Question

The height of the solid cone is 12cm and the area of the base is 81 π sqcm , find the curved surface area of the cone.[Take π =3.14]​

Answer 1

Solution

Given that,

• Height of cone, h = 12 cm

• Area of base = 81 π sq. cm.

Let assume that

Radius of cone = r cm

Slant height of cone = l cm

Now, as it is given that

Area of base of cone = 81πr

πr² = 81π

r² = 81

r = 9 cm

Now, we know slant height of a cone, I is evaluated as

l² = r² + h²

l² = 9² + 12²

l² = 81 + 144

l² 225

l = 15 cm

Now, we know

Curved Surface Area of cone = πrl

= 3.14 × 9 × 15

= 423.9 cm²

Hence, Curved Surface Area of cone = 423.9 cm

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Answer 2

Answer:

Given :-

• The height of the solid cone is 12 cm and the area of the base is 81 π cm².

To Find :-

• What is the curved surface area of the cone.

Formula :-

$\clubsuit$ Base Area Of Cone Formula :

$\bigstar \: \: \sf\boxed{\bold{\pink{Base\: Area_{(Cone)} =\: {\pi}r^2}}}\: \: \: \bigstar$

$\clubsuit$ Slant Height Of Cone Formula :

$\bigstar \: \: \sf\boxed{\bold{\pink{l =\: \sqrt{r^2 + h^2}}}}\: \: \: \bigstar$

$\clubsuit$ Curved Surface Area Of Cone Formula :

$\bigstar \: \: \sf\boxed{\bold{\pink{Curved\: Surface\: Area_{(Cone)} =\: {\pi}rl}}}\: \: \: \bigstar$

where,

• π = Pie or 22/7
• r = Radius
• l = Slant Height
• h= Height

Solution :-

First, we have to find the radius of cone :

Given :

• Area of the base = 81 π cm²

According to the question by using the formula we get,

$\implies \bf Base\: Area_{(Cone)} =\: {\pi}r^2$

$\implies \sf 81 {\cancel{{\pi}}} =\: {\cancel{{\pi}}}r^2$

$\implies \sf 81 =\: r^2$

$\implies \sf \sqrt{81} =\: r$

$\implies \sf 9 =\: r$

$\implies \sf\bold{\blue{r =\: 9}}$

Hence, the radius of a solid cone is 9 cm .

Now, we have to find the slant height of cone :

Given :

• Radius = 9 cm
• Height = 12 cm

According to the question by using the formula we get,

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

$\implies \sf l =\: \sqrt{(9)^2 + (12)^2}$

$\implies \sf l =\: \sqrt{(9 \times 9) + (12 \times 12)}$

$\implies \sf l =\: \sqrt{81 + 144}$

$\implies\sf l =\: \sqrt{225}$

$\implies \sf\bold{\green{l =\: 15}}$

Hence, the slant height of cone is 15 cm .

Now, we have to find the curved surface area of the cone :

Given :

• Radius = 9 cm
• Slant Height = 15 cm
• π = 3.14

According to the question by using the formula we get,

$\dashrightarrow \bf Curved\: Surface\: Area_{(Cone)} =\: {\pi}rl$

$\dashrightarrow \sf Curved\: Surface\: Area_{(Cone)} =\: 3.14 \times 9 \times 15$

$\dashrightarrow \sf Curved\: Surface\: Area_{(Cone)} =\: 3.14 \times 135$

$\dashrightarrow \sf\bold{\red{Curved\: Surface\: Area_{(Cone)} =\: 423.9\: cm^2}}$

$\sf\bold{\purple{\underline{\therefore\: The\: curved\: surface\: area\: of\: the\: cone\: is\: 423.9\: cm^2\: .}}}$