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S3 Maths | Surface Area and Volume of Prisms, Cylinders and Spheres Lesson-Prep Guide
Students substitute the diameter for the radius, write the sphere surface area as 2 pi r squared, and switch between squared and cubed units without noticing. This lesson does three things: build one formula table, write the radius before any substitution, and label every answer with its unit.
Objectives — Students can: (1) state that the volume of a prism or a cylinder is base area times height, and compute a cylinder with V = pi r squared h; (2) explain that a cylinder surface consists of two circular bases plus one curved surface, giving 2 pi r squared plus 2 pi r h, and show that the curved surface unrolls into a rectangle; (3) use V = 4/3 pi r cubed and A = 4 pi r squared for a sphere and explain that a sphere has no base; (4) split a composite solid into two basic solids and add or subtract accordingly; (5) convert to a single unit before calculating and label answers as squared or cubed.
01 Prerequisites (3-minute check)
If any of these fails, fix it before the lesson starts.
| Skill | Quick check | Pass mark |
|---|---|---|
| Area and circumference of a circle | What is the area of a circle of radius 5 cm? | 25 pi square centimetres |
| Diameter and radius | What is the radius when the diameter is 10 cm? | 5 cm |
| Unit conversion | How many square centimetres are in one square metre? | 10 000 |
02 Lesson flow (45 minutes)
| Time | Stage | Tutor does | Student does |
|---|---|---|---|
| 0-5 min | Wrong line first | Write “diameter 10 cm, V = pi times 10 squared times h” and ask what is wrong | Say the diameter was used as the radius |
| 5-14 min | Formula table | Build three rows: prism, cylinder, sphere, each with volume and surface area | Copy the table |
| 14-25 min | Cylinder surface | Roll a sheet into a cylinder and unroll the curved face into a rectangle, 2 pi r by h | Work one example with you |
| 25-35 min | Sphere | Model 4/3 pi r cubed and 4 pi r squared, stressing one radius and no base | Do two questions |
| 35-41 min | Composite solids | Model a cylinder plus a hemisphere by adding and a cylinder minus a cone by subtracting | Split one composite solid |
| 41-45 min | Exit check | Collect three items, mark on the spot, name the act to reteach | Hand in three items |
03 Teaching script (read it out)
Scene 1: Act 1: write the radius before anything goes into a formula
“The first move on every solids question is not choosing a formula; it is writing the radius. Most questions give the diameter, so a diameter of twelve centimetres means a radius of six. Write it down, because every formula here uses r, and substituting twelve is the error. The second move is to note the value of pi the question wants: use 3.14 when it says so, use twenty-two over seven when it says so, and keep pi in the answer when it says nothing. The third move is to decide whether you are finding volume, the space inside, measured in cubic units, or surface area, the skin around it, measured in square units. Most lost marks come from the radius and the unit, not from the formula itself.”
Radius first, then the value of pi, then squared or cubed.
Scene 2: Act 2: base area times height, and the curved face unrolls into a rectangle
“Prisms and cylinders share one volume sentence: base area times height. A cylinder has a circular base, so V equals pi r squared h. For surface area, count the faces out loud. A cylinder has two circular bases and one curved surface: the bases give 2 pi r squared, and the curved surface unrolls into a rectangle whose length is the circumference 2 pi r and whose width is the height h, so its area is 2 pi r h. A prism works the same way: two identical bases plus rectangles whose total area is the perimeter of the base times the height. Write the working in three lines, bases, curved surface, total, so a piece never goes missing. The classic error is counting the curved surface only, or one base instead of two, which halves the answer.”
Volume is base area times height; count every face, three for a cylinder.
Scene 3: Act 3: a sphere has one radius, and composites are add or subtract
“A sphere has one radius and two formulas: volume 4/3 pi r cubed and surface area 4 pi r squared. Remember the pattern: volume carries the cube and the fraction, surface area carries the square and the four. Do not try the prism method on a sphere, because it has no base. For a composite solid, split it first: ask which two basic solids make it. If they are joined, add, for example a cylinder plus a hemisphere; if one is cut out, subtract, for example a cone from a cylinder. Calculate each part fully before adding or subtracting, and never merge the steps. If one part is in centimetres and another in metres, convert first; otherwise the answer is out by a hundred or a thousand.”
Sphere: 4/3 pi r cubed and 4 pi r squared; composite solids split before adding or subtracting.
