Surface Area & Volume of Composites
3D Shape → Volume Formula
Draw a line from each 3D shape to its volume formula.
3D Shape → Surface Area Formula
Draw a line from each 3D shape to its total surface area formula.
Identify the Shape from a Description
Circle the 3D shape that matches each description.
A solid with two parallel circular faces and a curved surface connecting them
A solid with one circular base and a curved surface tapering to a point
A solid where every point on the surface is the same distance from the centre
A solid with two identical triangular faces and three rectangular faces
Units for Volume vs Surface Area
Circle the correct unit for each measurement.
Volume of a water tank measured in metres
Surface area of a box measured in centimetres
Volume of a medicine capsule measured in millimetres
Area of fabric needed to cover a sphere, measured in metres
Prisms vs Pyramids vs Curved Solids
Sort each 3D shape into the correct column: Prism, Pyramid, or Curved Solid.
Volume of a Rectangular Prism
Circle the correct volume for each rectangular prism.
l = 6 cm, w = 4 cm, h = 3 cm
l = 10 m, w = 5 m, h = 2 m
l = 8 cm, w = 8 cm, h = 8 cm (a cube)
Volume of a Cylinder
Circle the correct volume for each cylinder. Leave answers in terms of π where shown.
r = 5 cm, h = 10 cm
r = 3 cm, h = 7 cm
r = 10 m, h = 4 m
Surface Area of a Cube
A cube has 6 identical square faces. SA = 6s². Circle the correct surface area.
Side length = 4 cm
Side length = 5 cm
Side length = 10 m
Volume of a Cone
V = ⅓πr²h. Circle the correct volume for each cone.
r = 3 cm, h = 12 cm
r = 6 cm, h = 9 cm
r = 5 m, h = 15 m
Volume of a Sphere
V = ⁴⁄₃πr³. Circle the correct volume for each sphere.
r = 6 cm
r = 3 cm
r = 10 m
Surface Area of a Cylinder
SA = 2πr² + 2πrh (two circular ends + curved surface). Circle the correct surface area.
r = 5 cm, h = 10 cm
r = 3 cm, h = 8 cm
r = 4 m, h = 6 m
Volume of a Triangular Prism
V = ½ × base × height of triangle × length. Circle the correct volume.
Triangle: base = 6 cm, height = 4 cm; prism length = 10 cm
Triangle: base = 8 m, height = 3 m; prism length = 5 m
Triangle: base = 10 cm, height = 12 cm; prism length = 15 cm
Volume of a Hemisphere
A hemisphere is half a sphere. V = ⅔πr³. Circle the correct volume.
r = 6 cm
r = 3 cm
r = 10 m
Composite Solid → Component Shapes
Draw a line from each composite solid to the shapes that make it up.
Steps: Volume of Cylinder + Hemisphere
Put the steps in order to find the volume of a silo (cylinder with a hemisphere on top), where r = 3 m and cylinder height = 8 m.
Steps: SA of a Composite Solid
Put the steps in order to find the total surface area of a cylinder (r = 4 cm, h = 10 cm) with a hemisphere (r = 4 cm) on top.
Which Formula for Each Component?
A composite solid is made of a cone sitting on top of a cylinder, both with radius r. Circle the correct formula for each part.
Volume of the cylinder component (radius r, height h₁)
Volume of the cone component (radius r, height h₂)
Curved surface area of the cylinder
Curved surface area of the cone (slant height l)
Volume of a Two-Part Composite
Calculate the total volume of each composite solid. Circle the correct answer.
A cylinder (r = 4 cm, h = 10 cm) with a hemisphere (r = 4 cm) on top. V = πr²h + ⅔πr³
A rectangular prism (5 × 5 × 8 cm) with a square pyramid (base 5 × 5, height 6 cm) on top. V = lwh + ⅓ × l × w × h₂
A cone (r = 3 cm, h = 10 cm) with a hemisphere (r = 3 cm) on top. V = ⅓πr²h + ⅔πr³
Water Tank — Cylinder + Cone Roof
Calculate the total volume and surface area of a water tank. Show all working.
A water tank is made from a cylinder with a cone-shaped roof. The cylinder has radius 2 m and height 3 m. The cone has the same radius and a height of 1 m. (a) Calculate the volume of the cylindrical part: V = πr²h (b) Calculate the volume of the cone roof: V = ⅓πr²h (c) Find the total volume of the tank. (d) Calculate the slant height of the cone: l = √(r² + h²) (e) Calculate the total external surface area (exclude the base): curved cylinder + curved cone + base circle of cone is internal. Hint: Cylinder curved SA = 2πrh, Cone curved SA = πrl, Cylinder base = πr².
Design a Container for a Specific Volume
Design a container and verify it holds the required volume.
