Derive the formula for the curved surface area and total surface area of the frustum of
a cone, given to you in Section 15.5, using the symbols as explained.
Answers
Step-by-step explanation:
In ∠ABG and ∠ADF, DF∥BG
Therefore, ∠ABG∼∠ADF
BG
DF
=
AG
AF
=
AB
AD
r
1
r
2
=
h
1
h
1
−h
=
1
1
l
1
−l
r
1
r
2
=1−
h
1
h
=1−
l
1
l
1−
l
1
l
=
r
1
r
2
1−
l
1
l
=1−
r
1
r
2
=
r
1
r
1
−r
2
l
l
1
=
r
1
−r
2
r
1
l
1
=
r
1
−r
2
r
1
l
CSA of frustum DECB= CSA of cone ABC− CSA cone ADE
=πr
1
l
1
−πr
2
(l
1
−l)
=πr
1
(
r
1
−r
2
r
1
)−πr
2
(
r
1
−r
2
r
1
−l)
r
1
−r
2
πr
1
2
l
−πr
2
(
r
1
−r
2
r
1
l−r
1
l+r
2
l
)
=
r
1
−r
2
πr
1
2
l
−
r
1
−r
2
πr
2
2
l
=πl[
r
1
−r
2
r
1
2
−r
2
2
]
CSA of frustum = π(r
1
+r
2
)l.
Total surface area of frustum = CSA of frustum + Area of upper circular end + Area of lower circular end.
=π(r
1
+r
2
)l+πr
2
2
+πr
1
2
=π[(r
1
+r
2
)l+r
1
2
+r
2
2
]