Cone

Properties

• 1 flat circular face and 1 curved face.
• 1 curved edge and 1 vertex.
• The radius, the height and the slant height form a right-angled triangle.
• Opened flat, the curved surface is a sector of a circle whose radius is the slant height.

Why the formulas work

A cone holds exactly one third of what a cylinder with the same base and height holds. You can test it by filling a paper cone with sand three times into such a cylinder. So the volume is ⅓πr²h. The curved surface opens into a sector with radius l and arc length 2πr, and the area of that sector is πrl.

Worked example

Take a cone with radius 7 cm and height 24 cm.

1. Slant height. Slant height l = √(r² + h²). Here: √(7² + 24²) = 25 cm.
2. Volume. Volume = ⅓ πr²h. Here: ⅓ × π × 7 × 7 × 24 = 1,231.5 cm³ (with π = 22/7: 1,232 cm³).
3. Curved surface area. Curved surface area = πrl. Here: π × 7 × 25 = 549.78 cm² (with π = 22/7: 550 cm²).
4. Total surface area. Total surface area = πr(r + l). Here: π × 7 × (7 + 25) = 703.72 cm² (with π = 22/7: 704 cm²).

Finding a missing measurement

From the slant height and the radius, h = √(l² − r²): l = 25 cm and r = 7 cm give h = 24 cm. From the volume, h = 3V ÷ πr²: 1232 cm³ and r = 7 cm give 3696 ÷ 154 = 24 cm, taking π = 22/7.

Mistakes to avoid

• Using the slant height in the volume formula, or the vertical height in the surface formula.
• Forgetting the one third.
• If the slant height is given and the height is needed, use h = √(l² − r²).

Where you see it

Ice-cream cones, traffic cones, party hats, funnels, the tip of a sharpened pencil and a heap of sand or grain are cones.

Try these

Work each one out, then tap to check.

More shapes