Prism: Grade 9 Math – Science Playground

A ward office is building a covered concrete tank to hold 8,000 liters of drinking water, 8 m³. The mason has two plans. One is long and shallow: 4 m long, 2 m wide and 1 m deep. The other is a cube: 2 m every way. Both hold exactly the same water. But the walls, floor and lid have to be built, plastered and paid for, and the two plans don’t need the same amount. Which one needs less, and by how much?

A water tank, a matchbox, a tent and a pencil are all prisms. In this unit you will learn to find a prism’s cross-section, its side faces, its whole surface and the space inside it. Then, in Back to the tank, you will answer the mason’s question, and see why the answer is no accident.

Sources: CDC’s Mathematics, Grade 9; every number in this unit is worked out here.

On a forested hillside, a man crouches at the edge of the flat concrete lid of a village water tank and lifts its small metal hatch — **Figure 6.1** Opening the hatch of the intake tank of a village water supply in Puware Shikhar, Udayapur. A tank like this is a prism: its volume is how much water it holds, and its surface is how much concrete and plaster it needs. Photo: Department of Foreign Affairs and Trade, CC BY 4.0

For the exam

The question: Grade 9’s final exam follows CDC’s specification grid for Compulsory Mathematics, Grades 9–10, where Mensuration is 13 marks in three questions. In the grid’s model questions, the prism question is a tent shaped like a triangular prism, 4 marks: a. the area of its triangular ends (2 marks, understanding), b. how long a tent the same cloth makes, and how much longer or shorter it is (2 marks, higher ability).
What it needs:

Cross-section (base) A: the area of one end, by the formula for its shape.
Lateral surface = P × h, where P is the perimeter of the cross-section and h the prism’s length (or height).
Total surface = 2A + P × h; a cuboid 2(lb + bh + hl); a cube $6l^2$ .
Volume = A × h; 1000 cm³ = 1 liter, 1 m³ = 1000 liters.

How the exam asks it has a full question marked part by part; Exam practice has every exercise question to mark yourself.

6.0 Review

Keep the pile in mind. Every idea in this unit comes from it: the same flat shape, piled straight up.

6.1 Introduction to prism

A prism is a solid with two opposite faces that are congruent, parallel polygons, joined by flat side faces. Take any polygon, slide it straight out without turning it, and the space it sweeps through is a prism. The two polygons are its bases (or ends), and a prism is named after them:

triangle at each end: a triangular prism, like a tent or a box of chocolate bars;
rectangle: a rectangular prism, a cuboid, like a brick, a matchbox or a room;
square: a square prism, and if every edge is equal, a cube;
six sides: a hexagonal prism, like most pencils.

Every side face of a prism is a parallelogram, and in the prisms of this unit, a rectangle.

Which of these is a prism?

Pick an answer to see why.

A prism has a 7-sided polygon at each end. How many vertices does it have?

Pick an answer to see why.

Count the faces F, vertices V and edges E of any prism, and F + V − E = 2 every time: a cuboid has 6 + 8 − 12 = 2, a triangular prism 5 + 6 − 9 = 2. A prism with an n-sided end has n + 2 faces, 2n vertices and 3n edges, and (n + 2) + 2n − 3n = 2. The great Swiss mathematician Leonhard Euler noticed in 1750 that this works for every solid with flat faces and no holes, pyramids too: try it on the table above.

6.2 Cross-section area of prism

A cut through a prism parallel to its ends is a cross-section, and it is always congruent to the ends. That is the whole point of a prism: the same shape, all the way along. So the cross-sectional area of a prism is just the area of one end, its base area A.

Characteristics of a prism

a.It has two parallel, congruent opposite polygons, its bases. A prism whose base is a triangle is a triangular prism; one whose base is a rectangle is a rectangular prism.
b.A face cut parallel to the base is a cross-section, and it is congruent to the base.
c.In a right prism, every side face stands at right angles to the bases. If the far end is pushed sideways, the side faces lean over and the prism is oblique. In this class we work with right prisms only.
d.The faces other than the two bases are the lateral faces, and together their areas make the lateral surface area.
e.The perpendicular distance between the two bases is the prism’s height, also called its length when the prism lies on its side.
f.The volume of a prism is its base area times its height: V = A × h.

Figure 6.5 An oblique prism: the top has slid sideways. Its cross-sections are still the same rectangle, but its side faces lean.

A right prism is the kind you meet every day: a box, a brick, a tank. An oblique prism is like a stack of exercise books given a push from one side. The pile leans, but each book is still the same book, so the pile holds the same paper; its volume is still base area × height, if the height is measured straight up between the ends. Its side faces are now parallelograms, though, which is why we keep to right prisms here.

Areas of the bases

To find a prism’s cross-section, find the area of its base polygon. These are the shapes you will meet most often, with the formulas you know from Grade 8 and from Unit 5.

