Question

The volume of a right circular cylinder is 4487 pie cm sq. and height 7 cm. Find the
lateral surface area and total surface area.​

Answer 1

☆ To Find :

• The Lateral surface Area of the Cylinder.

• The Total Surface Area of the Cylinder.

☆ We Know :

☞ Volume of the Cylinder :

$\purple{\sf{\underline{\boxed{V = \pi r^{2}h}}}}$

Where ,

• r = Radius of the Cylinder
• h = Height of the Cylinder
• V = Volume of the Cylinder

☞ Curved Surface Area of the Cylinder :

$\purple{\sf{\underline{\boxed{CSA = 2\pi rh}}}}$

Where ,

• r = Radius of the Cylinder

• h = Height of the Cylinder

• CSA = Curved Surface Area of the Cylinder

☞ Total Surface Area of the Cylinder :

$\purple{\sf{\underline{\boxed{TSA = 2\pi r(h + r)}}}}$

Where ,

• r = Radius of the Cylinder

• h = Height of the Cylinder

• V = Volume of the Cylinder

☆ Concept :

To Find the Curved Surface Area and the Total Surface Area of the Cylinder , first we have to find the Radius of the Cylinder.

☞ Radius of the Cylinder :

Let the Radius be r cm.

Given ,

• Volume = 4487 π cm³

• Height = 7 cm

Using the formula for Volume of a Cylinder , and Substituting the values in it , we get :

$\purple{\bigstar\:\sf{V = \pi r^{2}h}\:\bigstar} \\ \\ \\ \implies \sf{4487\pi = \pi r^{2} \times } \\ \\ \\ \implies \sf{\dfrac{4487\pi}{\pi} = r^{2} \times 7} \\ \\ \\ \implies \sf{\dfrac{4487\not{\pi}}{\not{\pi}} = r^{2} \times 7} \\ \\ \\ \implies \sf{4487 = r^{2} \times 7} \\ \\ \\ \implies \sf{\dfrac{4487}{7} = r^{2}} \\ \\ \\ \implies \sf{641 = r^{2}}$

By Square Rooting on both the sides , we get:

$\implies \sf{\sqrt{641} = \sqrt{r^{2}}} \\ \\ \\ \implies \sf{\sqrt{641} = r} \\ \\ \\ \implies \sf{25.32(approx.) cm = r} \\ \\ \\ \therefore \purple{\sf{r = 25.32 cm}}$

Hence , the Radius of the Cylinder is 25.32 cm

Now by this information , we can find the Total Surface Area and the Curved Surface Area of the Cylinder.

☆ Solution :

☞ Curved surface area of the Cylinder :

• Radius = 24.32 cm

• Height = 7 cm

Using the formula for CSA of a Cylinder and putting the values in it , We get :

$\purple{\bigstar\:\sf{CSA = 2\pi rh}\:\bigstar} \\ \\ \\ \implies \sf{CSA = 2\pi \times 25.32 \times 7} \\ \\ \\ \implies \sf{CSA = 2\pi \times 172.24} \\ \\ \\ \implies \sf{CSA = 354.48\pi cm^{2}} \\ \\ \\ \therefore \purple{\sf{CSA = 354.48\pi cm^{2}}}$

Hence , the Curved Surface Area of the Cylinder is 354.48π cm².

☞ Total surface area of the Cylinder :

• Radius = 25.32 cm

• Height = 7 cm

Using the formula for Total Surface Area of a Cylinder , and Substituting the values in it , we get :

$\purple{\bigstar\:\sf{TSA = 2\pi r(h + r)}\:\bigstar} \\ \\ \\ \implies \sf{TSA = 2\pi \times 25.32(7 + 25.32)} \\ \\ \\ \implies \sf{TSA = 2\pi \times 25.32 \times 32.32} \\ \\ \\ \implies \sf{TSA = 2\pi \times 818.34} \\ \\ \\ \implies \sf{TSA = 1636.68\pi cm^{2}} \\ \\ \\ \therefore \purple{\sf{TSA = 1636.68\pi cm^{2}}}$

Hence , the Total Surface Area of the Cylinder is 1636.68π cm².

☆ Answer :

• Curved surface area = 354.48π cm²

• Total Surface Area = 1636.68π cm²

☆ Additional information :

• Area of a Equilateral Triangle = √3a²/4

• Height of a Equilateral Triangle = √3a/2

• Curved Surface Area of a Cuboid = 2(l + b)h