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 slant height for a cone's lateral surface area.
- 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 4 cm. Find its total surface area including the circular base.
Why this math matters
Choose covering formulas by inspecting the net and naming which faces are present. 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.
Unroll a cone to find its covering area
PausedQuestion: Start with the question. Paused.
Question
Start with the question
A right cone has radius 3 cm and perpendicular height 4 cm. Find its total surface area including the circular base.
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
Slant height l=√(3²+4²)=5 cm
The surface generator is the hypotenuse of a radial right triangle.
Develop the calculation
Lateral area = πrl=15π cm²
The unrolled lateral sector has radius l and arc length 2πr; its area is half their product.
Check and interpret
Total area = 15π+9π=24π cm²
Add the base disk exactly once because the cone is closed.
The result
Total area = 15π+9π=24π cm²
Add the base disk exactly once because the cone is closed.
Common mistakes to catch
- Do not substitute perpendicular height for slant height in πrl.
- Add a base only when the requested surface includes it.
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
For a cone with r=5 and l=13, find the lateral area only.
Show a hint
Multiply π, radius, and slant height.
Reveal answer and explanation
65π square units
The circular base is excluded by the wording.
Practice 2
Why does the volume height differ from the surface-area slant height?
Show a hint
One measures depth perpendicular to the base, the other distance along the side.
Reveal answer and explanation
They measure different geometric directions
A net uses the sloping surface radius, whereas volume uses perpendicular height.
Take the idea with you
Choose covering formulas by inspecting the net and naming which faces are present.
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: Change the denominator when a condition is given
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