Base	Area A
a.equilateral triangle, side a	$\frac{\sqrt{3}}{4} a^2$
b.isosceles triangle, equal sides a, third side b	$\frac{b}{4}\sqrt{4a^2 - b^2}$
c.scalene triangle, sides a, b, c	$\sqrt{s(s - a)(s - b)(s - c)}$ , $s = \frac{a + b + c}{2}$
d.right-angled triangle, perpendicular p, base b	$\frac{1}{2} p b$
e.right-angled isosceles triangle, equal sides p	$\frac{1}{2} p^2$
f.rectangle, length l, breadth b	l × b
g.square, side l	$l^2$

The first two aren’t new formulas: both are $\frac{1}{2} \times \text{base} \times \text{height}$ , with the height found by Pythagoras’ theorem.

A triangular prism 20 cm long is cut across, parallel to its ends, 5 cm from one end and again 15 cm from it. How do the two cut faces compare?

Pick an answer to see why.

6.3 Lateral surface area of prism

Activity

4: the side faces of a box

Distribute prisms with triangular, rectangular and square ends to the groups, and draw round each face in your copy. Here is group 3’s work.

Figure 6.6 Group 3’s box: length l, breadth b and height h. Its bases are the bottom ABCD and the top EHGF.

They took a prism with six faces and called its length, breadth and height l, b and h. Apart from its two bases, its faces are ABHE, DCGF, ADFE and BCGH, all rectangles:
Face ABHE: A₁ = l × h = lh
Face DCGF: A₂ = l × h = lh
Face ADFE: A₃ = b × h = bh
Face BCGH: A₄ = b × h = bh
Together:
$\begin{aligned} A_1 + A_2 + A_3 + A_4 &= lh + lh + bh + bh \\ &= 2lh + 2bh \\ &= 2h(l + b) \\ &= 2(l + b) \times h \end{aligned}$

2(l + b) is the perimeter P of the base ABCD. So the four side faces together are the perimeter of the base times the height. They are the box’s lateral surface, all its faces except the two bases (the walls of a room, in Unit 5).

This isn’t special to boxes. Every lateral face of a right prism is a rectangle: one side of the base for its width, the prism’s height h for its height. Stand them side by side and they make one long rectangle, as wide as all the base’s sides put together, which is the perimeter, and h high.

$\text{Lateral surface area (LSA)} = P \times h \qquad \text{for a cuboid: } 2h(l + b)$

A prism’s end is a triangle with sides 5 cm, 6 cm and 7 cm, and the prism is 10 cm long. What is its lateral surface area?

Pick an answer to see why.

6.4 Total surface area of prism

The total surface area of a prism is the area of all its faces: the two bases and the lateral surface. That is the cardboard in a box, the metal in a tin, the plaster on a tank.

$\text{Total surface area (TSA)} = 2A + P \times h$

For a cuboid, A = lb and P = 2(l + b), so
$\text{TSA} = 2lb + 2h(l + b) = 2(lb + bh + hl)$
and for a cube with edge l, each face is $l^2$ , so TSA $= 6l^2$ .

Two cuboids: one is 4 cm × 4 cm × 4 cm, the other 8 cm × 4 cm × 2 cm. Both hold 64 cm³. Which has the smaller total surface?

Pick an answer to see why.

6.5 Volume of prism

The layers don’t have to be square-shaped. Any prism is a pile of identical slices, each a copy of the base; a slice 1 cm thick holds A cm³, and h cm of them hold A × h.

$\text{Volume of a prism } V = \text{base area} \times \text{height} = A \times h$

Capacity and liters. The capacity of a tank, box or bottle is how much it can hold: the volume of the space inside it. A liquid’s volume, and so a tank’s capacity, is usually given in liters:
$1000 \text{ cm}^3 = 1 \text{ liter}, \qquad 1 \text{ m}^3 = 1000 \text{ liters}$
A cube 10 cm on each side holds exactly 1 liter, and a cube 1 m on each side holds 1000 liters, the size of the big plastic tanks on Nepal’s roofs.

Back to the tank

The ward’s tank must hold 8 m³, and it has a lid. Both plans hold 8 m³, so the question is only about the surface, 2(lb + bh + hl):

long and shallow, 4 m × 2 m × 1 m: 2(8 + 2 + 4) = 28 m²;
the cube, 2 m × 2 m × 2 m: 6 × 4 = 24 m².

The cube needs 4 m² less to build and plaster, about 14% less, for exactly the same water. That is not luck. Play with the tank below, and see if you can beat the cube.

For a closed box that must hold a given volume, the cube has the least surface; the further the box is from a cube, long, flat or tall, the more material it takes. That is why water tanks, grain bins and shipping boxes are often close to cubes, unless something else matters more (a tank on a roof must be low; a box must fit a shelf).

A story from maths

Why bees build hexagons

$A honeybee standing on a honeycomb whose six-sided wax cells are filled with honey$

Figure 6.15 A honeybee on its comb. Each cell is a hexagonal prism of wax. Photo: Matthew T Rader, CC BY-SA 4.0

A honeycomb is a wall of hexagonal prisms. Wax is costly for bees: they eat a lot of honey to make it. Around 36 BC, the Roman scholar Marcus Varro wrote that the hexagon holds the most honey for the least wax. Triangles, squares and hexagons are the only regular shapes that fit together with no gaps, and of those the hexagon has the least wall for the same area. But is there some cleverer shape, not regular at all, that does better? The “honeycomb conjecture” said no, and stayed unproved for two thousand years, until the American mathematician Thomas Hales proved it in 1999. The bees, like the mason, had found the shape that uses least material.

