L20 — Volume of Solids
Key Terms
- Volume
- The amount of 3D space a solid occupies, measured in cubic units (cm³, m³, etc.).
- Prism formula
- V = Abase × h for any solid with a constant cross-section.
- Pyramid / cone
- V = ⅓ × Abase × h — always one-third the volume of the enclosing prism or cylinder.
- Capacity
- The amount of liquid a container holds. Key conversions: 1 mL = 1 cm³, 1 L = 1000 cm³, 1 m³ = 1000 L.
- Composite solid
- A shape made by combining or subtracting simpler solids. Add volumes for joined parts; subtract for hollowed-out parts.
- Scale factor (volume)
- If all dimensions scale by k, volume scales by k³.
Volume Formulas
| Shape | Formula | Key idea |
|---|---|---|
| Prism (any cross-section) | V = Abase × h | Cross-section is constant |
| Cylinder | V = πr²h | Circular prism |
| Pyramid | V = ⅓ Abase × h | ⅓ of enclosing prism |
| Cone | V = ⅓πr²h | ⅓ of enclosing cylinder |
| Sphere | V = &frac43;πr³ | Radius is everything |
| Hemisphere | V = ⅔πr³ | Half a sphere |
Capacity
1 mL = 1 cm³ 1 L = 1000 cm³ 1 m³ = 1000 L
Prisms and Cylinders
For any shape with a constant cross-section, volume = (cross-section area) × length.
Worked Example 1 — Volume of a triangular prism
Right-triangle cross-section: legs 5 m and 8 m. Prism length 10 m.
Abase = ½ × 5 × 8 = 20 m². V = 20 × 10 = 200 m³.
Pyramids and Cones
A pyramid or cone has ⅓ the volume of the prism or cylinder with the same base and height. This can be verified by filling a cone-shaped cup exactly three times to fill an equal cylinder.
Worked Example 2 — Volume of a cone
Cone: radius 6 cm, height 9 cm.
V = ⅓π(6)²(9) = ⅓ × 36π × 9 = 108π ≈ 339.29 cm³.
Spheres
Worked Example 3 — Volume of a sphere
Sphere: diameter 10 cm, so r = 5 cm.
V = &frac43;π(5)³ = &frac43;π(125) = &frac{500}{3}π ≈ 523.60 cm³.
Composite Solids
Add or subtract volumes of simpler shapes. Identify each component carefully.
Worked Example 4 — Cone on a cylinder
A traffic cone: cylinder r=10 cm, h=5 cm; cone r=10 cm, h=30 cm on top.
Vcyl = π(100)(5) = 500π. Vcone = ⅓π(100)(30) = 1000π. Total = 1500π ≈ 4712.4 cm³.
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Volume of prisms. Fluency
Give answers to 2 decimal places where not exact.
- (a) Rectangular prism: l=9 cm, w=6 cm, h=4 cm.
- (b) Triangular prism: right-triangle legs 5 m and 12 m, length 8 m.
- (c) Trapezoidal prism: parallel sides 4 cm and 7 cm, height 5 cm, length 10 cm.
- (d) A prism whose cross-section is a right triangle with hypotenuse 13 cm and one leg 5 cm, length 6 cm.
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Volume of cylinders. Fluency
Give answers in terms of π and as a decimal (2 d.p.).
- (a) r = 4 cm, h = 9 cm.
- (b) r = 7 m, h = 3 m.
- (c) Diameter = 12 mm, h = 15 mm.
- (d) r = 2 cm, h = 20 cm. Then convert to millilitres.
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Volume of pyramids and cones. Fluency
Give answers to 2 decimal places.
- (a) Square pyramid: base side 6 cm, height 8 cm.
- (b) Rectangular pyramid: base 5×9 m, height 6 m.
- (c) Cone: r = 5 cm, h = 12 cm.
- (d) Cone: diameter = 8 m, h = 15 m.
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Volume of spheres and hemispheres. Fluency
Give answers in terms of π and as a decimal (2 d.p.).
- (a) Sphere: r = 6 cm.
- (b) Sphere: diameter = 10 m.
- (c) Hemisphere: r = 9 cm.
- (d) Sphere: V = 972π cm³. Find the radius.
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Volume of a composite solid. Understanding
The diagram shows a solid made of a hemisphere sitting on top of a cylinder. Both have radius 4 cm. The cylinder has height 9 cm.
- (a) Write down the formula for the volume of the hemisphere.
- (b) Calculate the volume of the hemisphere. Give your answer in terms of π.
- (c) Calculate the volume of the cylinder. Give your answer in terms of π.
- (d) Find the total volume of the composite solid to 2 decimal places.
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Find the missing dimension. Understanding
- (a) Cylinder: V = 200π cm³, r = 5 cm. Find h.
- (b) Rectangular prism: V = 360 cm³, l = 10 cm, w = 6 cm. Find h.
- (c) Cone: V = 100π cm³, h = 12 cm. Find r.
- (d) Sphere: V = 288π cm³. Find the radius.
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Compare containers. Understanding
Three containers each hold liquid.
- (a) Cylinder A: r=5 cm, h=10 cm. Cylinder B: r=10 cm, h=5 cm. Which holds more?
- (b) A cube of side 8 cm vs a sphere of diameter 10 cm. Which has greater volume?
- (c) A cone and a cylinder both have r=6 cm and h=12 cm. What fraction of the cylinder’s volume does the cone hold?
- (d) Two spheres: one has radius r, one has radius 2r. What is the ratio of their volumes?
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Liquid capacity problems. Understanding
- (a) A cylindrical water tank: r=1.2 m, h=2 m. Find its capacity in litres. (1 m³ = 1000 L)
- (b) A rectangular swimming pool: 25 m × 10 m × 1.8 m deep. Find its capacity in kilolitres (kL = 1000 L = 1 m³).
- (c) A conical funnel has r=5 cm and h=12 cm. How many millilitres does it hold? (1 mL = 1 cm³)
- (d) Water drains from the conical funnel at 20 mL/s. How long (in seconds) to empty it?
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Sphere and cylinder relationship. Problem Solving
A sphere is inscribed inside a cylinder (the sphere just touches the top, bottom, and curved sides). The sphere has radius r.
- (a) Write the height and radius of the cylinder in terms of r.
- (b) Find the volume of the cylinder in terms of r.
- (c) Find the volume of the sphere in terms of r.
- (d) What fraction of the cylinder’s volume is occupied by the sphere? (Archimedes proved this result over 2000 years ago.)
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Concrete footings. Problem Solving
A builder pours concrete footings for a house. Each footing is a rectangular prism (0.4 m × 0.4 m × 0.6 m deep) with a cylindrical hole (r=0.08 m, running the full 0.6 m depth) through the centre.
- (a) Find the volume of one footing (prism minus hole) to 4 decimal places in m³.
- (b) The house requires 16 footings. Find the total concrete volume.
- (c) Concrete costs $180/m³ (delivered). Find the cost of concrete for all footings.
- (d) If the cylindrical hole is later filled with a steel post of the same dimensions, find the volume of steel used for all 16 posts.