Volume and Surface Area of a Cone

Slant height: $l = \sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = 5 \text{ cm}$

Volume: $V = \dfrac{1}{3} \times \pi \times 9 \times 4 = 12\pi \approx 37.7 \text{ cm}^3$

Surface area: $SA = \pi(9) + \pi(3)(5) = 9\pi + 15\pi = 24\pi \approx 75.4 \text{ cm}^2$

A waffle cone has radius 3 cm and height 12 cm. How much ice cream fits inside?

$V = \dfrac{1}{3} \times \pi \times 3^2 \times 12 = \dfrac{1}{3} \times \pi \times 9 \times 12 = 36\pi \approx 113 \text{ cm}^3$

Ice cream manufacturers use this to size portions and calculate how many scoops fit in a standard cone.

A traffic cone has base radius 15 cm and slant height 45 cm. The curved surface area (amount of plastic needed):

$SA_{\text{lateral}} = \pi \times 15 \times 45 = 675\pi \approx 2{,}120 \text{ cm}^2$

Highway maintenance teams multiply this by the number of cones to order the right amount of reflective plastic sheeting.

An industrial funnel has base radius 20 cm and height 30 cm. Its volume determines the maximum liquid it can hold at any moment:

$V = \dfrac{1}{3} \times \pi \times 400 \times 30 = 4{,}000\pi \approx 12{,}566 \text{ cm}^3 \approx 12.6 \text{ litres}$

Engineers use this to size pumps that keep up with the flow through the funnel.