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Maths / Surface Area and Volume / Total Surface Area of Cone
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# Total Surface Area of Cone

### Total Surface Area of Cone:

Total surface area of cone = Curved surface area + Area of the base $=\pi r^{2}+\pi rl$$=\pi r(r+l)$

Illustration: Find the total surface area of a cone, if its slant height is 12 m and the radius of its base is 9 m.

Solution: We know that the total surface area S of a right circular cone of radius r and slant height l is given by S= $\dpi{80} \large \\ \pi rl+\pi r^2=\pi r(l+r)$.

Here, r = 9 m and  l = 12 m.

Therefore $\large \dpi{80} \large S= \pi r(l+r)=\frac{22}{7} \times 9 \times (12 +9)=\frac{22}{7} \times 9 \times 21 = 594m^2$

Illustration: The total surface area of a cone is $=60\pi \;cm^{2}.$ If the slant height of the cone be 8 cm, find the radius of the base.

Solution: Given: Area of the curved surface of a cone is $=60\pi \;cm^{2}.$ and slant height of the cone = 8 cm

Also, we know that the Total surface area of the cone $=\pi r^{2}+\pi rl$ $=\pi r(r+l)$

$60\pi= \pi r(r+l)$

$60= r(r+l)$

As l = 8, hence we have

$60= r(r+8)$

$60= r^{2}+8r$

$r^{2}+8r-60=0$

$r=\frac{-8\pm \sqrt{64+240}}{2}$

$r=\frac{-8\pm \sqrt{304}}{2}$

$r=\frac{-8\pm 4\sqrt{19}}{2}\;cm$

As r cannot be negative, So we must have

$r=\frac{-8+ 4\sqrt{19}}{2}\;cm$

Hence, radius of the base is  $\frac{-8+ 4\sqrt{19}}{2}\;cm$

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