# The Right Circular Cone

Any cone with circular right section is a circular cone. Right circular cone is a circular cone whose axis is perpendicular to its base.

**Properties of Right Circular Cone**

- The
*slant height*of a right circular cone is the length of an*element*. Both the slant height and the element are denoted by L. - The
*altitude*of a right circular is the perpendicular drop from vertex to the center of the base. It coincides with the*axis*of the right circular cone and it is denoted by h. - If a right triangle is being revolved about one of its legs (taking one leg as the axis of revolution), the solid thus formed is a right circular cone. The surface generated by the hypotenuse of the triangle is the lateral area of the right circular cone and the area of the base of the cone is the surface generated by the leg which is not the axis of rotation.
- All
*elements*of a right circular cone are equal. - Any section parallel to the base is a circle whose center is on the axis of the cone.
- A section of a right circular cone which contains the vertex and two points of the base is an isosceles triangle.

**Formulas for Right Circular Cone**

_{b}

The bases of a right circular cone are obviously circles

_{L}

The lateral area of a right circular cone is equal to one-half the product of the circumference of the base c and the slant height L.

$A_L = \frac{1}{2}cL$

Taking c = 2πr, the formula for lateral area of right circular cone will be more convenient in the form

The relationship between base radius r, altitude h, and slant height L is given by

The volume of the right circular cone is equal to one-third the product of the base area and the altitude.

$V = \frac{1}{3}A_b h$

**Derivation of Formula for Lateral Area of the Right Circular Cone**

$c = 2 \pi r$

$s = L \theta_{rad}$

$s = c$

$L \theta_{rad} = 2 \pi r$

$\theta_{rad} = \dfrac{2 \pi r}{L}$

$A_L = A_{sector}$

$A_L = \frac{1}{2} sL$

$A_L = \frac{1}{2}(L \theta_{rad})L$

$A_L = \frac{1}{2} L^2 \theta_{rad}$

$A_L = \frac{1}{2} L^2 \left( \dfrac{2 \pi r}{L} \right)$

$A_L = \pi rL$ (*okay!*)