The Cone (Leaving Cert Mathematics): Revision Notes

Worked Example: Finding Volume and Curved Surface Area

Problem: The height of a cone is 12 cm and the radius of the base is 5 cm. Find correct to 1 decimal place: (i) the volume (ii) the curved surface area of the cone.

Solution:

First, we need to find the slant height using Pythagoras' theorem:

$l^2 = h^2 + r^2$

$l^2 = 12^2 + 5^2 = 144 + 25 = 169$

$l = \sqrt{169} = 13 \text{ cm}$

(i) Volume of cone:

$V = \frac{1}{3}\pi r^2 h$

$V = \frac{1}{3} \times \pi \times 5^2 \times 12$

$V = \frac{1}{3} \times \pi \times 25 \times 12 = \frac{300\pi}{3} = 100\pi$

$V = 314.2 \text{ cm}^3$

(ii) Curved surface area:

$\text{Curved surface area} = \pi r l$

$\text{Curved surface area} = \pi \times 5 \times 13 = 65\pi$

$\text{Curved surface area} = 204.2 \text{ cm}^2$

Worked Example: Composite Rocket Ship Model

Problem: A rocket ship model consists of a solid cylinder and solid cone, both with circular base diameter 20 cm. The cone height is 24 cm. Find: (i) the volume of the cone in terms of π (ii) the height of the cylinder if its volume is 4 times the cone volume (iii) the value of k if the cylinder of height k has half the volume of the whole solid model.

Solution:

The radius is 10 cm (diameter ÷ 2).

(i) Volume of cone: $V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (10^2)(24) = \frac{2400\pi}{3} = 800\pi \text{ cm}^3$

(ii) Finding cylinder height: If cylinder volume = 4 × cone volume: $\text{Cylinder volume} = 4(800\pi) = 3200\pi \text{ cm}^3$

Using cylinder volume formula:

$\pi r^2 h = 3200\pi$

$\pi(100)h = 3200\pi$

$100h = 3200$

$h = 32 \text{ cm}$

(iii) Finding k: Total solid volume = cone volume + cylinder volume = 800π + 3200π = 4000π cm³

If cylinder of height k has half this volume: $\text{Required volume} = \frac{4000\pi}{2} = 2000\pi \text{ cm}^3$

$\pi r^2 k = 2000\pi$

$\pi(100)k = 2000\pi$

$100k = 2000$

$k = 20 \text{ cm}$

Worked Example: Finding Volume from Slant Height

Problem: Find the volume of a cone with slant height 8 cm and base radius 6 cm.

Solution:

First find the perpendicular height using Pythagoras:

$l^2 = h^2 + r^2$

$8^2 = h^2 + 6^2$

$64 = h^2 + 36$

$h^2 = 28$

$h = \sqrt{28} = 2\sqrt{7} \text{ cm}$

Now calculate volume:

$V = \frac{1}{3}\pi r^2 h = \frac{1}{3}\pi (6^2)(2\sqrt{7}) = \frac{1}{3}\pi (36)(2\sqrt{7}) = 24\sqrt{7}\pi \text{ cm}^3$