Surface Area & Volume of Prisms and Cylinders
Volume Formulae
Draw a line from each solid to its volume formula.
Choose the Correct Formula
Circle the correct formula for each surface area calculation.
Total surface area of a cube with side length s
Curved surface area of a cylinder
Total surface area of a cylinder
Volume of Rectangular Prisms
Calculate the volume of each rectangular prism.
l = 8 cm, w = 5 cm, h = 3 cm: V =
l = 12 m, w = 4 m, h = 2.5 m: V =
A cube with side 7 cm: V =
Volume of Cylinders
Calculate the volume of each cylinder. Use pi approx 3.14. Round to 2 decimal places.
r = 4 cm, h = 10 cm: V =
Diameter = 6 m, h = 5 m: V =
r = 2.5 cm, h = 8 cm: V =
Surface Area of Rectangular Prisms
Calculate the total surface area using SA = 2(lw + lh + wh).
l = 6 cm, w = 4 cm, h = 3 cm: SA =
l = 10 m, w = 5 m, h = 2 m: SA =
Surface Area of Cylinders
Calculate total surface area: SA = 2 pi r^2 + 2 pi r h. Use pi approx 3.14.
r = 3 cm, h = 7 cm: SA =
r = 5 m, h = 12 m: SA =
Practical Problems
Apply surface area and volume to solve these problems.
A rectangular fish tank is 60 cm long, 30 cm wide and 40 cm tall. What volume of water (in litres) does it hold? (1 litre = 1000 cm^3)
A cylindrical can has radius 4 cm and height 11 cm. A label wraps around the curved surface only. What is the area of the label? Use pi approx 3.14.
Challenge: Which Container Holds More?
Compare volumes and decide which holds more. Show all calculations.
Container A: cylinder, r = 5 cm, h = 8 cm. Container B: rectangular prism, 8 x 8 x 8 cm. Which holds more? By how much?
Surface Area of a Cylinder
Calculate the total surface area of each cylinder using SA = 2 pi r^2 + 2 pi r h. Use pi approx 3.14.
A tin can: r = 4 cm, h = 12 cm. Curved surface area = ___, Two circular ends = ___, Total SA = ___
A storage drum: r = 0.5 m, h = 1.2 m. Total SA =
A pipe (open at both ends, no circular faces): r = 3 cm, h = 50 cm. Curved surface area only =
Volume of a Cone
Use V = (1/3) pi r^2 h to find the volume of each cone. Use pi approx 3.14.
r = 3 cm, h = 9 cm: V =
r = 5 m, h = 12 m: V =
An ice-cream cone has r = 2 cm and h = 10 cm. What is its volume? (Round to 1 decimal place.)
Comparing Volumes
Calculate the volume of each solid and compare them. Show all working.
Cone A: r = 6 cm, h = 10 cm. Cylinder B: r = 6 cm, h = 10 cm. How many times does Cone A fit inside Cylinder B?
Which has more volume: a cube with side 6 cm, or a cylinder with r = 3 cm and h = 8 cm? By how much?
Unit Conversions: cm^3 to Litres
Convert each volume. Remember: 1 L = 1000 cm^3 and 1 mL = 1 cm^3.
5000 cm^3 = ___ litres
2.5 litres = ___ cm^3
A tank holds 72,000 cm^3 of water. How many litres is this? If water costs $0.002 per litre, what does a full tank cost?
A cylinder has r = 10 cm and h = 30 cm. Calculate its volume in cm^3 and then convert to litres.
Solid Shapes and Their Formulae
Draw a line from each solid to the correct volume formula.
Surface Area — Cylinder
Calculate the surface area of each cylinder.
Cylinder: r = 5 cm, h = 10 cm. Calculate the total surface area. Use π ≈ 3.14.
Cylinder: r = 3 m, h = 8 m. Calculate the total surface area.
A cylinder has diameter 14 cm and height 20 cm. What is its surface area?
Surface Area and Volume — Comparison
Calculate and compare surface area and volume.
A rectangular box: l = 12 cm, w = 8 cm, h = 5 cm. Find both surface area and volume.
