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Question

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


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Solution

Consider a frustum of a cone with h as height, l as the slant height, r₁ and r as radii of the ends where r>r

Construction:

Extended side BC and AD of the frustum of cone to meet at O

The frustum of a cone can be viewed as a difference of two right circular cones OABandOCD.

Let h₁ and l₁ be the height and slant height of cone OAB and h₂ and l₂ be the height and slant height of cone OCD respectively.

InΔAPOandΔDQO

APO=DQO=90° (Since both cones are right circular cones)

AOP=DOQ (Common)

Therefore, ΔAPO~ΔDQO(AA criterion of similarity)

APDQ=AODO=OPOQ (Corresponding sides of similar triangles are proportional)

rr=ll=hh

Wecansayrr=llorrr=ll

Subtracting 1 from both sides we get

rr-1=ll-1

(r-r)r=(l-l)l

(r-r)r=ll[Fromdiagram,l-l=l]

l=l×r(r-r)....(i)

or rr=ll

Subtracting 1 from both sides we get

rr-1=ll-1

(r-r)r=(l-l)l

(r-r)r=(l-l)l

(r-r)r=ll

l=l×r(r-r)....(ii)

(i)CSAoffrustumofcone=CSAofconeOAB-CSAofconeOCD

=πrl-πrl=π(rl-rl)=π(r×l×r(r-r)-r×l×r(r-r)[Using(i)and(ii)]=π(lr²-lr²)(r-r)=πl(r²-r²)(r-r)=πl(r-r)(r+r)(r-r)[Since,a²-b²=(a-b)(a+b)]=πl(r+r)

TSAoffrustumofcone=CSAoffrustum+Areaoflowercircularend+Areaoftopcircularend

=πl(r+r)+πr²+πr²

Therefore, CSA of the frustum of the cone =πl(r+r)

TSA of the frustum of the cone =πl(r+r)+πr²+πr²

Hence Proved.


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