Overview
What you’ll learn
- Calculate the volume of a pyramid using one third × base area × height.
- Calculate the volume and curved surface area of a cone, using the slant height where needed.
- Use Pythagoras’ theorem to find a cone’s slant height from its radius and height.
- Calculate the volume and surface area of a sphere.
- Find volumes and surface areas of composite solids by adding or subtracting parts.
- Convert between cubic centimetres and litres in filling and capacity problems.
In 30 seconds
- 1Pyramid volume: V = ⅓ × base area × height — exactly one third of the prism on the same base.
- 2Cone volume: V = ⅓ × pi × r² × h — a pyramid with a circular base.
- 3A cone’s slant height l comes from Pythagoras: l² = r² + h².
- 4Cone curved surface: pi × r × l; add the circular base pi × r² for a closed cone.
- 5Sphere: V = 4/3 × pi × r³ and S = 4 × pi × r².
- 6Composite solids split into parts: add volumes of joined pieces, subtract for holes; scale factors of 3 on radius multiply volume by 27.
Notes · 1 of 6
Pyramids: the one-third rule
A pyramid is a prism with a taper — and the taper costs two thirds of the space.
A pyramid has a flat base and triangular faces rising to one point, the apex. Its volume is ⅓ × base area × height, where the height is the perpendicular drop from the apex to the base plane. Pour water into a pyramid-shaped container and into a prism of the same base and height: the prism takes exactly three pours.
| Base | Base area | Height | Volume |
|---|---|---|---|
| Square 6 × 6 | 36 cm² | 10 cm | ⅓ × 36 × 10 = 120 cm³ |
| Rectangle 5 × 8 | 40 cm² | 9 cm | ⅓ × 40 × 9 = 120 cm³ |
| Triangle, area 30 cm² | 30 cm² | 12 cm | ⅓ × 30 × 12 = 120 cm³ |
Any base shape works — the base area does the talking. Compute the base area first, write ⅓ × B × h, and only then multiply; inventing a “slant height” into the volume formula is the classic error (the slant height belongs to surface area questions).
Notes · 2 of 6
Cones: circular pyramids with a slant
One third of a cylinder — plus a curved surface that needs the slant height.
A cone is a pyramid whose base is a circle, so its volume is ⅓ × pi × r² × h. Its slant height l — the distance from apex to rim — is the hypotenuse of a right triangle with legs r and h: l² = r² + h².
| Quantity | Formula | Needs |
|---|---|---|
| Volume | ⅓ × pi × r² × h | radius and height |
| Slant height | l = √(r² + h²) | radius and height |
| Curved surface | pi × r × l | radius and slant height |
| Total (closed cone) | pi × r × l + pi × r² | all three |
Watch the hand-off: the volume and the slant height both start from r and h, but the curved surface uses l, not h. With r = 3 and h = 4: l = √(9 + 16) = 5, so the curved surface is pi × 3 × 5 = 15 pi — a 3-4-5 triangle wearing a hat.
Notes · 3 of 6
Spheres
Two formulas, one letter: everything depends on the radius alone.
| Quantity | Formula |
|---|---|
| Volume | V = 4/3 × pi × r³ |
| Surface area | S = 4 × pi × r² |
With r = 7 and pi = 22/7: volume = 4/3 × 22/7 × 343 = 4 312/3 ≈ 1 437.3 cm³, and surface = 4 × 22/7 × 49 = 616 cm². Notice how the 7s cancel beautifully with 22/7 — question setters choose radii of 7, 14 or 21 for exactly this reason.
Radius cubed means radius changes hit volumes hard: doubling r multiplies V by 8, and tripling by 27. Surface area, growing with r², “only” multiplies by 4 or 9 — a favourite conceptual check.
Notes · 4 of 6
Surface area strategy
Unwrap the solid into flat pieces, then add every piece once.
| Solid | Flat pieces | Area pieces |
|---|---|---|
| Square pyramid | 1 square + 4 triangles | base + 4 × (½ × base × slant height) |
| Cone (closed) | 1 circle + a curved wrap | pi × r² + pi × r × l |
| Sphere | one curved skin | 4 × pi × r² |
Two cautions. First, a pyramid’s triangular faces use the slant height of the face, not the pyramid’s vertical height — for a square pyramid the face height comes from Pythagoras on half the base and the vertical height. Second, an open cone or tank loses a piece: the instruction “no base” deletes pi × r² from the total.
Notes · 5 of 6
Composite solids
Build big solids from cones, spheres and prisms — volumes add, surfaces need care.
A composite solid is glued from simple ones: a hemisphere on a cylinder, a cone on a box. Volumes simply add (or subtract, for holes). Surface areas need thought: where two pieces join, the hidden circle or face disappears from the total.
| Step | Action |
|---|---|
| 1. Split | name the parts and their radii/heights |
| 2. Volume | add the separate volumes |
| 3. Surface | add exposed faces only; delete any covered circle or face |
| 4. Check | does each formula use the right letter (r, h or l)? |
Notes · 6 of 6
Capacity and filling
Cubic centimetres become litres, and litres meet rates.
| Conversion | Value |
|---|---|
| 1 litre | 1 000 cm³ |
| 1 m³ | 1 000 000 cm³ = 1 000 litres |
| Filling rate | litres ÷ (litres per minute) = minutes |
Capacity questions run: compute the volume in cm³, divide by 1 000 for litres, then chain the rate if asked how long a filling takes. Keep the units written on every line — the arithmetic is easy; the units are where marks go missing.