A cube has the same volume as the box above. Find its side length. Is the cube's surface area larger or smaller? (Hint: cube side = ∛V)
What does this tell you about the relationship between shape and surface area for a fixed volume?
Match Formula to Shape
Match each formula to its solid.
Composite Solids
Calculate volume and surface area of composite (combined) solids.
A solid is formed by placing a cylinder (r = 3 cm, h = 5 cm) on top of a rectangular prism (10 cm × 10 cm × 4 cm). Calculate the total volume.
Calculate the external surface area of the composite solid above. (Note: the base of the cylinder sits on top of the rectangular prism — those faces are not external.)
Identifying the Correct Unit
Circle the correct unit for each measurement.
Volume of a swimming pool
Surface area of a bedroom wall
Capacity of a water tank in litres
Amount of paint needed to cover a fence
Surface Area — Real-World Problems
Solve each real-world surface area problem.
A cylindrical water tank is open at the top, has diameter 2 m and height 3 m. How much sheet metal is needed to build it? (Calculate the curved surface + base only.)
A gift box is a rectangular prism: 30 cm × 20 cm × 15 cm. How much wrapping paper is needed (with 10% extra for overlaps)?
A swimming pool is a rectangular prism: 12 m × 5 m × 1.8 m. How many square metres of tiles are needed for the floor and all 4 walls? (Not the top — the pool is open.)
Capacity and Volume
Convert between volume and capacity.
A fish tank is 80 cm × 40 cm × 50 cm. Calculate the volume in cm³ and the capacity in litres.
Water flows into the tank at 2 litres per minute. How long until it is 90% full?
A cylindrical rainwater tank has diameter 1.2 m and height 2 m. What is its capacity in litres?
Design Challenge — Optimum Container
Apply surface area and volume to a design problem.
You need to design a cylindrical can to hold exactly 500 cm³ of liquid. The formula for volume is V = πr²h = 500. Express h in terms of r.
Write the surface area formula SA = 2πr² + 2πrh, substituting your expression for h. What are you trying to minimise?
Try r = 4 cm and r = 5 cm. Calculate SA for each. Which gives a smaller surface area?
Research: for a fixed volume, the cylinder with minimum surface area has h = 2r (height equals diameter). Verify this for V = 500 cm³ by calculating r and h.
Volume and Surface Area — Word Problems
Solve practical problems requiring both formulas.
A concrete pillar is cylindrical: diameter 0.6 m, height 4 m. What volume of concrete is needed? What area of formwork (the mould surface) is needed for the curved side?
Grain is stored in a cylindrical silo with internal radius 3 m and height 12 m. What is the maximum grain capacity in m³? Convert to litres.
A swimming pool is 25 m × 10 m and has a uniform depth of 2 m. How many litres of water to fill it? How long at 500 L per minute to fill it?
Measurement in the Kitchen and Garden
Apply surface area and volume to real household contexts.
- 1Find three containers in your kitchen (tin can, box, jar). Measure their dimensions. Calculate the volume (capacity in mL) and compare to the label. How close is the geometry to the actual contents?
- 2Measure a room in your home. Calculate its volume in m³ and surface area (all 6 walls/ceiling/floor). Research how many litres of paint is needed per m².
- 3Design a raised garden bed (rectangular prism shape) to hold 0.5 m³ of soil. Find the cheapest dimensions by minimising the surface area of the 5 sides (no top face).
Surface Area and Volume — Reflection and Review
Reflect on your learning in this worksheet.
List the formulas for volume of: rectangular prism, triangular prism, cylinder.
List the formulas for surface area of: rectangular prism, cylinder.
Describe one real-world situation where knowing both the surface area and volume would be important.
What was the most challenging part of this worksheet? How did you work through it?
Error Analysis — Finding the Mistake
Find and correct the error in each worked solution.
Student calculates surface area of cylinder (r = 3, h = 5): SA = πrh = π × 3 × 5 = 47.1 cm². What is wrong? What is the correct answer?
Student calculates volume of triangular prism (base 4, height 3, length 10): V = 4 × 3 × 10 = 120 cm³. What is wrong? What is the correct answer?
Student says: 'To find the volume of a cube with side 5 cm: V = 5 × 3 = 15 cm³.' What did they do wrong?
