Learn: covering a party hat
A party hat needs material for its curved side, usually with an open bottom. Covering a closed model cone also requires a circular base. Decide which surfaces count before calculating.
These formulas describe a right circular cone. Its tip is directly above the base centre. The radius r goes from the centre to the edge; h is the perpendicular height; and the slant height ℓ follows the sloping side from the tip to the circular rim. All three are lengths.
Lateral area L = πrℓ covers the curved side. The base area is B = πr². The total area of the closed cone is A = πrℓ + πr² = πr(ℓ + r). Areas use square units such as cm².
Why does slant height appear?
Imagine cutting the curved surface along one sloping edge and flattening it. It becomes a sector of a circle with radius ℓ. Its arc length equals the base circumference, 2πr. Its fraction of a full circle is (2πr)/(2πℓ) = r/ℓ. Multiplying that fraction by πℓ² gives πrℓ.
When r and perpendicular h are known, ℓ = √(r² + h²) by the Pythagorean theorem. The square-root symbol √ asks for the positive length whose square equals r² + h². Do not substitute h directly into πrℓ, or apply this single-slant-height formula to an arbitrary oblique cone.
Worked example: radius 3 cm, height 4 cm
- Find ℓ = √(3² + 4²) = √25 = 5 cm.
- Curved side: L = π × 3 × 5 = 15π cm² ≈ 47.12 cm².
- Base: B = π × 3² = 9π cm² ≈ 28.27 cm².
- Closed cone: A = 24π cm² ≈ 75.40 cm².
An open-bottom hat uses the lateral result. The total result includes the base. These are ideal geometric areas; any seam allowance must be specified separately.
Worked example: radius 4 cm, slant height 9 cm
The side area is 36π cm². Adding the base, 16π cm², gives 52π cm² ≈ 163.36 cm². Here 9 cm is already a slant height, so no square-root calculation is necessary.
Explore: change one dimension
The diagram is schematic, not to scale. Quantities below it update when you move the sliders. Predict the change first, then compare the result.
Try radius 3 cm, then 6 cm. For volume, keep the height fixed where available. Explain why the change differs from simply doubling the result. The shape drawing stays fixed so you can focus on the labelled dimensions and numerical comparison.
Practice: explain before checking
Work each question on paper. Open the explanation to compare your setup, calculation and units.
1. A right cone has r = 5 cm and ℓ = 13 cm. Find lateral and total area.
L = 65π cm². The base adds 25π cm², so A = 90π cm². Approximate values are 204.20 cm² and 282.74 cm².
2. A right cone has r = 5 cm and h = 12 cm. What is its slant height?
ℓ = √(25 + 144) = √169 = 13 cm. The perpendicular height and slant height describe different segments.
3. A hat has no base. Should you add πr²?
No. Count only the surfaces the problem asks you to cover. For the ideal open-bottom hat, use πrℓ.
Review
Use slant height for curved area, perpendicular height for volume, and add the base only when it is included. Area uses square units.
Recall without looking: What does each symbol measure? Which units belong in the answer? Which surfaces or spaces are included?