What is the Formula for the Surface Area of a Cone?

Answer – The formula for the surface area of a cone is $π r (r + l)$ , where $l = \sqrt{h^{2} + r^{2}}$ .

Explanation:

Consider a right circular cone with a radius r, height h, and slant height l as shown below:

Cone with height h, radius r, and slant height l

A right triangle is formed by the sides h, r, and l. Thus, by the Pythagoras theorem,

$l = \sqrt{h^{2} + r^{2}}$

$S u r f a c e a r e a o f t h e c o n e = A r e a o f t h e c i r c u l a r b a s e + L a t e r a l a r e a o f t h e c o n e$ – Equation 1

$A r e a o f t h e c i r c u l a r b a s e = π r^{2}$

$L a t e r a l a r e a o f t h e c o n e = \frac{1}{2} \times C i r c u m f e r e n c e o f t h e b a s e \times L a t e r a l h e i g h t$

$= \frac{1}{2} \times 2 π r \times l$

$= π r l$

Substituting for the lateral area and area of the base in Equation 1:

$S u r f a c e a r e a o f t h e c o n e = π r^{2} + π r l$

$= π r (r + l)$

$S u r f a c e a r e a o f t h e c o n e = π r (r + \sqrt{h^{2} + r^{2}})$

Textbook Solutions

Area Cone Geometry Math

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