DSE Maths | Mensuration of Solids — Teacher Prep Guide

DSE Maths | Mensuration of Solids — Teacher Prep Guide | Cone nets, one third and k2/k3

2026-11-05

Tutor lesson-prep guides · Secondary Maths

Students can recite the formulas but cannot tell slant height from perpendicular height, so they substitute the wrong number the moment a question gives only r and h. This lesson starts from a net and links area, volume and similarity into one line.

Objectives — Students will be able to: (1) find the curved surface area of a cone from its net and explain that the sector radius is the slant height 13 cm, not the height 12 cm; (2) justify one third of a prism for cone volume and compute 314.2 cm^3; (3) state that similar solids have area ratio k^2 and volume ratio k^3, giving 9 and 27 when k = 3.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Circle area and circumference	Area and circumference of a circle of radius 5 cm	78.5 cm^2 and 31.4 cm
Pythagoras	Legs 5 and 12, find the hypotenuse	13
Units	How many cm^3 in 1 m^3	1,000,000

02 Lesson flow (45 minutes)

Time	Stage	Tutor does	Student does
0-6 min	Slant versus height	Draw both on one cone; which becomes the sector radius	Point to the sector radius
6-18 min	Area from the net	r = 5, l = 13, arc 31.4, angle 138.5 degrees	Compute the curved area
18-30 min	Cone volume	Pour three cones into one matching cylinder	Get 314.2 cm^3
30-40 min	Similar solids	k = 3 gives area x9 and volume x27	Pairs explain why
40-45 min	Displacement and units	Water rises 1.2 cm in a jar of radius 5 cm	Work alone

03 Teaching script (read it out)

Scene 1: Separate slant height from perpendicular height

The height 12 cm runs from the apex to the centre of the base; the slant height 13 cm runs from the apex to any point on the rim. Cut the curved surface open and the sector radius is the slant height, not the height.
Check with r = 5 and h = 12: 25 + 144 = 169, whose square root is 13. If a question gives only r and h, the first move is always to find the slant height.

Ask students: mark the height as a dashed line and the slant as a solid line, and write both lengths before substituting anything.

Scene 2: Curved area comes from the arc length

The sector arc must equal the base circumference, 31.4 cm, so the sector angle is arc divided by radius: 31.4 divided by 13 equals 2.417 radians, or 138.5 degrees.
Sector area is half times arc times radius: 0.5 x 31.4 x 13 = 204.2 cm^2, which is pi r l. Add the base of 78.5 cm^2 for a total of 282.7 cm^2.

Ask students: why the arc must equal the base circumference, and what a mismatch would mean.

Scene 3: One third is poured, not memorised

The matching cylinder holds 942.5 cm^3, and three cones of water fill it exactly, so a cone is one third: 314.2 cm^3. Students should pour it themselves once so the formula does not invert.
Similar solids: when the length ratio is k = 3, every length triples, area grows by 3^2 = 9 and volume by 3^3 = 27, because area multiplies two lengths and volume three.

Ask students: write the area and volume ratios next to the formula and justify them as two lengths and three lengths.

04 Question types and worked steps

Type	Example	Steps	Answer
Net and curved area	Given r and h, find curved or total surface area	Find slant first, then pi r l, add pi r^2 only for total	Slant always exceeds height
Cone and cylinder volume	Single or composite solid	Prism is base times height, cone then divides by three	Only cones divide by three
Similar solids	Given k, find area or volume ratio	Area k^2, volume k^3; reverse needs a square or cube root	k is a length ratio
Displacement and units	Object dropped in water, level rises	Rise volume equals base area times rise; unify units	1 m^3 = 10^6 cm^3

Sanity check (teach this): Sanity checks: the slant 13 exceeds the height 12 and can never be shorter; curved area 204.2 exceeds the base 78.5, and the total 282.7 is their sum; with k = 3 the ratios are 9 and 27 as k^2 and k^3 require; 2500 mL is 2500 cm^3, so a unit change never moves the answer by three orders of magnitude.

05 Independent practice (3 questions)

Question	Answer	What to watch
Net	Cone with r = 5 cm and h = 12 cm: curved and total surface area	Slant first, then pi r l
Volume	Volume of the same cone to one decimal place	Base area times height divided by three
Similar solids	Two similar cones have volumes 27 : 8, find the area ratio	Cube root for k, then square
Displacement	Jar of radius 5 cm, level rises 1.2 cm, find the stone volume	pi r^2 times rise, one decimal place

06 Common mistakes and what to say

Mistake	Student writes	Say this
Using height inside pi r l	Writing pi x 5 x 12	The sector radius is the slant, not the height. The height is inside the solid and disappears when you flatten it.
Forgetting to divide by three	Answering 942.5 cm^3	Three cones fill one cylinder, so your answer is three cones worth.
Treating k as the volume ratio	Answering 3 : 1 when k = 3	Every length triples, and volume multiplies three lengths, so it is 27.
Mixing units mid-calculation	Centimetres and metres in one line	One equation carries one unit; convert everything to centimetres before you calculate.

07 Homework (paper, section and question numbers)

Secondary Maths chapter index — Every chapter in the subject
All teacher prep guides — Guides for other subjects
Redo this chapter — Two net questions and two displacement questions, showing every line

08 Exit check (last 5 minutes)

Question	Answer	Decision
Which length is the sector radius	The slant 13 cm, not the height 12 cm	Answering height means restart act one
What fraction of a cylinder is a cone	One third	Answering one half means repeat act three
Volume ratio when k = 3	27 : 1	Answering 3 : 1 means repeat act three

09 Tutor FAQ

Must I always use Pythagoras?

Only when the question gives r and h. If the slant is given, use it directly; finding it is usually where the method mark sits.

Why area k^2 and volume k^3?

Area multiplies two lengths and volume three, and every length is scaled by k, so the factors are k^2 and k^3.

How do I split a composite solid?

Break it into cylinders, cones and spheres, total them, then subtract any overlapping part once.

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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.