Question

$\huge \bf Question$

● The height of a cone is 16cm and its base radius is 12cm. Find the curved surface area and the total surface area of the cone (Use π = 3.14).

▪︎Note:-

Wrong answer will reported on spot..!!!
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Answer 1

★ Concept :-

Here the concept of CSA and TSA of the conce has been used. We see that we are given the height and radius of the cone. So firstly we can find its slant height. After finding this, we can apply required values and thus find the answer.

Let's do it !!

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★ Formula Used ::

$\\\;\boxed{\sf{\pink{L^{2}\;=\;\bf{H^{2}\;+\;r^{2}}}}}$

$\\\;\boxed{\sf{\pink{CSA\;of\;Cone\;=\;\bf{\pi rL}}}}$

$\\\;\boxed{\sf{\pink{TSA\;of\;Cone\;=\;\bf{\pi rL\;+\;\pi r^{2}}}}}$

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★ Solution :-

Given,

» Height of Cone = H = 16 cm

» Radius of Cone = r = 12 cm

» Value of π = 3.14

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~ For the Slant Height of the Cone ::

• Let the Slant Height of the cone be L

By relationship of Slant Height, we know that

$\\\;\sf{\rightarrow\;\;L^{2}\;=\;\bf{H^{2}\;+\;r^{2}}}$

By applying values, we get

$\\\;\sf{\rightarrow\;\;L^{2}\;=\;\bf{(16)^{2}\;+\;(12)^{2}}}$

$\\\;\sf{\rightarrow\;\;L^{2}\;=\;\bf{256\;+\;144}}$

$\\\;\sf{\rightarrow\;\;L^{2}\;=\;\bf{400}}$

$\\\;\sf{\rightarrow\;\;L\;=\;\bf{\sqrt{400}}}$

$\\\;\bf{\rightarrow\;\;Slant\;Height,\;L\;=\;\bf{\green{20\;\;cm}}}$

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~ For the CSA of Cone ::

We know that,

$\\\;\sf{\Longrightarrow\;\;CSA\;of\;Cone\;=\;\bf{\pi rL}}$

By applying values, we get

$\\\;\sf{\Longrightarrow\;\;CSA\;of\;Cone\;=\;\bf{3.14\:\times\:12\:\times\:20}}$

$\\\;\bf{\Longrightarrow\;\;CSA\;of\;Cone\;=\;\bf{\red{753.6\;\;cm^{2}}}}$

$\\\;\underline{\boxed{\tt{CSA\;\:of\;\:Cone\;=\;\bf{\purple{753.6\;\;cm^{2}}}}}}$

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~ For the TSA of the Cone ::

We know that,

$\\\;\sf{\Longrightarrow\;\;TSA\;of\;Cone\;=\;\bf{\pi rL\;+\;\pi r^{2}}}$

By applying values, we get

$\\\;\sf{\Longrightarrow\;\;TSA\;of\;Cone\;=\;\bf{CSA\;+\;(3.14\:\times\:12\:\times\:12)}}$

$\\\;\sf{\Longrightarrow\;\;TSA\;of\;Cone\;=\;\bf{753.6\;+\;452.16}}$

$\\\;\bf{\Longrightarrow\;\;TSA\;of\;Cone\;=\;\bf{\orange{1205.76\;\;cm^{2}}}}$

$\\\;\underline{\boxed{\tt{TSA\;\:of\;\:Cone\;=\;\bf{\purple{1205.76\;\;cm^{2}}}}}}$

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★ More to know :-

$\\\;\sf{\leadsto\;\;Volume\;of\;Cone\;=\;\dfrac{1}{3}\pi r^{2}H}$

$\\\;\sf{\leadsto\;\;Volume\;of\;Cylinder\;=\;\pi r^{2}H}$

$\\\;\sf{\leadsto\;\;TSA\;of\;Cylinder\;=\;2\pi rH\;+\;2\pi r^{2}}$

$\\\;\sf{\leadsto\;\;CSA\;of\;Cylinder\;=\;2\pi rH}$

$\\\;\sf{\leadsto\;\;Volume\;of\;Hollow\;Cone\;=\;\dfrac{1}{3}\pi (R^{2}\;-\;r^{2})H}$

Answer 2

Step-by-step explanation:

${\large{\bold{\rm{\underline{Let's\ understand\ the\ concept\ 1^{st}:-}}}}}$

★ Through the given data we know that here we will use the concept of C.S.A and T.S.A of cone. As we see that we are given the height and the base radius of the cone. So first, we will find out the slant height of the cone. After that, we will easily find the answer by applying the required values.

