonal
Divide by 6 before the square root
→ Cube: volume, surface and diagonal
Lateral surface leaves out top and bottom
→ Cuboid: volume, surface, diagonal and the open tank
An open tank has no lid, and wet walls stop at the water
→ Cuboid: volume, surface, diagonal and the open tank
Square for area, cube for volume
→ Scaling a cube, and comparing a cube with a cuboid
A 10% longer edge is not 30% more volume
→ Scaling a cube, and comparing a cube with a cuboid
Do not forget the third
→ Prisms and pyramids: base area times height
Slant height for faces, vertical height for volume
→ Prisms and pyramids: base area times height

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