Exercise 6.1

Question 1

What is the volume of a prism whose cross-sectional area is 35 cm² and height 10 cm?

Answer

Question 2

Find the lateral surface area of a prism whose base has a perimeter of 36 cm and whose height is 8 cm.

Answer

Question 3

Find the total surface area of a prism whose cross-sectional area is 40 cm² and lateral surface area 175 cm².

Answer

Question 4

Find the cross-sectional area, lateral surface area, total surface area and volume of each prism.

Figure 6.16

Answer

Figure 6.17

Answer

Figure 6.18

Answer

Figure 6.19

Answer

Figure 6.20

Answer

Figure 6.21

Answer

Question 5

A cuboid is 6 cm long, 5 cm wide and 8 cm high.

Find its cross-sectional area.

Answer

Find its lateral surface area.

Answer

Find its volume.

Answer

Question 6

A rectangular tank is 2 m long, 1.5 m wide and 1 m high. Find its capacity in liters.

Answer

Question 7

A wooden cube with edges 20 cm is cut into 8 equal cubes. Find the edge of each small cube.

Answer

Question 8

Find the cross-sectional area, lateral surface area, total surface area and volume of each prism.

Figure 6.22

Answer

Figure 6.23

Answer

Figure 6.24

Answer

Figure 6.25

Answer

Question 9

The volume of a prism whose ends are right-angled triangles is 864 cm³. The two sides of the triangle that make the right angle are 8 cm and 9 cm. Find the height of the prism.

Answer

Question 10

10.

A prism 30 cm high has right-angled triangles for its ends, with base 4 cm and perpendicular 3 cm. Find the areas of its rectangular faces.

Answer

Question 11

11.

The total surface area of a triangular prism is 660 cm², and its base is a right-angled triangle with base 12 cm and hypotenuse 13 cm. Find the prism’s length.

Answer

Question 12

The volume of a triangular prism is 480 cm³. It is l cm long, and the sides of its ends are 4 cm, 13 cm and 15 cm.

Find its length l.

Answer

Find the areas of its rectangular faces.

Answer

Find its total surface area.

Answer

Question 13

The lateral surface area of a triangular prism is 2160 cm². The perimeter of its base is 54 cm and its cross-sectional area 126 cm².

Find the height of the prism.

Answer

Find its volume.

Answer

Question 14

14.

Santaman plans to make a bin of corrugated metal sheet to store rice, in one of the two shapes below, each with a lid. Which holds more rice? Which needs less sheet, and how much less?

Figure 6.26 Bin A.

Figure 6.27 Bin B.

Answer

Question 15

A school needs a tent of the shape below for its mathematics exhibition hall. It will be carpeted, and one end (across the breadth) will be left fully open as the entrance.

Figure 6.28

How much cloth is needed to make the tent?

Answer

What area of carpet is needed?

Answer

Each exhibition cabin takes 7 m² of floor. How many cabins fit in the hall?

Answer

Question 16

16.

A village will build a reserve tank for drinking water, to one of the two designs below, each closed with a lid. Which design is better, judged by how much it holds and how much material it needs?

Figure 6.29 Tank A.

Figure 6.30 Tank B.

Answer

Quiz yourself on this unit →

Practice every question in the book, and keep track of the ones you miss →

How the exam asks it

Grade 9’s final exam is set to CDC’s specification grid for Compulsory Mathematics, Grades 9–10. There, Mensuration is 13 marks in three questions; the grid’s model prism question is 4 marks on a tent:

Part	Level	Marks	What it asks
a.	understanding	2	the area of the tent’s triangular ends
b.	higher ability	2	the length of a tent made from a given amount of cloth, with the same ends, and how much longer or shorter it is

Here is one in that shape, with how it is marked.

Where these facts come from

The Giant’s Causeway (6.1, the fact box): Wikipedia, “Giant’s Causeway” (about 40,000 interlocking basalt columns, from lava about 50–60 million years ago), read November 2026.
Euler’s formula (6.1): Wikipedia, “Euler characteristic” (Euler’s 1750 letter to Goldbach), read November 2026.
The liter (6.5, the fact box): Wikipedia, “Litre” (defined as one cubic decimeter since 1964; from 1901 the volume of 1 kg of water, 1.000028 dm³), read November 2026.
The honeycomb (Back to the tank, the story): Wikipedia, “Honeycomb conjecture” (Varro, about 36 BC; proved by Thomas Hales in 1999), read November 2026.
The photos: Wikimedia Commons, credited under each picture.
The exam’s prism question (For the exam, How the exam asks it): CDC, Specification Grid, Compulsory Mathematics, Grades 9–10, and its model questions (the triangular tent question: U2, HA2); our question is our own.

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