Right Circular Cone

Rotate a right triangle ABC right angled at B about one of its side AB containing the right angle. The solid generated as a result of this rotation is called a right circular cone. In daily life, we come across many objects of this shape, such as Joker's cap, tent, ice cream cones.

The end (or base) of a right circular cone is a circle. BC is the radius of the base with centre B and AB is the height of the cone and it is perpendicular to the base. Further, A is called the vertex of the cone and AC is called its slant height.

From the Pythagoras Theorem,

Slant height = radius² + height²

$$ l = \sqrt{r^2 + h^2} $$

where r, h and l are respectively the base radius, height and slant height of the cone.

The surface formed by the base of the cone is flat and the remaining surface of the cone is curved.

Surface Area

Take a hollow right circular cone of radius r and height h and cut it along its slant height. Now spread it on a piece of paper. You obtain a sector of a circle of radius l and its arc length is equal to 2πr.

$$ \text{Area of the sector} = \frac{\text{Arc length}}{\text{Circumference of the circle}} \times \text{Area of circle} $$

$$ = \frac{2 \pi r}{2 \pi l} \times \pi l^2 $$

$$ = \pi r l $$

Curved surface of the cone = Area of the sector = πrl

If the area of the base is added to the above, then it becomes the total surface area.

Total surface area of the cone = πrl + πr²

= πr(l + r)

Volume

Take a right circular cylinder and a right circular cone of the same base radius and same height. Now, fill the cone with sand (or water) and pour it in to the cylinder. Repeat the process three times. You will observe that the cylinder is completely filled with the sand (or water). It shows that volume of a cone with radius r and height h is one-third the volume of the cylinder with radius r and height h.

$$ \text{volume of cone} = \frac{1}{3} \text{volume of cylinder} $$

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