Question

Derive the formula for the curved surface area and total surface area of the
frustum of a cone.​

Answer 1

The lateral area of frustum of a right circular cone is given by the formula

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where

R = radius of the lower base

r = radius of the upper base

L = length of lateral side

Derivation of Formula

Lateral area of right circular cone is the difference of the areas of sectors of a circle of radii L1 and L2 and common central angle θ. See the figure below

frustum-of-cone-developement.jpg Figure for the Derivation of Formula for Lateral Area of Right Circular Cone

By ratio and proportion:

L1R=LR−r

L1=RLR−r

From the figure:

L2=L1−L

L2=RLR−r−L

L2=RL−(R−r)LR−r

L2=rLR−r

The length of arc is the circumference of the base:

s1=2πR

s2=2πr

From the figure:

A=12s1L1−12s2L2

A=12(2πR)(RLR−r)−12(2πr)(rLR−r)

A=πR2LR−r−πr2LR−r

A=πR2L−πr2LR−r

A=π(R2−r2)LR−r

A=π(R−r)(R+r)LR−r

A=π(R+r)L

Answer 2

Answer:

ᴛᴏ ғɪɴᴅ ᴛʜᴇ ᴄᴜʀᴠᴇᴅ sᴜʀғᴀᴄᴇ ᴀʀᴇᴀ ᴏғ ᴀɴʏ ᴄᴏɴᴇ, ᴍᴜʟᴛɪᴘʟʏ ᴛʜᴇ ʙᴀsᴇ ʀᴀᴅɪᴜs ᴏғ ᴛʜᴇ ᴄᴏɴᴇ ʙʏ ᴘɪ. ɴᴏᴡ ᴍᴜʟᴛɪᴘʟʏ ʏᴏᴜʀ ᴀɴsᴡᴇʀ ʙʏ ᴛʜᴇ ʟᴇɴɢᴛʜ ᴏғ ᴛʜᴇ sɪᴅᴇ ᴏғ ᴛʜᴇ ᴄᴏɴᴇ. ɪғ ʏᴏᴜ ᴡᴀɴᴛ ᴛᴏ ᴛᴏᴛᴀʟ sᴜʀғᴀᴄᴇ ᴀʀᴇᴀ ʀᴇᴍᴇᴍʙᴇʀ ᴛᴏ ᴀᴅᴅ ᴏɴ ᴛʜᴇ ᴀʀᴇᴀ ᴏғ ᴛʜᴇ ʙᴀsᴇ ᴏғ ᴛʜᴇ ᴄᴏɴᴇ.

Thanks ❤️

Ashley ❤️