Measuring the Unfamiliar
Estimate and calculate the volume of irregular objects.
Describe the 'water displacement' method for measuring the volume of an irregular solid. Draw a diagram.
A rock is placed in a 10 cm × 10 cm × 10 cm container of water. The water level rises by 3 cm. What is the volume of the rock?
How could you use displacement to measure the volume of a person's hand? What problems might arise?
Cross-Sections and Prisms
Explore how cross-sections define prisms.
A solid has a cross-section that is a trapezium with parallel sides 4 cm and 6 cm and height 3 cm. The solid is 10 cm long. Calculate its volume.
The cross-section of a swimming pool is an L-shape (a composite rectangle). If the pool is 20 m long and the cross-section has dimensions that give an area of 14 m², what is the pool's volume?
Why must the cross-section be measured perpendicular to the length of the prism?
Surface Area vs Volume — When to Use Each
Match each situation to whether surface area or volume is needed.
Measurement Project — Designing a Box
Design the optimal box for a given volume.
You need a rectangular box to hold exactly 1000 cm³ (1 litre). The box must be square at the base (l = w). Express the surface area SA in terms of just one variable.
Calculate SA for l = 5 cm, l = 8 cm, and l = 10 cm. Which gives the smallest SA?
The optimal box is actually a cube (l = w = h ≈ 10 cm). Calculate the SA of this cube and compare to your other values.
Surface Area and Volume — Comprehensive Problem
Solve this multi-part measurement problem.
A silo consists of a cylinder (r = 2 m, h = 6 m) topped by a cone (r = 2 m, h = 3 m). Calculate the volume of the cylinder.
The volume of a cone is V = ⅓πr²h. Calculate the volume of the cone.
Calculate the total storage volume of the silo in m³. Convert to litres.
Calculate the surface area of the cylindrical section only (curved surface + base). How many square metres of metal is needed?
Surface Area and Volume — Final Portfolio Task
Demonstrate mastery of measurement.
Calculate the volume and surface area of a cylinder with diameter 12 cm and height 15 cm. Show all steps.
A factory makes cylindrical tins with r = 4 cm and h = 10 cm from sheet metal. How many tins can be made from 10 m² of sheet metal? (1 m² = 10000 cm²)
Explain, in words a parent could follow, how to find the volume of any prism.
Surface Area — Drawing Nets
Draw and use nets to find surface area.
Draw the net of a rectangular prism with dimensions 5 cm × 3 cm × 2 cm. Label each face with its area.
Use your net to calculate the total surface area.
Draw the net of a cylinder with r = 4 cm and h = 6 cm. Label each part with its area or dimensions.
Calculate the total surface area of the cylinder from your net.
Real World: Concrete and Construction
Apply volume to construction contexts.
A concrete slab for a driveway: 10 m × 4 m × 0.15 m. Calculate the volume of concrete needed in m³.
Concrete costs $250 per m³. What is the total cost? Add 10% for wastage.
A concrete post is cylindrical: diameter 30 cm, height 1.2 m. How many m³ of concrete does one post need? How many for 20 posts?
Volume in Context — Match to Application
Match each context to the volume measurement needed.
Measurement Investigation — Cylinders vs Prisms
Compare the efficiency of cylindrical and rectangular containers.
A rectangular carton: 7 cm × 7 cm × 12 cm. Calculate its volume and surface area.
A cylinder with the same volume: find r and h = 12 cm. Calculate the surface area. (Hint: V = πr²h, so r = √(V/(πh)).)
Which shape has less surface area? Why does this matter for packaging?
Surface Area — Painting and Coating
Apply surface area to painting and finishing problems.
A garden shed: 4 m × 3 m × 2.5 m (rectangular prism). The shed needs painting on all 4 sides and the roof only (not the floor). Calculate the surface area to be painted.
Paint covers 12 m² per litre and costs $28 per litre. How many litres needed and what is the total cost for one coat? For two coats?
A cylindrical water tower (r = 5 m, h = 10 m) needs to be painted on the curved side and top only. Calculate the surface area and the cost at $8/m².
Measurement — Making Connections
Connect surface area and volume to other areas of maths.
Explain how calculating the volume of a prism uses the concept of area from your earlier maths studies.