04 Question types and worked steps
| Type | Example | Steps | Answer |
|---|---|---|---|
| Cylinder volume | Find the volume of a cylinder of radius 7 cm and height 10 cm, using pi = 22/7. | (1) Write r = 7 and h = 10 (2) Use V = pi r squared h (3) V = 22/7 times 49 times 10 (4) Answer 1540 cubic centimetres | 1540 cubic centimetres |
| Cylinder surface area | Find the total surface area of the same cylinder. | (1) Two bases: 2 pi r squared = 2 times 22/7 times 49 (2) Curved: 2 pi r h = 2 times 22/7 times 7 times 10 (3) Add 308 and 440 (4) Answer 748 square centimetres | 748 square centimetres |
| Sphere | Find the volume and surface area of a sphere of radius 6 cm, using pi = 3.14. | (1) V = 4/3 times 3.14 times 216 (2) About 904.32 cubic centimetres (3) A = 4 times 3.14 times 36 (4) About 452.16 square centimetres | V is about 904.32 cubic centimetres and A about 452.16 square centimetres |
| Composite solid | A cylinder of radius 5 cm and height 12 cm is topped by a hemisphere of the same radius. Find the total volume, using pi = 3.14. | (1) Split into cylinder and hemisphere (2) Cylinder: 3.14 times 25 times 12 = 942 (3) Hemisphere: 2/3 times 3.14 times 125, about 261.67 (4) Add to get about 1203.67 cubic centimetres | About 1203.67 cubic centimetres |
Sanity check (teach this): Reasonableness checks: (1) Halve the diameter first; using it directly gives four times the volume. (2) Volume is cubic and surface area is square; the wrong unit loses the mark. (3) A cylinder total surface area must exceed its curved area; a smaller total means a base is missing. (4) A sphere surface area is four times a circle of the same radius; a value near pi r squared means the wrong formula. (5) When a composite solid is built by subtraction, the answer must be smaller than the larger part.
05 Independent practice (3 questions)
| Question | Answer | What to watch |
|---|---|---|
| Find the volume of a cylinder of radius 3 cm and height 8 cm, in terms of pi. | 72 pi cubic centimetres | Was the radius written first, and is the unit cubic? |
| Find the surface area of a sphere of diameter 20 cm, using pi = 3.14. | r = 10, A = 4 times 3.14 times 100 = 1256 square centimetres | Was the diameter halved first? |
| A tank measures 40 cm by 25 cm by 30 cm. How many litres can it hold? | 40 times 25 times 30 = 30 000 cubic centimetres = 30 litres | Was cubic centimetres converted to litres by dividing by 1000? |
06 Common mistakes and what to say
| Mistake | Student writes | Say this |
|---|---|---|
| Diameter used as radius | “Diameter 10 cm, so V = pi times 10 squared times h” | Formulas use r, so halve the diameter first. |
| Missing bases | Cylinder surface given as 2 pi r h only | A cylinder has three faces: two bases plus the curved surface. |
| Wrong sphere formula | “A = 2 pi r squared” | Sphere surface area is 4 pi r squared; volume is 4/3 pi r cubed. |
| Units not converted | Adding a centimetre part to a metre part | Convert first: one metre is a hundred centimetres, so areas differ by 10 000. |
07 Homework (paper, section and question numbers)
- Solids practice, questions 1-4 on this page — Write r and the chosen formula before substituting, then label the unit
- Prep-guide master index — The same splitting method works for cones and pyramids
- Redo this lesson — Design one composite solid of your own, then find its volume and surface area with the splitting steps shown
08 Exit check (last 5 minutes)
| Question | Answer | Decision |
|---|---|---|
| What is the volume formula for a cylinder? | V equals pi r squared h, base area times height | No answer means reteach Act 2 |
| Give the sphere surface area and volume formulas. | A equals 4 pi r squared and V equals 4/3 pi r cubed | Swapping them means reteach Act 3 |
| The question gives a diameter of 18 cm. What is the first move? | Halve it to a radius of 9 cm | Substituting 18 means reteach Act 1 |
09 Tutor FAQ
Should pi be 3.14 or twenty-two over seven?
Follow the question. With no instruction, keep pi in the answer or use 3.14 and write approximately. Mixing both gives an answer that will not match the mark scheme.
Must students draw composite solids?
Yes. Ask them to separate the two parts with a dashed line and label each radius and height, so a shared face is not counted twice.
Should cones be taught here?
One line about V equals one third pi r squared h is enough for contrast; the depth belongs to S4.
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A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.