Mensuration：練習問題

The diagram shows a sector of a circle with a radius of \( 10 \text{ cm} \). The angle subtended at the centre is \( 0.8 \) radians. Calculate the area of the sector.

In the diagram below, a sector of a circle is shown. Calculate the area of the sector, giving your answer in terms of \(\pi\).

A sector of a circle has a radius of \(9\text{ cm}\) and an arc length of \(6\pi\text{ cm}\). If this sector is folded to form the curved surface of a right circular cone, find the base radius of the cone.

A circle has a radius of \(7\) cm. Using \(\pi = \frac{22}{7}\), calculate the area of the circle.

A solid cylinder has a radius of \( 4 \text{ cm} \) and a total surface area of \( 112\pi \text{ cm}^2 \). Calculate the volume of the cylinder, giving your answer in terms of \( \pi \).

A cylindrical water tank has a radius of \(0.7 \text{ m}\) and a height of \(2 \text{ m}\). Calculate the volume of the tank in litres. (Take \(\pi = \frac{22}{7}\) and note that \(1 \text{ m}^3 = 1000 \text{ litres}\))

A solid right circular cylinder has a total surface area of \( 480\pi \text{ cm}^2 \). Given that the radius of its base is \( 8 \text{ cm} \), calculate the volume of the cylinder, leaving your answer in terms of \( \pi \).

A solid metal sphere of radius \( 6 \) cm is melted down and recast into a solid right circular cone with a base radius of \( 8 \) cm. Calculate the total surface area of the cone, leaving your answer in terms of \( \pi \).

A solid trophy is formed by a cylinder of radius \( r \) cm and height \( 10 \) cm, surmounting a hemisphere of the same radius \( r \) cm. The total volume of the trophy is \( 156\pi \) cm\(^3\).
(a) Show that the radius \( r \) satisfies the equation \( 2r^3 + 30r^2 - 468 = 0 \).
(b) Given that \( r = 3 \), calculate the total surface area of the trophy, including the base, leaving your answer in terms of \( \pi \).