Prisms and cylinders: a week of ready-to-teach maths

Five lessons for Year 9 Measurement. Print this pack and the week is prepared: each day has a one-page plan and a student worksheet with practice and a stretch section, plus measure sort cards, two nets to fold at actual size, a mini-check and every answer with its working.

Start here: five minutes to Monday

1. Skim the week at a glance on the next page.
2. Print the five days. Each day is two A4 sheets: a plan and a worksheet.
3. Print the two cut-out sheets at 100% (actual size), one copy per pair: sheet 1 has the measure sort cards (Day 1) and a cylinder net (Day 4), sheet 2 a triangular prism net (Day 3). Printed at actual size, every marked length measures true with a ruler.
4. Collect 12 connecting cubes per pair for Day 1, and scissors, tape and rulers for Days 3 and 4. Calculators are needed from Day 2.
5. Open the free interactive unit on the board. Every plan says which picture to show and which buttons to press.

The idea in plain words

Volume is how much space a solid fills, counted in cubic units (cm³, m³) or as capacity (mL, L, kL); surface area is how much covers it, counted in square units (cm², m²). A right prism has the same cross-section all along its length, so its volume is the area of that slice times the length, V = Al. Its surface is the two ends plus the side faces, which unroll into one rectangle as wide as the perimeter: SA = 2A + Pl. A cylinder is a prism with a circular slice, so V = πr²h and SA = 2πr² + 2πrh. The problem decides which measure is wanted, which unit suits its size, and how to round.

One day, one lesson

The five lessons fill a week of maths, one lesson of about 60 minutes a day. Run them in order: each day stands on the one before. Every plan marks a split point for two shorter sessions, and every worksheet ends with a stretch section for students who finish early.

How we write these answers

Every answer carries its unit, with the power on the unit (cm², m³). An answer with π is given exactly, such as 28π cm³, and then to one decimal place, 88.0 cm³; π is rounded only at the last step. The conversions used all week are 1 cm³ = 1 mL, 1000 cm³ = 1 L and 1 m³ = 1000 L = 1 kL.

The week at a glance

One lesson a day for a week. Each day stands on the day before, so run them in order.

How the week builds

Day 1 separates the two measures and their units; Days 2 and 3 take the volume and then the surface area of right prisms; Day 4 treats the cylinder as a prism with a circular slice; and Day 5 puts it all to work on real problems. It builds on Year 8, where students found the volume and capacity of right prisms and the area and circumference of a circle. It prepares the surface area and volume of composite objects in Year 10.

Materials for the week (one trip)

• From the classroom: 12 connecting cubes per pair, calculators, rulers, scissors, sticky tape, envelopes for the cards, this pack printed.
• Cut-out sheet 1, one per pair: 12 measure sort cards (Day 1) and a cylinder net (Day 4).
• Cut-out sheet 2, one per pair: a triangular prism net (Day 3).
• If you have them: an empty can or cardboard tube to measure on Day 4. No other equipment to buy.

Words for the word wall

volume, capacity, surface area, cubic unit, square unit, millilitre, litre, kilolitre, right prism, cross-section, net, face, edge, perimeter, lateral surface, cylinder, radius, diameter, circumference, curved surface, exact value, decimal approximation

Dear families

This week in maths, Year 9 is measuring solids: how much a box, a tank or a can holds (its volume) and how much card, paint or metal covers it (its surface area). We work with boxes, other prisms and cylinders, and choose units to suit, from millilitres to kilolitres.

Try this at home

• Measure a cereal or tissue box. How much does it hold, and how much card is it made of?
• Ask: a 1 L milk carton holds how many cubic centimetres? (1000 cm³: a litre is a 10 cm cube.)
• Ask: which holds more, a can 7 cm across and 12 cm tall, or one 10 cm across and 6 cm tall? (About 462 cm³ and 471 cm³: the wide one, just.)
• Ask: why leave π in an answer? (It is exact; any decimal of it is only close.)

