The slant across the join
A toy order gives two numbers. The radius is 21. The total height is 49.
Step 1 of 4
- The toy stands 49 tall, all one length.
- The dome takes 21; the cone stands 28.
- The cone's own triangle: base 21, height 28.
- Pythagoras finds the slant: l = 35.
Split the height at the join
The cone's own height is not given. Only the total is. So split it at the join. Here r is the radius. The cone's height is h. The slant height is l.
The cone's height, radius and slant height form a right triangle. The slant height is the longest side. Pythagoras says l² = r² + h².
- The dome is as tall as its radius: 21.
- The cone takes the rest: 49 − 21 = 28.
- Slant: l² = 21² + 28² = 1225, so l = 35.
- Here 21² = 441 and 28² = 784. Also 441 + 784 = 1225, and 35 × 35 = 1225.
- Cone skin: 22/7 × 21 × 35 = 2310.
- Dome skin: 2 × 22/7 × 441 = 2772.
- Surface area: 2310 + 2772 = 5082.
The same idea works for a cone on a cylinder. The cone takes the total height minus the cylinder's height.
In these toys both solids share one radius r. The hemisphere takes r of the total height. The cone takes the rest. Then l² = r² + h².
Slide the cone height to match the total.
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