A juice company wants a bottle that holds exactly 600 mL (600 cm³). The bottle is shaped as a cylinder with a hemisphere dome on top (same radius). Choose a radius r, then calculate the cylinder height h needed so the total volume equals 600 cm³. Use: V_total = πr²h + ⅔πr³ = 600 Solve for h: h = (600 − ⅔πr³) ÷ (πr²) Try r = 4 cm. Show all working and check your answer.
Compare Surface Areas — Same Volume, Different Shapes
Investigate which shape uses the least material for a given volume.
Three containers each hold a volume of 1000 cm³. (a) A cube: Find the side length s where s³ = 1000, then calculate SA = 6s². (b) A cylinder with height equal to diameter (h = 2r): Find r where πr²(2r) = 1000, then calculate SA = 2πr² + 2πr(2r) = 6πr². (c) A sphere: Find r where ⁴⁄₃πr³ = 1000, then calculate SA = 4πr². Which shape has the smallest surface area? What does this tell us about efficient packaging?
Swimming Pool — Rectangular + Semicircular End
Calculate the volume of a swimming pool with a composite cross-section.
A swimming pool is 25 m long with a uniform depth of 2 m. The main section is a 25 m × 10 m rectangle. At one end, there is a semicircular extension with diameter 10 m (radius 5 m), also 2 m deep. (a) Calculate the volume of the rectangular section: V = 25 × 10 × 2 (b) Calculate the volume of the semicircular section: V = ½ × π × 5² × 2 (c) Find the total volume of the pool. (d) If 1 m³ = 1000 litres, how many litres does the pool hold?
Calculate Paint Needed for a Room
Use surface area to calculate how much paint is needed.
A room is 5 m long, 4 m wide and 2.7 m high. It has: • 1 door (0.9 m × 2.1 m) • 2 windows, each 1.2 m × 1.0 m (a) Calculate the total wall area: 2 × (5 × 2.7) + 2 × (4 × 2.7) (b) Subtract the door and windows. (c) Add the ceiling area: 5 × 4 (d) If one litre of paint covers 12 m², how many litres are needed? (Round up to a whole number.)
Ice Cream Cone + Hemisphere Scoop
Calculate the total volume of an ice cream cone with a scoop on top.
An ice cream cone has radius 3 cm and height 10 cm. A hemispherical scoop of ice cream (radius 3 cm) sits on top. (a) Calculate the volume of the cone: V = ⅓πr²h = ⅓ × π × 9 × 10 (b) Calculate the volume of the hemisphere: V = ⅔πr³ = ⅔ × π × 27 (c) Find the total volume. (d) If the ice cream melts and runs into the cone, will the cone overflow? Compare the hemisphere volume to the empty space in the cone above the ice cream line.
True or False — SA and Volume Relationships
Circle TRUE or FALSE for each statement.
Doubling every dimension of a solid multiplies its volume by 8
Doubling every dimension of a solid multiplies its surface area by 4
A sphere has the smallest surface area of any shape for a given volume
If two solids have the same volume, they must have the same surface area
When you join two solids, the composite's total SA is less than the sum of the individual SAs (because the joined faces are hidden)
Volume scales with the cube of the linear dimension (×k³), while SA scales with the square (×k²)
Optimisation — Least Material for a Given Volume
Investigate which composite design uses the least material.
A storage container must hold 500 cm³. Compare two designs: Design A: A cylinder (r = 4 cm) with height chosen to give V = 500 cm³. → h = 500 ÷ (π × 16). Calculate h, then find the total SA = 2πr² + 2πrh. Design B: A cylinder (r = 4 cm) topped with a hemisphere (r = 4 cm), with cylinder height chosen so total V = 500 cm³. → V_hemi = ⅔π(64) = 128π/3. Cylinder V = 500 − 128π/3. Calculate h, then find total SA (cylinder curved + bottom circle + hemisphere curved). Which design uses less material? Why might the hemisphere design be more efficient?
Measure Household Objects
Find composite 3D shapes around your home and calculate their volume and surface area.
- 1Find a drink bottle or thermos. Sketch it as a combination of simple shapes (cylinder + dome, etc.). Measure the dimensions and calculate its approximate volume. Compare with the stated capacity on the label.
- 2Measure a saucepan (cylinder) with its lid (could be a hemisphere or flattened dome). Calculate the total volume the pan can hold.
- 3Find a tin can. Measure its radius and height, then calculate both the volume (πr²h) and the total surface area (2πr² + 2πrh). Compare your calculated volume with the volume printed on the label.
Composite Shapes in Architecture
Explore how composite solids are used in buildings and structures around you.
- 1Look at buildings in your neighbourhood. Sketch one that uses composite shapes (e.g. a rectangular building with a triangular or domed roof). Estimate the dimensions and calculate the approximate volume.
