The slant height of the frustum of a cone is 4 cm and the perimeters (i.e. , circumferences) of its circular ends are 18 cm and 6 cm. Find the curved surface area of the frustum.
Given: perimeter of upper circle = 18 cm
Perimeter of lower circle = 6 cm
Slant height of frustum = l = 4 cm
Formula: Total surface area of frustum = πr2 + πR2 + π(R + r)l cm2
Let r be the radius of lower circle and R be the radius of upper circle
Now perimeter of circle = circumference of circle = 2π × radius
∴ Perimeter of upper circle = 2πR
18 = 2 × π × R
R = 9/π cm
Perimeter of lower circle = 2πr
6 = 2 × π × r
r = 3/π cm
Now we have asked curved surface area, so we should subtract the top and bottom surface areas which are flat circles.
Surface area of top = πR2
Surface area of bottom = πr2
∴ Curved surface area = total surface area - πr2 - πR2 cm2
= πr2 + πR2 + π(R + r)l - πr2 - πR2 cm2
= π(R + r)l cm2
= π × [(9/π) + (3/π)] × 4 cm2
= (9 + 3) × 4 cm2
= 48 cm2
∴ curved surface area = 48 cm2