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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= $large pi rl+pi r^2=pi r(l+r)$ .

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 $=60pi ;cm^{2}.$ 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 $=60pi ;cm^{2}.$ and slant height of the cone = 8 cm

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

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

$60= r(r+l)$

As l = 8, hence we have

$60= r(r+8)$

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

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

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

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

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

As r cannot be negative, So we must have

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

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

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