Surface Area of a Right Cone

Surface Area of a Right Cone

How to find the surface area of a right cone: definition and its parts, formula, proof, example, and its solution.

Cone

This shape is called a cone.

This shape is called a cone.

A cone has a base (a circle) and a vertex.

A cone has a base (a circle) and a vertex.

The lateral face of a cone is the non-base face.

The lateral face of a cone is the non-base face.

Right Cone

A right cone is a cone whose height meets at the center of the base circle.

A right cone is a cone
whose height meets at the center of the base circle.

The slant height is the distance between the vertex and the endpoint of the base circle.

The slant height is the distance between
the vertex and the endpoint of the base circle.

Formula

A = pi*r^2 + pi*r*h_s. A: surface area of a right cone, r: radius of the base, h_s: slant height of the right cone

A = πr2 + πrhs

A: surface area of a right cone
r: radius of the base
hs: slant height of the right cone

πr2: base area
πrhs: lateral area

Proof

Surface Area of a Right Cone: Proof of the Formula

Draw the net of a right cone.

(lateral area's arc) = (base circle's circumference)

2πhs⋅(m∠[blue] / 360) = 2πr

Then, after cancelling 2π,
m∠[blue] / 360 = r / hs.

(base area) = πr2
(lateral area) = πhs2⋅(m∠[blue]/360)

Area of a circular sector

A = πr2 + πhs2⋅(m∠[blue]/360)

m∠[blue]/360 = r/hs

So A = πr2 + πhs2⋅(m∠[blue]/360)
= πr2 + πhs2⋅(r/hs)
= πr2 + πrhs.

Example

Find the surface area of the given right cone. r = 3, h = 4.

See the right triangle in the cone.
Its legs are 3 and 4.

So it's a (3, 4, 5) triangle.

So hs = 5.

r = 3, hs = 5

A = π⋅32 + π⋅3⋅5