Surface Area of a Cone

The surface area of a cone is made of two main components, the area of the circular base and the lateral area.

Let

L A

be the lateral area of a cone and

B A

the area of the circular base. The surface area

S A

of a cone is made of the sum of the area of the circular base and the lateral area.

S A = B A + L A

The area of the circular base is found using the formula for the area of a circle.

B A = π r^{2}

The lateral area can be better visualized in two dimensions. Suppose that the cone is cut and the lateral area is expanded.

It should be noted that the figure obtained is a sector of a circle of radius

ℓ,

the slant height of the cone. Then, the lateral area can be obtained using the formula for the area of a sector of a circle of radius

ℓ .

Area of a Sector = \frac{θ}{3 6 0 ^{\circ}} \cdot π ℓ^{2}

But from the image it should also be noted that the arc of the sector has the same length as the circumference of the circular base of radius

r .

Using the formula for the length of an arc relative to its measure, it is possible to write an equation to equate these quantities.

Circumference of Base = Arc Length ⇓ 2 π r = \frac{θ}{3 6 0 ^{\circ}} \cdot 2 π ℓ

Dividing both sides of the equation by

2 π,

this equation can be simplified.

r = \frac{θ}{3 6 0 ^{\circ}} \cdot ℓ

Now it is possible to write an expression for the lateral area. First, the expression is the same as the formula for the area of a sector.

L A = \frac{θ}{3 6 0 ^{\circ}} \cdot π ℓ^{2}

CommutativePropMult

Commutative Property of Multiplication

L A = (\frac{θ}{3 6 0 ^{\circ}} \cdot ℓ) \cdot π ℓ

Substitute

$\frac{θ}{3 6 0 ^{\circ}} \cdot ℓ = r$

L A = r π ℓ

Finally, the expressions for the area of the circular base and the lateral area can be substituted to find the expression for the surface area of a cone.

S A = B A + L A ⇓ S A = π r^{2} + π r ℓ

Surface Area of a Cone

Proof

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