${\large{\bold{\rm{\underline{Given\ that:-}}}}}$

$\sf {\bigstar Height\ of\ the\ cone\ =\ 16cm.}$

$\sf {\bigstar Radius\ of\ the\ cone\ =\ 12cm.}$

$\sf {\bigstar Value\ of\ the\ {\pi}\ =\ 3.14.}$

${\large{\bold{\rm{\underline{To\ find:-}}}}}$

★ In this question we have to find the T.S.A and C.S.A of the cone ?

${\large{\bold{\rm{\underline{Using\ formula:-}}}}}$

$\star\;\underline{\boxed{\bf{\orange{C.S.A\ of\ Cone\ =\ {\pi}rl.}}}}$

▪︎Where,

• Value of π is 3.14.
• r, denotes radius of cone.
• l, denotes slant height of cone.

___________________

$\star\;\underline{\boxed{\bf{\orange{T.S.A\ of\ Cone\ =\ {\pi}rl\ +\ {\pi}r^2.}}}}$

▪︎Where,

• Value of π is 3.14.
• r, denotes radius of cone.
• l, denotes slant height of cone.

${\large{\bold{\rm{\underline{Solution:-}}}}}$

★ C.S.A of Cone = 753.6cm².

★ T.S.A of Cone = 1205.76cm².

${\large{\bold{\rm{\underline{Full\ solution:-}}}}}$

• Let the slant height of the cone be L.

~ As we know that, by relationship of slant height:-

$\sf \mapsto {L^2\ =\ \bf H^2\ +\ r^2}$

By substituting the values, we get:-

$\sf \mapsto {L^2\ =\ \bf H^2\ +\ r^2}$

$\sf \mapsto {L^2\ =\ \bf (16)^2\ +\ (12)^2}$

$\sf \mapsto {L^2\ =\ \bf 256\ +\ 144}$

$\sf \mapsto {L^2\ =\ \bf 400}$

$\sf \mapsto {L\ =\ \bf \sqrt{400}}$

$\bf \therefore\ {Slant\ height\ =\ L\ =\ {\pink {20cm.}}}$

~ Now finding the value of Curved Surface Area of the Cone :-

$\sf \rightarrow {C.S.A\ of\ Cone\ =\ \bf {\pi}rl}$

~ By substituting the values, we get:-

$\sf \rightarrow {C.S.A\ of\ Cone\ =\ \bf {\pi}rl}$

$\sf \rightarrow {C.S.A\ of\ Cone\ =\ \bf 3.14 \times 12 \times 20}$

$\sf \rightarrow {C.S.A\ of\ Cone\ =\ \bf 753.6cm^2.}$

$\bf \therefore\ {C.S.A\ of\ Cone\ =\ {\orange {753.6cm^2.}}}$

~ Now finding the value of Total Surface Area of the Cone:-

$\sf \leadsto {T.S.A\ of\ Cone\ =\ \bf {\pi}rl\ +\ {\pi}r^2}$

~ By substituting the values, we get:-

$\sf \leadsto {T.S.A\ of\ Cone\ =\ \bf {\pi}rl\ +\ {\pi}r^2}$

$\sf \leadsto {T.S.A\ of\ Cone\ =\ \bf C.S.A\ +\ (3.14 \times 12 \times 12)}$

$\sf \leadsto {T.S.A\ of\ Cone\ =\ \bf 753.6\ +\ 452.16}$

$\sf \leadsto {T.S.A\ of\ Cone\ =\ \bf 1205.76cm^2.}$

$\bf \therefore\ {T.S.A\ of\ Cone\ =\ {\blue {1205.76cm^2.}}}$

${\large{\bold{\rm{\underline{\underline{More\ to\ know:-}}}}}}$

★ Formulas related to Cone:-

$\boxed{\begin{minipage}{6 cm}\bigstar \:\underline{\textbf{Formulae Related to Cone :}}\\\\\sf {\textcircled{\footnotesize\textsf{1}}} \:Area\:of\:Base =\pi r^2 \\\\ \sf {\textcircled{\footnotesize\textsf{2}}} \:\:Curved \: Surface \: Area = \pi rl\\\\\sf{\textcircled{\footnotesize\textsf{3}}} \:\:TSA = Area\:of\:Base + CSA=\pi r^2+\pi rl\\ \\{\textcircled{\footnotesize\textsf{4}}} \: \:Volume=\dfrac{1}{3}\pi r^2h\\ \\{\textcircled{\footnotesize\textsf{5}}} \: \:Slant \: Height=\sqrt{r^2 + h^2}\end{minipage}}$