Area and Volume

Two ideas carry the whole unit, and you have just used both. A surface area is the area of every face you could paint, laid out flat and added. A volume is how much the solid holds, in cubes: cm³, m³, or liters. Prisms and cylinders keep the same shape all the way up, so their volume is base × height. This unit is about solids that narrow to a point, pyramids and cones, and about solids built from several pieces.

What a pyramid is

Fold your own net on screen. The slider folds all four triangles at once; the second slider changes how tall the triangles are.

A pyramid is a solid with a polygon for its base and triangles for all its other faces, the lateral faces, which meet at one point, the apex. It is named by its base: triangular, square, pentagonal. In a right pyramid the apex is straight above the center of the base; with a regular base too, all its lateral faces are congruent. The vertical height is the distance from the apex straight down to the base, at right angles to it.

In this unit, as in the SEE, every pyramid is a right square pyramid: a square base with the apex straight above its center.

The parts of a square pyramid

A square pyramid has four lengths that matter:

• a, the base edge, the side of the square ABCD;
• h, the vertical height, PO, straight down from the apex to the center of the base;
• l, the slant height, PM, down the middle of a triangular face to the middle of a base edge: the height of each triangle;
• e, the lateral edge, such as PB, from the apex down to a corner.

They are tied together by two right-angled triangles. O is the center of the square, so OM is half the base edge, a/2. In triangle POM the right angle is at O, and Pythagoras’ theorem gives

$$PM^2 = PO^2 + OM^2, \quad\text{so}\quad l^2 = h^2 + \left(\frac a2\right)^2$$

M is the middle of BC, so MB = a/2, and the slant line PM meets BC at a right angle (the face is an isosceles triangle). In triangle PMB:

$$PB^2 = PM^2 + MB^2, \quad\text{so}\quad e^2 = l^2 + \left(\frac a2\right)^2$$

Each formula has its longest side on the left: the edge is longer than the slant height, and the slant height longer than the height. A pyramid’s face slopes, so the path down it is longer than the path straight down.

Surface area of a pyramid

The net is the surface, laid flat: folding changes no face’s size. So the surface area is the area of the net. The slant height l is the height of each triangle, which is why it matters so much.

$$\text{one triangle} = \frac12 \times \text{base} \times \text{height} = \frac12 a l$$

$$\begin{aligned} \text{lateral surface area (LSA)} &= 4 \times \frac12 a l = 2al \\ \text{total surface area (TSA)} &= a^2 + 2al = a(a + 2l) \end{aligned}$$

The lateral surface is the four triangles alone; the total surface adds the base. A tent’s cloth or a roof’s tin is a lateral surface; a box made all round, a total surface.

Volume of a pyramid

A pyramid narrows to a point, so it must hold less than a prism on the same base with the same height. How much less?

Pouring shows it; a cube cut into three shows why. Take a cube and pick one top corner, V. Join V to every corner of the three faces that don’t touch it: the bottom, the right and the back. The cube falls apart into three square pyramids, all with apex V.

Each piece has a base $a^2$ (a face of the cube) and height a (V is a full edge away from that face), and the three pieces are equal. Together they make the cube, $a^3$, so each is $\tfrac13 a^3$: a third of base × height. The same holds for any pyramid, tall or flat:

$$V = \frac13 \times \text{area of base} \times \text{height} = \frac13 a^2 h$$

Question 1

Study the square pyramid below and answer.

Question 2

Question 3

Question 4

Find the total surface area and the volume of each square pyramid.

Question 5

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Question 7

Question 8

Question 9

A hotel plans a tent shaped like a square pyramid for a group of foreign tourists: each side of the base 32 m, height 50 m.

Question 10

Nepal Telecom has built a mobile tower in Asharam’s field. The tower is a square pyramid: its slant height is 50 ft and its vertical height 40 ft.

Question 11

What a cone is

A cone has a circle for its base and one curved surface that narrows to a point, the vertex (or apex). In a right circular cone, the cone of this unit, the vertex is straight above the center of the base.

$$l^2 = h^2 + r^2, \qquad l = \sqrt{h^2 + r^2}$$

The slant height is always the longest side of that triangle: longer than the height and longer than the radius. It is the same all the way round the cone.

Surface area of a cone

The curved surface is the sector you get when you cut the cone open, so its area is the sector’s area. But a sector’s area needs its angle, which the cone doesn’t tell you directly. There is a neater way.

So the curved surface of a cone is a rectangle $\pi r$ long and l high, in disguise:

$$\begin{aligned} \text{curved surface area (CSA)} &= \pi r \times l = \pi r l \\ \text{total surface area (TSA)} &= \pi r^2 + \pi r l = \pi r(r + l) \end{aligned}$$

The base is a circle, $\pi r^2$; add it when the cone is closed underneath (a solid wooden cone), leave it out when it is open (a birthday cap, a tent, an ice-cream cone).

