Frusta from similar cones
Subtract a small similar cone and find the exposed curved and circular surfaces of the remaining frustum.
Extended only.
Before you begin
- Use similar lengths and cone formulas.
- Use Pythagoras for a right cone's slant height.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Small-to-large cone radius ratio is 1 : 2. What is the corresponding height ratio?
What you will learn
- Use similarity to recover the removed cone's dimensions.
- Calculate a conical frustum's volume and surface area.
Extend the sides back to the original apex
A conical frustum is formed by cutting a cone parallel to its base. The removed small cone is similar to the original large cone. If the frustum radii are R and r, its perpendicular height is H − h, and r/R = h/H. For R = 6, r = 3 and frustum height 4 cm, h = H/2 and H − h = 4, so H = 8 and h = 4 cm.
Subtract small-cone volume from large-cone volume. Here the large volume is π × 36 × 8/3 = 96π and the small volume is π × 9 × 4/3 = 12π, leaving 84π cm³. This method establishes the dimensions rather than guessing the full cone height from the frustum height. The shortcut π(H − h)(R² + Rr + r²)/3 follows from the same subtraction.
Subtract curves, then add the exposed end discs
The large cone slant is √(6² + 8²) = 10 cm and the small slant is 5 cm. Subtracting their curved areas gives π × 6 × 10 − π × 3 × 5 = 45π cm². The frustum's own slant length is 10 − 5 = 5 cm, so π(R + r)l gives the same curved area.
A closed frustum also has two exposed circular discs: πR² + πr² = 45π cm² here. Total area is therefore 90π cm². An open vessel omits the top disc; a cut-out lampshade may omit both discs. Its height, slant and radius difference form a right triangle, l² = (R − r)² + (H − h)², for a right circular frustum.
Pause and explain
A right frustum has radii 6 and 3 cm and height 4 cm. Find its slant length.
Worked example
A right conical frustum has radii 6 cm and 3 cm and height 4 cm. Find its volume and closed total area.
Show the worked solution
- Similarity gives full cone height 8 cm and removed cone height 4 cm.
- Subtract volumes: 96π − 12π = 84π cm³.
- Frustum slant is √(3² + 4²) = 5 cm; curved area is π(6 + 3) × 5 = 45π cm².
- Add both end discs, 36π + 9π, giving total area 90π cm².
Answer Volume 84π cm³; closed area 90π cm².
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
For radii 6 and 3 cm and height 4 cm, recover both complete cone heights.
Give me a hint
The removed cone's height is half the original's.
Compare my reasoning
- Let original height be H and removed height h = H/2.
- H − h = 4, so H/2 = 4.
- Thus H = 8 cm and h = 4 cm.
Original height 8 cm; removed height 4 cm.
Look for these in your work
- Used corresponding length similarity.
- Treated frustum height as a difference.
A right frustum has radii 4 and 2 cm and height 6 cm. Find its exact volume.
Give me a hint
The full cone height is twelve and the removed height six.
Compare my reasoning
- Radius ratio 1/2 gives original height 12 cm and small height 6 cm.
- Large volume is 64π cm³ and small volume is 8π cm³.
- Subtract to get 56π cm³.
56π cm³.
Look for these in your work
- Recovered full cone dimensions.
- Subtracted the removed volume.
The radius-6, radius-3, height-4 cm frustum is open at the smaller top and closed at the bottom. Find its material area, ignoring thickness.
Give me a hint
Keep the curved wall and larger bottom only.
Compare my reasoning
- Its slant is 5 cm, so curved area is 45π cm².
- The large bottom area is 36π cm².
- Total material area is 81π cm²; exclude the open small disc.
81π cm².
Look for these in your work
- Included the correct end disc.
- Excluded the open top.
Common mistakes
- Using frustum height as the original full-cone height.
- Using radius sum instead of difference for slant height.
- Counting a disc over an open end.
What can you explain now?
Why is the removed cone similar to the original?
Compare with the explanation
The cut is parallel to the base.
The parallel cross-section preserves corresponding angles and scales every length consistently.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Tomorrow, extend a frustum to its apex, use similar heights and subtract the two cone volumes.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
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Number contains 19 sequenced lessons; Algebra and graphs 21; Coordinate geometry 10; Geometry 15; Mensuration 13; Trigonometry 13; Transformations and vectors 11; Probability 7; Statistics 13. All 72 syllabus section entries now link to teaching and staged practice. Seven mixed checkpoints sample skills and suggest review lessons. Nine earlier overviews remain available. Full cumulative assessment and human teacher review remain pending. Written lesson practice, charts and constructions are self-checked, not automatically graded. Checkpoints do not save an exam grade or certify mastery.
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