 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. A cone has a base (a circle) and a vertex. 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. The slant height is the distance between
the vertex and the endpoint of the base circle.

Formula 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 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 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