The cone and its slant
A heap of sand has a round base and a sharp point.
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- A cone: base radius r, height h.
- Apex, centre and rim form a right triangle.
- The slant l runs apex to rim.
- l is the longest side, never the height.
The rule
A cone here is a right circular cone. It has one round base. Its curved side rises to a point, the apex. The apex is straight above the centre of the base.
The height h runs straight down from the apex. It meets the base at the centre. The slant height l runs from the apex to the rim.
The radius, height and slant make a right triangle. The slant is the triangle's longest side. This is Pythagoras' rule. The longest side squared equals the other two sides squared, added.
l² = r² + h²
For r = 7 and h = 24: 7² = 49 and 24² = 576. The sign √ means square root.
l = √(49 + 576) = √625 = 25. Check: 25 × 25 = 625.
curved surface = π × r × l
total surface = π × r × l + π × r²
volume = ⅓ × π × r² × h
| Solid | Curved surface | Total surface | Volume |
|---|---|---|---|
| Cone | 22/7 × 7 × 25 = 550 | 550 + 154 = 704 | 3696 ÷ 3 = 1232 |
| Cylinder | 154 × 24 = 3696 |
Three cones of sand fill that cylinder. So one cone holds one third of the cylinder.