- 2Measure a room in your home. If you wanted to paint the walls and ceiling, calculate the total surface area (subtracting windows and doors). Estimate how many litres of paint you would need if 1 litre covers 12 m².
- 3Find a structure with curved and flat surfaces (e.g. a letterbox, water tank, or garden feature). Identify the component shapes, estimate dimensions, and calculate the total volume.
Surface Area of Composite Solids
Calculate the surface area of composite 3D shapes.
A cylinder (radius 4 cm, height 10 cm) has a hemisphere (radius 4 cm) placed on top. Calculate the total surface area. Note: the circular face where they join is internal, so exclude it from both shapes.
A rectangular prism (15 cm × 8 cm × 6 cm) has a square pyramid (base 8 cm × 6 cm, slant height 7 cm) on top. Find the external surface area. Which faces are hidden by the join?
3D Shape to Surface Area Formula
Draw a line from each 3D shape to its surface area formula.
Calculate Volume — Which Formula?
Circle the correct formula for calculating volume of each shape.
Volume of a cylinder
Volume of a cone
Volume of a sphere
Volume of a pyramid
Volume of Composite Solids
Calculate volumes of composite 3D shapes.
An ice cream cone has: a cone of radius 3 cm and height 12 cm, with a hemisphere of radius 3 cm on top. Find the total volume in cm³ to 2 decimal places.
A swimming pool has a rectangular section (20 m × 10 m × 2 m deep) and a semicircular end section (radius 5 m, depth 2 m). Calculate the total volume of water the pool can hold. Convert to litres (1 m³ = 1,000 L).
Units — Area vs Volume
Sort each unit: Area Unit or Volume Unit.
Surface Area Calculations — Check the Answer
Circle the correct surface area for each shape.
Sphere with radius 5 cm. SA = 4πr²:
Closed cylinder, radius 3 cm, height 8 cm. SA = 2πr² + 2πrh:
Cone (including base), radius 6 cm, slant height 10 cm. SA = πr² + πrl:
Optimisation — Minimising Surface Area
Explore how to minimise material used for a given volume.
A cylindrical can must hold exactly 500 cm³ of liquid. Express the surface area S in terms of radius r only (eliminate h using V = πr²h = 500). For r = 3, 4, 5, 6 cm, calculate the total surface area each time. Which radius gives the minimum surface area?
For a cylinder with fixed volume, why is the optimal shape (minimum surface area) the one where height = 2 × radius? Explain the real-world significance for packaging design.
Steps for Finding Surface Area of a Composite Solid
Put the steps in the correct order.
Scale Factor and Volume
Explore the relationship between scale factor and volume.
A model car is built at 1:20 scale. The original car has a volume of 8,000,000 cm³. What is the volume of the model? Explain why you cube the scale factor.
A tank is scaled up by a factor of 3 in all dimensions. By what factor does: (a) the surface area increase? (b) the volume increase? State the general rule for scale factor k.
3D Shape to Real-World Object
Draw a line from each 3D shape to a real-world object it models.
Area and Volume — Unit Conversions
Convert between units for area and volume.
Convert each: (a) 3.5 m² to cm² (b) 450 cm³ to mL (c) 2.4 m³ to litres (d) 12,500 cm² to m² (e) 0.008 m³ to cm³. Show all conversion factors used.
Composite Volume — Add or Subtract?
Circle whether you ADD or SUBTRACT volumes for each composite solid.
A rectangular block with a cylindrical hole drilled through it
A cone placed on top of a cylinder (like a rocket shape)
A hemispherical bowl (solid hemisphere minus inner hemisphere)
Surface Area in the Kitchen
Measure and calculate surface areas of real containers at home.
- 1Choose 3 tins from the pantry (e.g., soup, beans, tomatoes). Measure radius and height. Calculate the surface area of each. If the metal costs $0.002 per cm², estimate the material cost of each tin.
- 2Compare a cylindrical cup and a rectangular juice box that hold the same volume. Calculate the surface area of each. Which uses less material?
- 3Design your own container to hold exactly 1 litre (1,000 cm³). Choose any shape. Calculate its surface area. Try two different shapes and compare which uses less material.
Archimedes' Principle — Volume by Displacement
Connect geometry to real measurement methods.
A rock is placed in a rectangular tub of water (30 cm × 20 cm base). The water level rises by 0.4 cm. Calculate the volume of the rock in cm³. If the rock has a mass of 120 g, find its density in g/cm³.
Explain why Archimedes' water displacement method works for measuring the volume of irregular objects but not for measuring the volume of a porous sponge (which absorbs water).
Effect of Doubling a Dimension
Sort each statement: True or False when only one dimension of a cylinder is doubled.
