Question

find the volume , curved surface area and total surface area of cone whose height and slant height is 6 cm and 10 cm respectively​

Answer 1

Answer :-

$\: \: \boxed{\boxed{\rm{\mapsto \: \: \: Firstly \: let's \: understand \: the \: concept \: used}}}$

Here the concept of Volume, Total Surface Area (TSA) and Curved Surface Area (CSA) of the Cone has been used. Volume is the measure of amount of space in a 3D solid. TSA is the measure of area of surface of solid including the bases. And CSA is the measure of only the front surface of the solid excluding bases.

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★ Formula Used :-

$\: \boxed{\boxed{\bf{Volume\:of\:Cone\: = \: \dfrac{1}{3} \: \times \: \pi r^{2}h}}}$

$\: \boxed{\boxed{\bf{TSA \: of \: Cone \: = \: \pi rl \: + \: \pi r^{2}}}}$

$\: \boxed{\boxed{\bf{CSA \: of \: Cone \: = \: \pi rl}}}$

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★ Question :-

Find the Volume, Curved Surface Area (CSA) and Total Surface Area (TSA) of the cone whose height and slant height is 6 cm and 10 cm respectively.

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★ Solution :-

Given,

» Height of the Cone = h = 6 cm

» Slant Height of the Cone = l = 10 cm

• Let the radius of the base of the cone be 'r' cm.

We know that,

→ l² = h² + r²

→ (10)² = (6)² + r²

→ r² = 100 - 36

→ r² = 64

$\: \: \bf{\longrightarrow \: \: r \: = \: \sqrt{64} = 8 cm}$

$\: \: \huge{\boxed{\boxed{\rm{r \: = \: 8 cm}}}}$

• Hence, the radius of the base of cone = r = 8 cm

*Note = Here we will be using the value of π = 22/7

Now let us find the required things :-

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$\: \: \boxed{\boxed{\underline{\tt{Volume \: of \: Cone}}}}$

⌬ Volume of Cone = ⅓(πr²h)

⌬ Volume of Cone = ⅓ × ((22/7) × 8 × 8 × 6)

⌬ Volume of Cone = (22/7) × 8 cm × 8 cm × 2 cm

⌬ Volume of the cone = 402.29 cm³

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$\: \: \boxed{\boxed{\underline{\tt{TSA \: of \: Cone}}}}$

⌬ TSA of Cone = πrl + πr²

⌬ TSA of Cone = ((22/7) × 8 × 10) + ((22/7) × 8 × 8)

⌬ TSA of Cone = 251.43 cm² + 201.14 cm²

⌬ TSA of Cone = 452.57 cm²

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$\: \: \boxed{\boxed{\underline{\tt{CSA \: of \: Cone}}}}$

⌬ CSA of Cone = πrl

⌬ CSA of Cone = ((22/7) × 8 × 10)

⌬ CSA of Cone = 251.43 cm²

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$\: \: \boxed{\rm{\leadsto \: \: The \: Volume \: of \: cone \: is \: \underline{402.29 \: cm^{3}} \: , \: the \: CSA \: is \: \underline{251.43 \: cm^{2}} \: and \: the \: TSA \: is \: \underline{452.57 \: cm^{2}}}}$

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$\: \: \boxed{\underline{\mapsto \: \: Aid \: to \: memory \: :-}}$

• Volume of Cuboid = Length × Breadth × Height

• Volume of Cube = (Side)³

• Volume of Cylinder = πr²h

• TSA of Cylinder = 2πr² + πrh

• CSA of Cylinder = πrh

• TSA of Cube = 6 × (side)²

• CSA of Cube = 4 × (side)²

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» Note :- The values will be different if we use the value of π = 3.14

When, we use π = 3.14 , instead of (22/7) then the values will be =

=> Volume of Cone = 401.92 cm³

=> TSA of Cone = 452.16 cm²

=> CSA of Cone = 251.2 cm²

Answer 2

Given :

• Height of cone (h) = 6 cm
• Slant height (l) = 10 cm

To find :

• Volume of cone
• CSA of cone
• TSA of cone

Solution :

Given that height (h) = 6 cm

Slant height (l) = 10 cm

As we know,

⇒ Slant height² (l²) = Height²(h²) + Radius²(r²)

⇒ 10² = 6² + r²

⇒ 100 = 36 + r²

⇒ r² = 100 - 36

⇒ r² = 64

⇒ r = √64

⇒ r = 8 cm

Now we know,

Volume of cone = 1/3 πr²h

⇒ Volume of cone = 1/3 * 22/7 * 8² * 6

⇒ Volume of cone = 22/7 * 64 * 2

⇒ Volume of cone = 2816/7

⇒ Volume of cone = 402.29 cm³              (ANSWER)

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Now we know,

CSA of cone = π * r * l

⇒ CSA of cone = 22/7 * 8 * 10

⇒ CSA of cone = 1760/7

⇒ CSA of cone = 251.43 cm²               (ANSWER)

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Now we know,

TSA of cone = πr(l + r)

⇒ TSA of cone = 22/7 * 8 (10 + 8)

⇒ TSA of cone = 22/7 * 8 * 18

⇒ TSA of cone = 3168/7

⇒ TSA of cone = 452.57 cm²           (ANSWER)