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.

Right Circular Cone with Unrolled Lateral Area

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

Area of the base, A_b
The bases of a right circular cone are obviously circles

$A_b = \pi r^2$

Lateral Area, A_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

$A_L = \pi rL$

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

$r^2 + h^2 = L^2$

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

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

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

Derivation of Formula for Lateral Area of the Right Circular Cone

Click here to read or hide the derivation

The unrolled surface shown in the above figure is in the form of a sector of a sector with radius L, central angle θ, and length of arc s. The length of arc s is the circumference of the base of the cone which is denoted by c. Thus s = c.

$c = 2 \pi r$

$s = L \theta_{rad}$

$s = c$

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

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

$A_L = A_{sector}$

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

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

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

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

$A_L = \pi rL$ (okay!)

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