In this unit $\pi = 22/7$, unless a question says otherwise (some SEE questions give $\pi = 3.14$: use what the question gives); an answer in terms of $\pi$ (like $65\pi$ cm²) is exact.

Volume of a cone

$$V = \frac13 \times \text{area of base} \times \text{height} = \frac13 \pi r^2 h$$

It is the pyramid’s rule again, with a circle for a base: any solid that narrows evenly from a base to a point holds a third of base × height.

Question 1

Look at the cone and answer.

Question 2

Question 3

Question 4

Find the curved surface area, total surface area and volume of each cone.

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Solids built from pieces

To find a combined solid’s volume, add the volumes of its parts: the parts fill different spaces. Its surface needs more care. Where two parts are joined, the circle (or square) between them is inside the solid: nobody can paint it, so it is not part of the surface. Add only the faces you can see.

The five shapes and their formulas

All the parts of one combined solid share the same radius r (or base edge a).

Cylinder and cone (a tent): the cylinder has height h₁; the cone height h₂ and slant height l.

$$\begin{aligned} \text{TSA} &= \pi r^2 + 2\pi r h_1 + \pi r l \\ V &= \pi r^2 h_1 + \frac13 \pi r^2 h_2 \\ &= \pi r^2 \left(h_1 + \frac{h_2}{3}\right) \end{aligned}$$

The $\pi r^2$ is the floor; leave it out for a tent with no floor.

Cone and hemisphere (an ice cream): the cone has height h and slant height l.

$$\begin{aligned} \text{TSA} &= \pi r l + 2\pi r^2 = \pi r(l + 2r) \\ V &= \frac13 \pi r^2 h + \frac23 \pi r^3 \\ &= \frac13 \pi r^2 (h + 2r) \end{aligned}$$

Cylinder and hemisphere (a grain silo with a dome): the cylinder has height h.

$$\begin{aligned} \text{TSA} &= \pi r^2 + 2\pi r h + 2\pi r^2 \\ &= 2\pi r h + 3\pi r^2 \\ V &= \pi r^2 h + \frac23 \pi r^3 \\ &= \pi r^2 \left(h + \frac{2r}{3}\right) \end{aligned}$$

Two cones on one base (a spinning top, a toy rocket’s ends): heights h₁ and h₂, slant heights l₁ and l₂.

$$\begin{aligned} \text{TSA} &= \pi r l_1 + \pi r l_2 = \pi r(l_1 + l_2) \\ V &= \frac13 \pi r^2 h_1 + \frac13 \pi r^2 h_2 \\ &= \frac13 \pi r^2 (h_1 + h_2) \end{aligned}$$

Prism and pyramid (a pillar with a pointed top): a square base of edge a; the prism has height h₁, the pyramid height h₂ and slant height l.

$$\begin{aligned} \text{TSA} &= a^2 + 4a h_1 + 2al \\ V &= a^2 h_1 + \frac13 a^2 h_2 \\ &= a^2 \left(h_1 + \frac{h_2}{3}\right) \end{aligned}$$

Here 4a is the perimeter p of the base, so the prism’s sides are ph₁, as for any prism.

Question 1

Answer from what is given for each combined solid.

Question 2

Find the volume of each solid.

Question 3

Question 4

Find the total surface area of each solid.

Question 5

Question 6

A pyramid 12 cm high stands on top of a square cuboid whose base has area 100 cm² and which is 10 cm high. Find:

Question 7

Question 8

Question 9

A tower is a square prism 6 ft high on a base 1 ft square, with a square pyramid 1 ft high on top.

Question 10

Every job of building, painting, tiling, fencing or filling is priced per unit: so many rupees a square foot of paint, a meter of wire, a liter of water. Its cost is

$$\text{cost} = \text{quantity} \times \text{rate}$$

and the whole skill is finding the right quantity, an area, a length or a volume, in the unit the rate uses.

Why is the walls’ area 2h(l + b)? Unfold them.

The four walls are two l × h and two b × h rectangles. Laid side by side they make one rectangle h high and 2(l + b) long: the room’s perimeter. So

$$\text{four walls} = 2h(l + b), \qquad \text{four walls and ceiling} = 2h(l + b) + lb$$

In the SEE’s Compulsory Mathematics paper, Mensuration (area and volume) carries 13 marks in three questions, with knowledge, understanding, application and higher-ability parts spread among them (CDC’s specification grid). CDC’s model paper asks:

Question	Part	Level	Marks	What it asks
a square pyramid	a.	knowledge	1	how many triangular faces
	b.	understanding	1	its volume
	c.	application	2	the cost of covering its triangles
a cone and hemisphere	a.	knowledge	1	the slant height formula
	b.	understanding	2	its volume
	c.	higher ability	1	melted into a cylinder: its height
a water tank	a.	application	3	the cost of painting its inner walls
	b.	higher ability	2	the cost of filling it

Here is a question in the tank’s shape (3 + 2 marks, as the model’s), and one in the pyramid’s to practice with.

