A sector of a circle of radius 7cm subtending an angle of 270° at the centre of the circle is used to form a cone.
(a) Find the base radius of the cone.
(b) Calculate the area of the base of the cone to the nearest square centimetre.
[Take \(\pi = \frac{22}{7}\)]
Explanation
(a) Length of the arc XY = Circumference of the base.
\(\frac{\theta}{360°} \times 2\pi r\)
where \(\theta\) = 270° ; r = 7cm.
Length of arc XY = \(\frac{270°}{360°} \times 2 \times \frac{22}{7} \times 7 = 33cm\)
Let the radius of the base of the cone = x
\(2 \pi r = 33cm\)
\(2 \times \frac{22}{7} \times x = 33\)
\(\frac{44}{7} \times x = 33 \implies x = \frac{33 \times 7}{44}\)
\(x = 5.25cm\)
(b) Area of the base of the cone = \(\pi r^{2}\)
= \(\frac{22}{7} \times \frac{21}{4} \times \frac{21}{4}\)
= \(\frac{693}{8} cm^{2} = 86.625 cm^{2}\)
\(\approxeq 87 cm^{2}\)