My measuring week

Fill one row a day. Write a solid from class, its volume and its surface area, each with its unit.

Fill it, cover it

Volume counts the cubic units that fill a solid; surface area counts the square units that cover it. Every box has both, in different units, and boxes that hold the same can need very different amounts of card.

We are learning to

• find the volume and surface area of a box built from cubes, each in its own unit,
• convert between cm³, mL and L, and between m³, L and kL,
• decide whether a question asks for a volume, a surface area or a length.

Success criteria

• I can give the volume and the surface area of a box made of cubes, with the right unit on each.
• I can convert 2.5 L to cm³ and say why 1 cm³ is 1 mL.

You need

The worksheet; 12 connecting cubes per pair; the measure sort cards (cut-out sheet 1), one set per pair, cut out.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second by building the demo box again.

Watch for these ideas

• Counting only the three faces in view: a box has six, in three equal pairs.
• Thinking boxes with the same volume have the same surface area.
• Writing 1 L = 100 cm³: a litre is a 10 cm cube, 10 × 10 × 10 = 1000 cm³.

Answers

• Part A: 1 × 1 × 12: 12 cm³, 50 cm²; 1 × 2 × 6: 12 cm³, 40 cm²; 1 × 3 × 4: 12 cm³, 38 cm²; 2 × 2 × 3: 12 cm³, 32 cm². The 2 × 2 × 3 box needs the least card: the closer to a cube, the less surface.
• Part B: 500 cm³ = 500 mL; 2.5 L = 2500 cm³; 3 m³ = 3000 L; 750 mL = 0.75 L; 4.8 kL = 4.8 m³; the box holds 1000 cm³ = 1 L.
• Part C: the card key is with the mini-check answers.
• Part D: edge 6 cm gives 216 cm³ and 216 cm²: the same number, but not equal, as one counts cubes and the other squares.

Fill it, cover it

A. Twelve cubes, four boxes

Build each box from 12 cubes. Each cube has 1 cm edges. Find its volume and its surface area.

Which box needs the least card? What do you notice about its shape? ____________________________________

B. Convert

Use 1 cm³ = 1 mL, 1000 cm³ = 1 L and 1 m³ = 1000 L = 1 kL.

A box measures 20 cm by 10 cm by 5 cm. How many cm³ does it hold? How many litres? _______________________

C. Volume, surface area or length?

Sort the measure cards into three piles. Write each card’s letter in its pile with a sensible unit.

D. Stretch: the same number?

Find the edge of a cube whose volume and surface area are the same number. Are they equal? Explain.

Volume of a right prism

A right prism has the same cross-section all along its length, so it is a stack of identical slices. Its volume is the area of one slice times the length, V = Al, whatever the shape of the slice.

We are learning to

• find the area of a rectangle, triangle, trapezium or L-shaped cross-section,
• multiply by the length to find the volume of the prism, and work backwards to a missing measurement,
• give a volume as a capacity in millilitres or litres.

Success criteria

• I can find the volume of a prism with a trapezium cross-section and show the working.
• I can find the length of a prism from its volume and the area of its cross-section.

You need

The worksheet; calculators; a stack of identical cards or coins to show slices.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with the stack of slices again.

Watch for these ideas

• Multiplying every number on a triangular prism: the triangle’s area needs ÷ 2.
• Using a sloping side as the height of a triangle or trapezium: the height is at right angles to the base.
• Mixing units: all lengths in cm give cm³; 1000 cm³ is 1 L, not 100.

