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 2pi 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})