Revise
Key terms
The words examiners expect you to use precisely.
- Pyramid
- A solid with a flat polygon base and triangular faces meeting at an apex.
- Apex
- The single point where a pyramid’s or cone’s faces meet.
- Cone
- A solid with a circular base tapering to an apex; volume = ⅓ pi r² h.
- Sphere
- A perfectly round solid; every surface point sits one radius from the centre.
- Hemisphere
- Half a sphere, cut through its centre; volume = 2/3 pi r³.
- Slant height
- A cone’s (or pyramid face’s) distance from apex to the base edge along the surface.
Show 6 more key termsShow fewer key terms
- Vertical height
- The perpendicular drop from apex to base plane; the h in volume formulas.
- Curved surface area
- The area of a cone’s sloping wrap: pi × r × l.
- Composite solid
- A solid built by joining simpler solids; volumes add.
- Capacity
- The amount a container holds, measured in litres; 1 L = 1 000 cm³.
- Scale factor (cubic)
- Scaling a length by k multiplies volumes by k³.
- Open (no base)
- An instruction to exclude a face from a surface-area total.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Dropping the ⅓ from a pyramid or cone volume.
⅓ × base × height is the whole point; the untapered prism answer is three times too big.
Instead: Cone: ⅓ pi r² h. Pyramid: ⅓ × base area × h.
Mistake: Using the vertical height h in the cone’s curved surface formula.
The wrap needs the slant height l, the rim-to-apex distance.
Instead: Curved surface = pi × r × l, with l² = r² + h².
Show 8 more mistakesShow fewer mistakes
Mistake: Using the slant height as the cone’s height in the volume.
Volume needs the perpendicular height from apex to base centre.
Instead: l and h are different lines; compute the one each formula asks for.
Mistake: Writing sphere surface area as 4/3 pi r² or volume as 4 pi r³.
The 4/3 belongs with the cube, the plain 4 with the square.
Instead: V = 4/3 pi r³; S = 4 pi r².
Mistake: Counting a hidden joining face in a composite solid’s surface area.
The circle or face where two parts meet is inside the solid — no longer exposed.
Instead: Add only exposed faces; delete the joint.
Mistake: Predicting volume changes by k instead of k³.
Volume scales with the cube of the length factor.
Instead: Tripling a radius multiplies volume by 27; doubling by 8.
Mistake: Leaving a litres answer in cm³ (or converting with 100 instead of 1 000).
1 litre = 1 000 cm³, not 100.
Instead: Divide cm³ by 1 000 for litres; check the size of the answer against the object.
Mistake: Feeding the diameter into a sphere formula, e.g. using r = 10 for a ball of diameter 10.
Every sphere formula takes the radius, which is half the diameter.
Instead: Halve first: d = 10 gives r = 5, then use 4/3 pi r³ or 4 pi r².
Mistake: Writing half a sphere’s volume as ½ pi r³.
Half of the sphere formula is half of 4/3, which is 2/3 — not 1/2.
Instead: Hemisphere volume = 2/3 pi r³.
Mistake: Saying a 1 m³ tank holds 1 000 cm³ of water.
The length factor is cubed for volume: 100³ = 1 000 000.
Instead: 1 m³ = 1 000 000 cm³ = 1 000 litres.
Check yourself
Quick quiz
18 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
State the pyramid rule
Compared with a prism on the same base and height, a pyramid’s volume is
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Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
- 5Foundation
- 6Standard
- 4Challenging
15 questions worth 58 marks, each with a model answer, mark scheme and examiner tips.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Chapter 9 Practice Paper
30 marks · 40 min · 5 pages
10 multiple-choice questions and structured questions with answer lines. Chapter 9.
Mark scheme
PROChapter 9 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
- Term 3Coming soonWeighted Assessment 3 (WA3)
- Term 4Coming soonEnd-of-Year Examination (EOY)
Questions
Frequently asked questions
What is Sec 2 Math Chapter 9 about?
Chapter 9 covers the volume of pyramids (⅓ × base area × height), cones and spheres, the cone’s slant height and curved surface, the sphere’s surface area, composite solids, and capacity conversions between cm³ and litres.
Why is a cone’s volume one third of a cylinder’s?
Filling experiments and the pyramid rule agree: a cone holds exactly one third of the cylinder sharing its radius and height. That is why V = ⅓ pi r² h — it is the circular version of the pyramid’s one-third rule.
How do I find a cone’s slant height?
Pythagoras: the radius and the vertical height are the legs, and the slant height is the hypotenuse, so l = √(r² + h²). The curved surface formula pi × r × l uses this slant height, never the vertical height.
How do I handle a composite solid?
Volumes of joined pieces simply add. For surface area, add only the exposed faces — the circle where a hemisphere meets its cylinder, for example, is hidden and must be left out of the total.
Up next
Chapter 10: Data Analysis — Averages