Total Surface Area of Frustum of Cone

Total Surface Area of Frustum of a Cone

The cone is the three-dimensional geometrical figure or object which has the base in the form of circle of some radius. It also has height and slant height. The surface area of curved surface of cone is called as the curved surface area of the cone.

If l is the slant height of cone and r is the radius of circular base then the curved surface area of cone is given by,

Curved surface area of cone = π*r*l

The colored portion of the cone shown in following figure is the curved surface area of cone.

Now, when we cut the cone from middle by a plane parallel to the base of cone and perpendicular to its height then the portion with base formed is the frustum of the cone as shown in figure below.

  • In following figure, the colored portion indicates the curved surface area of frustum of a cone.
  • Now, find the curved surface area of frustum of a cone.

Derivation for curved surface area of frustum of a cone:

  • Let us consider the solid right circular cone, when we cut into two parts by a plane parallel to its base then the portion with the base is the frustum of cone ABCD, as shown in figure.
  • We see that, in figure two cones are formed one is small and the other is the whole one and the frustum again.
  • Let us consider, r1 be the radius of circular base of whole big cone and r2 be the radius of circular base of small cone.
  • Let, L be the slant height of big cone and l be the height of the frustum of cone then the height of small cone will be (L – l).

  • Now, from figure in ΔPQC and ΔPRB,
  • < RPB is common and < PQC = < PRB = 900, so we can say that ΔPQC and ΔPRB are similar triangles.
  • And for similar triangles, the ratio of sides remains constant.

Thus,

QC/ RB = PC/ PB

Hence, r2/ r1 = (L -l)/ L

So, r2/ r1 = 1 – (l/ L)

l/ L = 1 – r2/ r1

l/ L = (r1 – r2)/ r1

l*r1 = L*(r1 – r2)

Thus, L = l*r1/ (r1– r2)

Now, we find (L – l):

(L – l) = l*r1/ (r1 – r2) – l

Hence, we can find the curved surface area of frustum of cone,

Curved surface area of frustum of cone = (curved surface area of big cone – curved surface           area of small cone)

Curved surface area of frustum of cone = π*r1*L – π*r2*(L – l)

= π*[r1*L – r2*(L – l)]

= π*[l*r12/ (r1 – r2) – l*r22/ (r1 – r2)]

= π*[r12/ (r1 – r2) – r22/ (r1 – r2)] *l

= π*[ (r12 – r22)/ (r1 – r2)] *l

= π*(r1 + r2) *l

Thus,

Curved surface area of frustum of cone = π*(r1 + r2) *l

Hence proved.

To find the total surface area of frustum of cone:

  • To find the total surface area of frustum of cone, we have to add the area of upper and lower circular base of frustum of cone to the curved surface area of frustum of cone.
  • In following figure, the coloured portion is showing the total surface area of frustum of cone.

Total surface area of frustum of cone = curved surface area of frustum of cone + area of lower circular base + area of upper circular base

Thus,

Total surface area of cone = π*(r1 + r2) *l + π*r12 + π*r22

 

Hence proved.


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