Answers

• Part A: rectangle 8 × 5 = 40, V = 40 × 3 = 120 cm³; triangle 10 × 6 ÷ 2 = 30, V = 30 × 12 = 360 cm³; trapezium (4 + 10) × 5 ÷ 2 = 35, V = 35 × 8 = 280 cm³; L-shape 7 × 5 − 4 × 2 = 27, V = 27 × 10 = 270 cm³.
• Part B: length 420 ÷ 35 = 12 cm; edge 270 ÷ (6 × 5) = 9 cm; cross-section 200 ÷ 8 = 25 cm².
• Part C: tank 60 × 30 × 40 = 72 000 cm³ = 72 L; trough 40 × 30 ÷ 2 = 600, × 150 = 90 000 cm³ = 90 L.
• Part D: doubling the length doubles the volume (120 cm³ to 240 cm³); doubling all three edges multiplies it by 2 × 2 × 2 = 8 (960 cm³).

Volume of a right prism

A. Slice, then stack

Each figure is the cross-section of a prism. Find its area, then the volume of the prism.

B. Work backwards

C. How many litres?

A fish tank is 60 cm long, 30 cm wide and 40 cm high. A water trough is a prism 150 cm long whose cross-section is a triangle with a 40 cm top and a depth of 30 cm. How many litres does each hold when full?

D. Stretch: double it

Take the first prism in Part A. What happens to its volume if you double its length? If you double all three of its edges? Explain why.

Surface area of a prism

Surface area adds the areas of all the faces, and a net shows them at once. For a right prism the side faces unroll into one rectangle, the perimeter of the end by the length, so SA = 2A + Pl: two ends, plus perimeter times length.

We are learning to

• draw or fold the net of a prism and find the area of every face,
• find the surface area of a box from its three pairs of equal faces,
• use SA = 2A + Pl for a triangular prism, and adapt it when a face is missing.

Success criteria

• I can explain why a box has three pairs of equal faces, and use them to find its surface area.
• I can use 2A + Pl for a triangular prism and check it against its net.

You need

The worksheet; the triangular prism net (cut-out sheet 2), one per pair; scissors, sticky tape, rulers; calculators.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after the shortcut; start the second with Part A.

Watch for these ideas

• Adding only three faces of a box: each face has an equal partner opposite.
• Using a sloping side as the triangle’s height, or forgetting ÷ 2 for its area.
• Giving surface area in cm³: it covers, so it is in square units.

Answers

• Part A: 6 × 2 × 5: 2 × (12 + 30 + 10) = 104 cm²; 10 × 10 × 3: 2 × (100 + 30 + 30) = 320 cm²; 4 × 2.5 × 8: 2 × (10 + 32 + 20) = 124 cm²; 7 × 7 × 7: 2 × (49 + 49 + 49) = 294 cm².
• Part B: A = 30, P = 30: 2 × 30 + 30 × 10 = 360 cm²; A = 12, P = 16: 2 × 12 + 16 × 10 = 184 cm²; A = 24, P = 24: 2 × 24 + 24 × 4 = 144 cm².
• Part C: base 60 × 30 + 2 × 60 × 40 + 2 × 30 × 40 = 9000 cm² of glass.
• Part D: 16 cm long, SA = 2 × 6 + 12 × 16 = 204 cm², not 2 × 108 = 216: the ends do not double.

Surface area of a prism

A. Three pairs of faces

Find the surface area of each box. Show the three different faces.

B. Triangular prisms: SA = 2A + Pl

Sketch the net, then find the area of the end (A), its perimeter (P) and the surface area.

C. Glass for a tank

The fish tank from Day 2 is 60 cm long, 30 cm wide and 40 cm high, with no lid. How much glass is it made of?

D. Stretch: twice as long

Make the prism from your net twice as long. Does its surface area double? Show why or why not.

Cylinders

A cylinder is a prism whose slice is a circle, so V = πr²h. Its curved side unrolls into a rectangle 2πr wide and h high, so SA = 2πr² + 2πrh. Answers keep π exactly, then round once, at the end.

We are learning to

• find the volume of a cylinder, as a multiple of π and to one decimal place,
• explain the curved surface 2πrh from a net, and find the surface area of a cylinder,
• use a diameter correctly and see what rounding π early costs.

