Cones & spheres · Ages 14–15
Solids at work
Recasting keeps the volume; frustum = big cone − small cone.
How to do it
Method
Recasting keeps the volume; frustum = big cone − small cone
63 ÷ 23 = 27 balls
Another way
A frustum by similarity
Heights 1 : 2 → volumes 1 : 8 → 78
In full
Melting and recasting: metal melted down and cast into another shape keeps its volume. A metal sphere of radius 6 cm melted into balls of radius 2 cm: 43 π × 216 ÷ (43 π × 8) = 27 balls; the π and the 43 cancel, so compare r³ alone: 63 ÷ 23 = 27. Pouring: the water's volume stays the same from one container to the next, and volume ÷ base area gives the new depth. A frustum (a cone with its top cut off): volume = the big cone − the small cone. Or by similarity: the small cone cut off is similar to the big one; heights in the ratio 1 : 2 make volumes 1 : 8, so the frustum is 78 of the big cone. A composite solid (an ice cream, a cone and a hemisphere; a pencil, a cylinder and a cone): part by part, then add; for a surface area count only the faces on the outside.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
An ice cream is a cone with a hemisphere on top, both of radius 6 cm, the cone 9 cm high. What is its total volume, in terms of π?
Show the answer and working
Answer: 252 π cm³
The cone 13 × π × 62 × 9 = 108π, the hemisphere 23 × π × 63 = 144π, 252π cm³ in all.
Example 2
A metal ball of radius 12 cm can be melted and cast into 27 small balls of radius 4 cm, none wasted.
True or false?
Show the answer and working
Answer: True
Volumes go as the radii cubed: (12 ÷ 4)3 = 33 = 27.
Example 3
The frustum in the picture has a base radius of 7 cm, a top radius of 4 cm and a height of 6 cm. Put back the cone it was cut from: the small cone cut off is cm high, and the frustum's volume, in terms of π, is π cm³.
Fill in each blank
Show the answer and working
Answer: 8, 186
Similar: small height : big height = 4 : 7, and the big height = the small + 6, so the small cone is 8 high and the big 14; volume 13 × π × (72 × 14 − 42 × 8) = 186π.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.A cone of base radius 6 cm and height 24 cm is cut by a plane parallel to its base, the small cone on top 12 as high as the whole. What fraction of the whole cone's volume is the small cone?
- A. 18
- B. 14
- C. 12
- D. 38
Choose the right answer
Answer:
2.The cone-shaped cup in the picture is full of water; it is 9 cm in radius and 21 cm high. All the water is poured into a cylindrical cup, also 9 cm in radius. How deep is the water, in cm?
Answer:
3.An ice cream is a cone with a hemisphere on top, both of radius 9 cm, the cone 9 cm high. What is its total volume, in terms of π?
Answer:
4.The frustum in the picture has a base radius of 9 cm, a top radius of 6 cm and a height of 10 cm. Put back the cone it was cut from: the small cone cut off is cm high, and the frustum's volume, in terms of π, is π cm³.
Fill in each blank
Answer:
5.A sharpened pencil is a cylinder of radius 5 mm and length 9 mm with a cone-shaped tip 4 mm high on one end, of the same radius. Correct to 1 decimal place, what is the pencil's volume, in mm³?
Answer:
All the answers
- 1. A. 18
- 2. 7 cm
- 3. 729 π cm³
- 4. 20, 570
- 5. 811.6 mm³
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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