Question 1

How much water can each tank hold when full? (1 ft³ = 28.317 liters)

Question 2

The parking area in front of an insurance company’s office is the shape ABCD below. It is to be paved with bricks; each brick covers 0.222 sq ft and costs Rs 16.

Question 3

A garden beside a new temple has the shape below. The temple committee will fence it with barbed wire, going round it 5 times. Three workers can do it in 2 days at Rs 1,500 a day each; the wire costs Rs 80 a meter.

Question 4

A cube-shaped room is 9 ft each way. It has two square windows 2 ft on a side and two doors 5 ft by 3 ft.

Question 5

A well needs 60 concrete rings, each 3.5 ft across and 1 ft high. A ring costs Rs 1,200; 2 workers set the rings in 12 days at Rs 1,500 a day each. (1 ft³ = 28.317 liters)

Question 6

A house’s gate has two pillars, each a square prism with a base 5 ft square and 10 ft high, with a square pyramid 1 ft high on top.

Question 7

A house’s gate has two pillars, each a square prism with a base 1 ft square and 6 ft high, with a square pyramid 1 ft high on top.

Question 8

A plant nursery builds a round tank 2.80 m across inside to collect rainwater. Its walls are 3 m high, and its top is a cone 0.72 m high.

Question 9

A pyramid tent is sewn from 8 equal triangles of cloth, each with sides 5 m, 6 m and 6 m.

These questions mix the whole unit, as the SEE’s Mensuration questions do.

Question 1

For a project, a Grade 10 student hands his mathematics teacher three paper square pyramids, each with base edge 12 cm and slant height 10 cm.

Question 2

A square pyramid has base edge 16 cm and volume 1,280 cm³.

Question 3

A square pyramid’s lateral surface area is 540 cm² and its slant height 15 cm.

Question 4

A toy is a cylinder with a hemisphere on one end.

Question 5

A solid is a square prism with base edge 8 cm, with a square pyramid of slant height 5 cm on top. Its volume is 1,024 cm³.

Question 6

A tent is a cylinder with a hemisphere on top, both of the same radius. It is 6.5 m high in all, and its cylindrical part is 3 m high.

Question 7

A solid is a hemisphere and a cone, both of radius 7 cm. Painting it at Rs 6 a square centimeter costs Rs 5,148.

Question 8

A toy is a cylinder and a cone on the same base, 10 cm across. The cylinder is 14 cm long and the cone 12 cm high.

Question 9

A pyramid 12 cm high stands on a square cuboid whose base has area 100 cm² and which is 9 cm high.

Question 10

A cylindrical drum of radius 14 cm and height 40 cm is full of cement. Poured onto the ground, the cement makes a conical heap 30 cm high.

Question 11

A stadium has two pillars, each with a base 6 ft square and 8 ft high, with a pyramid 4 ft high on top.

Question 12

A house’s gate has two cuboid pillars, each 1 ft long, 1 ft wide and 6 ft high, with a pyramid 1 ft high on top.

Question 13

A well is made of 32 concrete rings, each 3.5 ft across and 1 ft high, costing Rs 1,200 a ring.

Question 14

Two square pyramids on the same base are joined base to base. The base edge is 12 cm and the solid is 20 cm from tip to tip.

Question 15

Nandakishor came home from work abroad and started a farm of cows and goats in his village. His water tank is a cylinder 1.4 m across and 2.1 m high inside, with a cone 0.36 m high on top.

Question 16

A solid with volume $240\pi$ cm³ is a cone with a hemisphere on it. The cone’s slant height and radius are in the ratio 5 : 3.

Question 17

A solid is a cylinder 24 cm long with a cone on one end, and its total surface area is $840\pi$ cm². The cone’s slant height and vertical height are in the ratio 13 : 12.

Question 18

Figure 1 shows a cylinder and a cone on the same base, 14 cm across; the cylinder is 39 cm long and the cone 24 cm. Joined, they make the solid in Figure 2.

Question 19

A toy is a cylinder and a cone on the same base, 10 cm across; the cylinder is 14 cm long and the cone 12 cm high.

Question 20

A solid is a cylinder 28 cm long with a cone of slant height 17 cm on one end, both of the same radius. Painting the whole solid at Rs 140 for each 100 cm² costs Rs 2,851.20.

Question 21

A water tank is a cylinder with a hemisphere on top. It is 20 m high in all and its base has area 154 m². It is filled with water at 45 paisa a liter.