Learn with Amar
Teaching video
A teaching recording for this chapter has not been published yet. Start with the worked example below and explore the related animations where available.
Grade 10 chapters and video availability01 · Read and understand
What you will learn
- Use similarity and volume subtraction for a cone cut by a plane parallel to its base.
- Justify the method and check its domain, units, or logical conditions.
Before you start
Algebraic equations, ratios, angle and area facts, and basic coordinate geometry.
Keep paper nearby. Read the question once for the context, then again to identify what is known and what you need to find.
Start with a question
A right cone has radius 3 cm and perpendicular height 6 cm. Remove its tip by a cut parallel to the base, leaving a smaller cone of radius 1 cm and height 2 cm. Find the remaining frustum's volume.
Why this math matters
Recover a truncated solid's volume by embedding it in a simpler solid and subtracting a similar missing part. The example connects a stated condition to a conclusion and then checks whether the result satisfies that condition.
Set up the model
A useful answer starts with clear assumptions:
- Use Euclidean geometry and the angle, parallelism, similarity, or congruence conditions stated; a sketch alone does not establish them.
- Treat provided measurements as exact classroom-model values unless an approximation or uncertainty is stated.
02 · Work through the example
Follow the reasoning, one step at a time.
Try to predict the next step before reading it. After each calculation, explain why the operation makes sense and how it helps answer the original question.
Work through it with Amar
See this example unfold.
The complete worked example, one idea at a time.
Subtract similar cones to find a frustum's volume
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A right cone has radius 3 cm and perpendicular height 6 cm. Remove its tip by a cut parallel to the base, leaving a smaller cone of radius 1 cm and height 2 cm. Find the remaining frustum's volume.
Before you calculate
Read what is known and what you need to find. Make a prediction before moving to the first calculation.
Starts paused. Play advances through the full text at a reading pace; pause whenever you need more time. Previous, Next, and the phase buttons let you set your own pace. Playback pauses when this walkthrough leaves the screen or you switch tabs.
Your device’s reduced-motion setting keeps each phase still. Manual controls remain available. The full written solution stays below.
Represent the conditions
Full cone: (1/3)π(3²)(6)=18π cm³
Compute the original solid before subtracting the removed portion.
Develop the calculation
Removed cone: (1/3)π(1²)(2)=2π/3 cm³
Its radius and height are both one third of the originals, consistent with a parallel cut.
Check and interpret
Frustum volume=18π−2π/3=52π/3 cm³
The remaining height is four; the frustum formula πh(R²+Rr+r²)/3 also gives 4π(9+3+1)/3=52π/3.
The result
Frustum volume=18π−2π/3=52π/3 cm³
The remaining height is four; the frustum formula πh(R²+Rr+r²)/3 also gives 4π(9+3+1)/3=52π/3.
Common mistakes to catch
- The removed cone's height is measured from the apex; the frustum height is the difference of the two cone heights.
- A cut parallel to the base forces equal radius and height scale factors.
03 · Practice independently
Try it before revealing the answer.
Use paper or a calculator as needed. Write your units and reasoning, then open the hint or explanation to check your approach.
Practice 1
A cone has radius 4 and height 8. A parallel cut removes a tip cone of radius 2 and height 4. Find the remaining volume.
Show a hint
Subtract the small cone from the full cone.
Reveal answer and explanation
112π/3 cubic units
The full volume is 128π/3 and the removed volume is 16π/3, leaving 112π/3.
Practice 2
For the main cone, what tip-cone radius results from a parallel cut 3 cm from the apex?
Show a hint
The tip's height is half of six, so its radius has the same scale.
Reveal answer and explanation
1.5 cm
Similar cones require r/3=3/6; a cut radius cannot be chosen independently of its distance from the apex.
Take the idea with you
Recover a truncated solid's volume by embedding it in a simpler solid and subtracting a similar missing part.
04 · Reflect and continue
Can you explain it in your own words?
Before moving on, explain the main idea without looking at the worked example. Try both practice questions, check your reasoning, and name one mistake you now know how to avoid. Return to a step if you still need support.
Up next: Unroll a cone to find its covering area
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