Success criteria

• I can give the volume of a cylinder exactly, such as 28π cm³, and to one decimal place.
• I can roll the net and say why the curved side is 2πr wide.

You need

The worksheet; calculators; the cylinder net (cut-out sheet 1), scissors, tape; an empty can or tube if you have one.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after rolling the net; start the second with Part A.

Watch for these ideas

• Using the diameter as the radius: halve it first.
• Squaring the wrong thing: πr² squares only r, so 2πr is not 2πr².
• Rounding π to 3 or 3.14 at the start: the error grows with every step.

Answers

• Part A: π × 2² × 7 = 28π cm³ ≈ 88.0 cm³; π × 5² × 3 = 75π cm³ ≈ 235.6 cm³; r = 4, π × 4² × 6 = 96π cm³ ≈ 301.6 cm³; π × 1.5² × 10 = 22.5π cm³ ≈ 70.7 cm³.
• Part B: r = 3, h = 4: 18π + 24π = 42π cm² ≈ 131.9 cm²; r = 5, h = 10: 50π + 100π = 150π cm² ≈ 471.2 cm²; r = 1, h = 9: 2π + 18π = 20π cm² ≈ 62.8 cm².
• Part C: label = 2π × 3.5 × 11 = 77π cm² ≈ 241.9 cm²; the can holds π × 3.5² × 11 = 134.75π cm³ ≈ 423.3 cm³, about 423 mL.
• Part D: V = 360π cm³ ≈ 1130.97 cm³ to two places; with π ≈ 3, 1080 (out by 51.0); with π ≈ 3.14, 1130.4 (out by 0.6).

Cylinders

A. Volume: V = πr²h

Give each volume exactly, as a multiple of π, and then to one decimal place. All lengths are in cm.

B. Surface area: SA = 2πr² + 2πrh

Show the two ends and the curved side, exactly, then the total to one decimal place.

C. A can of tomatoes

A can has radius 3.5 cm and height 11 cm. How much paper is its label, which wraps the curved side once? How much does the can hold, in mL?

D. Stretch: round at the end

Find the volume of a cylinder with radius 6 cm and height 10 cm three ways: with π ≈ 3, with π ≈ 3.14, and exactly (then to two decimal places). How far out is each estimate?

Problems with the right units

Real problems mix the week. Students decide whether a question wants a volume, a surface area or a length, choose a unit that suits its size, convert where needed, and round to fit the situation, sometimes up.

We are learning to

• plan a problem in steps and say what each step finds,
• choose a unit to suit the answer and convert between m³, L and kL,
• round to fit the situation and judge whether an answer is sensible.

Success criteria

• I can turn a volume in m³ into litres and kilolitres.
• I can say why the number of soil bags is rounded up, not to the nearest.

You need

The worksheet; calculators; the measure sort cards from Day 1; the unit’s self-check quiz for the game.

Lesson flow (about 60 minutes)

Two shorter sessions instead? Split after Part A; start the second with Part B and the quiz.

Watch for these ideas

• Converting m³ to litres by × 100: 1 m³ is 1000 L.
• Rounding bags of soil to the nearest: 46 bags would leave the bed short.
• Painting the base of a tank that stands on the ground, or adding 10% as + 10.

Answers

• Part A: 1. 48 m³ = 48 kL (48 000 L). 2. 6π + 2.25π = 8.25π m² ≈ 25.9 m². 3. 1.152 m³ = 1152 L; 1152 ÷ 25 = 46.08, so 47 bags. 4. 2200 cm² × 1.1 = 2420 cm².
• Part B: both hold 128π cm³ ≈ 402.1 cm³. Metal: 96π ≈ 301.6 cm² and 136π ≈ 427.3 cm²: the radius 4 can uses less.
• Part C: not sensible: about 375 mL: a litre is a whole carton of milk; not sensible: an area is in square units: 60 m²; sensible: a brick is about 23 cm × 11 cm × 7.6 cm, close to 1900 cm³.
• Part D: r = 3: h ≈ 17.7, SA ≈ 389.9; r = 4: h ≈ 9.9, SA ≈ 350.5; r = 4.3: h ≈ 8.6, SA ≈ 348.7; r = 5: h ≈ 6.4, SA ≈ 357.1; r = 6: h ≈ 4.4, SA ≈ 392.9. The least metal is near r = 4.3, where the height is about the diameter.

