Name: Surface Area and Volume of Right Triangular Pyramid
Uploaded: 2019-07-19
Duration: 3 min 52 s
Description: To find the volume of a right triangular pyramid, we use the formula V = (1/3)AH, where A = area of the pyramid's base and H = height of the pyramid. Watch a...

Question

Prove that for a cone made by rolling up a semicircle, the area of curved surface is twice the base area.

Accepted Answer

We need to prove for a cone made by rolling the semicircle, the area of curved surface is twice the base area.

Let $R$ be the radius of the semicircle.

Since the semicircle is rolled up to form a cone, slant height of the cone $=$ radius of the semicircle.

$\Rightarrow l = R$

Since the cone is made from a semi-circular sector, the central angle $= 180 °$
Circumference of the base of the cone $=$ Arc length
Let $r$ be the radius of the cone.

$2 π r = 2 π R \times \frac{180}{360}$

So, base radius of the cone, $r = \frac{R}{2}$ units

Since the cone is made from a semi-circular sector, the central angle $= 180 °$

Circumference of the base of the cone $=$ Arc length

Let $r$ be the radius of the cone.

$2 π r = 2 πR \times \frac{180}{360}$

So, base radius of the cone, $r = \frac{R}{2}$ units.

We know that, Curved surface area of the cone $= π r l$

Curved surface area of the cone $= π \frac{R}{2} \times R = \frac{π R^{2}}{2}$ square units.

Base area of the cone $= \frac{π R^{2}}{4} = \frac{1}{2} (\frac{π R^{2}}{2})$ square units.

Hence, the area of curved surface is twice the base area.

Prove that for a cone made by rolling up a semicircle, the area of curved surface is twice the base area.

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