How does the idea of scale factor relate to surface area and volume? (For example: if a box is scaled by factor 3, what happens to its SA and V?)
Where else in Year 9 maths have you encountered the number π? What does it represent geometrically?
Surface Area and Volume at Home
Apply measurement concepts in everyday contexts.
- 1Collect 5 containers from your kitchen (boxes, cans, bottles). Measure each and calculate its volume and surface area. Compare your calculations to any volume stated on the label.
- 2Design a box that holds exactly 1 litre (1000 cm³). Try three different shapes — a cube, a flat tray, and a tall narrow box. Which uses the least cardboard (smallest SA)?
- 3Research how 3D printers determine the amount of material (filament) needed. The calculation is essentially finding the volume of the object being printed.
Surface Area and Volume — Final Self-Assessment
Review and assess your measurement skills.
List all the formulas from this worksheet that you know from memory.
Which type of problem was easiest for you? Which was hardest? Explain.
Write a measurement problem that you could not have solved before this worksheet. Provide the full solution.
Surface Area — Advanced Decomposition
Break complex surfaces into manageable parts.
An L-shaped prism is 10 m long. The cross-section is an L-shape made of two rectangles: (6 × 4) and (2 × 3). Calculate the volume.
Identify all the faces of this L-shaped prism. How many faces does it have?
Calculate the total surface area of the L-shaped prism.
Comparing Volumes Algebraically
Compare volumes using algebra.
A cylinder has radius r and height 2r. Write its volume in terms of r.
A cube has side length s = 2r. Write its volume in terms of r.
For which value of r are the two volumes equal? Show your algebra.
Volume Investigation — Cylinder Proportions
Investigate how changing proportions affects cylinder volume.
A cylinder has V = 1000π cm³. If h = 10 cm, what is r? If h = 40 cm, what is r?
Compare the surface areas of both cylinders. Which is more efficient (smaller SA for the same volume)?
What does this tell you about the most efficient cylinder shape?
Measurement — Research and Extension
Extend your understanding through research.
Research the volume of the Great Pyramid of Giza. What shape is it? What formula would you need for its volume?
Archimedes' principle states: the buoyant force on an object equals the weight of fluid displaced. How does this relate to volume? Give an example.
Find one more real-world application of volume or surface area that interests you. Describe the application and the calculation involved.
Sort by SA:V Efficiency
Sort these shapes from least efficient (highest SA:V) to most efficient (lowest SA:V) for the same volume.
Volume — Challenging Context Problem
Solve this extended real-world problem.
A water tank is a cylinder with a hemispherical dome on top. The cylinder has r = 3 m and h = 5 m. The hemisphere has radius 3 m. The volume of a hemisphere is ⅔πr³. Find the total volume of the tank.
The tank is 90% full. How many litres of water does it contain?
Water is drawn off at 50 litres per minute. How long until the tank is 20% full?
Surface Area and Volume — Comprehensive Year 9 Review
Complete this final review to demonstrate full mastery of measurement.
For a cylinder with r = 6 cm and h = 8 cm: (a) calculate volume, (b) calculate surface area. Show full working.
For a rectangular prism 12 × 8 × 5 cm: (a) calculate volume, (b) calculate surface area.
Convert both volumes to litres.
Which solid has the greater surface-area-to-volume ratio? Is it more or less efficient as a container?
Measurement Strategy — Which Formula?
Circle the correct formula for each context.
How much soil to fill a circular garden bed (cylinder)
How much paint to cover a rectangular shed
How much water a cylindrical pipe holds
Measurement — Peer Teaching
Prepare to teach surface area and volume to a younger student.
Write a clear explanation of the difference between surface area and volume, with a real-world analogy.
Write one worked example for each: (a) volume of a cylinder, (b) surface area of a rectangular prism.
Write one question at each level: easy, medium, hard. Provide answers.
Measurement — Capstone Reflection
Write a final reflection on surface area and volume.
Explain, in your own words, why surface area and volume are both important measurements and why neither alone is sufficient.
Give three examples from real life where knowing volume is more important than surface area, and three where surface area is more important.
What surprised you most about surface area or volume as you worked through this worksheet?