Problems with the right units

A. Plan, solve, check

For each problem, write what each step finds, then solve it. Give the answer with a unit that suits it.

B. Two cans

Can P has radius 4 cm and height 8 cm; can Q has radius 2 cm and height 32 cm. Show that they hold the same. Which needs less metal?

C. Sensible or not?

D. Stretch: the best 500 mL can

A can must hold 500 cm³. For each radius, find the height and the surface area. Which radius uses the least metal?

Measure sort cards and a cylinder net

Measure sort cards (Day 1), one set per pair: 12 cards

Does each question ask for a volume, a surface area or a length? Sort the cards into three piles and write a sensible unit on each.

Cylinder net (Day 4), one per pair

Cut along the lines. Roll the rectangle into a tube: its width goes once round an end. Tape the two circles on. The cylinder has radius 1.5 cm and height 5 cm.

Teacher note: print at 100% (actual size). The card key is with the mini-check answers, so this sheet can go to pairs whole.

Triangular prism net

For Day 3, one per pair

Before you cut, write the area of each face on it. Then cut along the solid lines, fold along the dashed ones and tape the edges. The end is a right-angled triangle with sides 3 cm, 4 cm, 5 cm, and the prism is 8 cm long.

Teacher note: print at 100% (actual size), so every marked length measures true. The worked areas are in the Day 3 plan.

What we know: Prisms and cylinders

Work on your own. Use a calculator from Question 4. Give every answer with its unit.

1. A box is 3 cm by 4 cm by 7 cm. Find its volume and its surface area. ______________________
2. Volume, surface area or length? (a) the milk in a carton, (b) the paint on a door, (c) the height of a door. ______________________
3. How many cm³ are in 4.2 L? ______________________
4. A prism has a right-angled triangle for its end, with sides 9 cm, 12 cm, 15 cm, and is 10 cm long. Find its volume. ______________________
5. Find the surface area of the prism in Question 4. ______________________
6. A cylinder has radius 3 cm and height 8 cm. Find its volume exactly and to one decimal place. ______________________
7. A cylinder has radius 2 cm and height 10 cm. Find its surface area exactly and to one decimal place. ______________________
8. A prism has volume 252 cm³ and a cross-section of area 18 cm². How long is it? ______________________

Answers and marking guide

Answers with working

1. V = 3 × 4 × 7 = 84 cm³; SA = 2 × (12 + 21 + 28) = 122 cm².
2. (a) volume, (b) surface area, (c) length.
3. 4.2 × 1000 = 4200 cm³.
4. A = 12 × 9 ÷ 2 = 54 cm²; V = 54 × 10 = 540 cm³.
5. SA = 2 × 54 + 36 × 10 = 108 + 360 = 468 cm².
6. V = π × 3² × 8 = 72π cm³ ≈ 226.2 cm³.
7. SA = 8π + 40π = 48π cm² ≈ 150.8 cm².
8. 252 ÷ 18 = 14 cm.

A quick three-level guide

Eight questions, four ideas. A student at standard answers most questions with working and the right unit on every answer.

Card key

Measure sort (Day 1): volume: A (kL), D (m³), F (L), G (m³), L (m³); surface area: B (m²), E (cm²), I (m²), K (cm²); length: C (m), H (cm), J (m).

Weekly class record

Jot a tick as you move around the room; the mini-check fills any gaps. A tick a day is plenty.

The five columns are the five days: measures and units, prism volume, prism surface area, cylinders, and problems.