Formula

Slant height and curved surface of a cone

Mensuration • Solids • Cone

Formula

l = r^{2} + h^{2}, CSA = π r l

A right circular cone of base radius

r

and vertical height

h

has slant height

l = r^{2} + h^{2}

, curved surface area

π r l

, and total surface area

π r l + π r^{2} = π r (l + r)

when the base is included.

How to use it

1Get $l$ before any surface is attempted: the radius, the vertical height and the slant height form a right triangle with $l$ as hypotenuse.
2A conical tent or roof made of canvas uses the curved surface only, since it has no fabric base.
3Recognise the triples: $(3, 4, 5)$ , $(5, 12, 13)$ and $(16, 63, 65)$ all turn up as radius, height and slant height.

A cone has base radius

7

cm and vertical height

24

cm. Find its curved surface area.

Step-by-step

1
$l = 7^{2} + 2 4^{2} = 625 = 25$ cm
Why: The slant height is the hypotenuse of the right triangle formed by $r$ and $h$ .
2
$CSA = \frac{22}{7} \times 7 \times 25 = 550$ cm $^{2}$

Answer

550

^{2}

Watch out

The curved surface takes

l

while the volume takes

h

; interchanging them is the standard error. The slant height always exceeds both

r

and

h

, which is a quick check on any value obtained.

Builds on

Also known as

slant height of a cone, curved surface area of a cone, pi r l, total surface area of a cone