Question

A lampshade in form of a frustum of cone has height of 12cm upper and lower diameter of 10cm and 20cm respectively what area of material is required to cover the curved surface area

Answer 1

Explanations & Calculations

• Lateral surface area of a cone is given by $\small A = \pi rL$
• Therefore, by careful inspection, the needed area is

$\qquad\qquad \begin{aligned} \small A_1 &=\small\pi R(L+x)-\pi rx\\ \small &= \small \pi \big[RL+x(R-r)\big]\cdots(1) \end{aligned}$ $\qquad\qquad \begin{aligned} \small R &=\small \text{Base face radius}\\ \small r &= \small\text{upper face radius of the fusrtum} \end{aligned}$

• Considering the equiangular triangles ABE & ACD,

$\qquad\qquad \begin{aligned} \small \frac{5}{10}&= \small \frac{h}{12+h}\\ \small h&= \small 12cm \end{aligned}$

• By Pythagoras theorem to ABE triangle,

$\qquad\qquad \begin{aligned} \small x^2 &=\small 12^2 +5^2\\ \small x &= \small 13cm \end{aligned}$

• Considering the equiangular triangles ACD & ABE,

$\qquad\qquad \begin{aligned} \small \frac{10}{5} &= \small \frac{L+x}{x}\\ \small L &= \small 13cm \end{aligned}$

• Therefore, by (1),

$\qquad\qquad \begin{aligned} \small A_1 &= \small \pi\big[10\times 13+13\times (10-5)\big]\\ &= \small \bold{612.61cm^2} \end{aligned}$