Errors in Surface Area Calculations
Tally the types of errors students made in a surface area assignment.
| Item | Tally | Total |
|---|---|---|
Wrong formula used | ||
Forgot to exclude hidden faces | ||
Unit conversion error | ||
Arithmetic error | ||
Used diameter instead of radius |
Packaging Design Challenge
Apply surface area and volume to a real design problem.
A company sells 500 mL of juice. They can package it as: (a) a cylinder, (b) a rectangular prism with a square base. For each shape, find dimensions that minimise surface area. Compare the minimum surface areas and suggest which design is better.
Volume Formulas — Match the Shape
Draw a line from each 3D shape to its correct volume formula.
Surface Area of Composite Solids
Calculate the surface area of shapes made from two or more basic solids.
A cylinder (r = 5 cm, h = 10 cm) has a hemisphere on top. Find total surface area, noting the joining circle is internal.
A rectangular prism (l=8, w=4, h=3 cm) has a triangular prism on top (base 4, height 2 cm). Find the total external surface area.
Explain why composite surface area is not simply the sum of the individual surface areas.
Units of Volume — Choose Correctly
Circle the most appropriate unit for each measurement.
Volume of a swimming pool
Volume of a grain of rice
Volume of water in a 600 mL bottle
Volume of a shipping container
Surface Area or Volume?
Sort each problem into whether you need Surface Area or Volume to solve it.
Volume and Capacity Conversions
Convert between units and solve capacity problems.
Convert 2.5 m³ to litres. (1 m³ = 1000 L)
A fish tank is 60 cm × 30 cm × 40 cm. Find its volume in cm³ and capacity in litres.
A cone has radius 7 cm and height 15 cm. Find its volume to 2 decimal places.
A factory fills cylindrical cans (r = 4 cm, h = 12 cm). How many cans does 10 L fill?
Formula Errors in Surface Area Problems
Tally common mistakes made when calculating surface area.
| Item | Tally | Total |
|---|---|---|
Wrong formula selected | ||
Forgot to double the base | ||
Used diameter not radius | ||
Incorrect unit conversion | ||
Added internal faces |
Spheres and Hemispheres
Apply sphere formulas to solve these problems.
Find the surface area of a sphere with radius 6 cm. Give your answer in terms of π and as a decimal.
Find the volume of a hemisphere with diameter 10 cm.
A sphere and a cube have the same surface area. Which has the greater volume? Test with numbers.
Surface Area and Volume at Home
Investigate surface area and volume using objects around your home.
- 1Find a tin can in your kitchen. Measure its radius and height, then calculate its surface area and volume.
- 2Compare two different-shaped containers that hold the same volume. Measure both and verify they hold the same amount.
- 3Design a box to hold 12 tennis balls in a 2×2×3 grid. What are the minimum dimensions? Calculate surface area.
- 4Research the surface-area-to-volume ratio and find out why it matters for cells, buildings, and packaging.
- 5Take photos of five 3D composite objects at home. Sketch and label each one.
Optimisation — Minimising Surface Area
Investigate how shape affects efficiency.
A cylindrical can must hold exactly 1 litre (1000 cm³). Write an expression for total surface area in terms of radius r only.
Use trial and error to find the radius that minimises surface area. What is the optimal height?
Why do aerosol cans and oil drums not follow the optimal proportions? Give two practical reasons.
Converting Between Units of Volume
Practise unit conversions for volume in real contexts.
How many millilitres are in 1 cm³? How many cm³ are in 1 L? Explain the relationship.
A swimming pool is 25 m long, 12 m wide, and 1.8 m deep. Find its volume in m³ and in kilolitres.
Convert 750 mm³ to cm³. Convert 2.4 m³ to mm³.
Surface Area Formula — Select the Right One
Circle the correct surface area formula for each shape.
Total surface area of a cylinder (radius r, height h)
Surface area of a sphere (radius r)
Lateral surface area of a cone (radius r, slant height l)
Total surface area of a cube (side a)
3D Shape Properties — Match the Fact
Match each 3D shape to one of its key properties.
3D Shapes in Architecture and Engineering
Explore how 3D geometry is used in design and construction.
- 1Research the Sydney Opera House. What 3D shapes make up its iconic roof? Estimate the surface area from publicly available dimensions.
- 2Look at three food packages at home. Identify the 3D shape of each and calculate which uses the least packaging material for a similar volume.
- 3Research how architects use software (like AutoCAD or Revit) to calculate surface areas and volumes. Write a paragraph about one use.
- 4Build a model of a composite solid using clay or paper. Measure its dimensions and calculate the surface area and volume.
- 5Find out why bees build hexagonal honeycombs. Research how the hexagonal prism minimises wax use for a given volume of honey.