Compound shapes and parts of shapes：練習問題

A sector of a circle has a radius of 7 cm and an angle of \(120^\circ\) . Calculate the perimeter of the sector in terms of \(\pi\) .

A prism has a cross-section in the shape of a trapezium with parallel sides of length 7 cm and 13 cm, and a perpendicular height between them of 6 cm. The length of the prism is 10 cm. Calculate the volume of the prism.

A solid is formed by a hemisphere placed on top of a cylinder. The cylinder has a radius of 4 cm and a height of 8 cm. The hemisphere has the same radius. Calculate the total volume of the solid in terms of \(\pi\).

A frustum is formed by removing a small cone from the top of a larger cone. The large cone has a radius of 12 cm and a height of 20 cm. The small cone removed has a radius of 3 cm and a height of 5 cm. Calculate the volume of the frustum, correct to 3 significant figures.

A square with side length 8 cm has a quarter circle cut out from each of its four corners. The radius of each quarter circle is 2 cm. Calculate the area of the remaining shape in terms of \(\pi\) .

A rectangular piece of card \(20 \text{ cm}\) by \(15 \text{ cm}\) has four identical squares of side length \(2 \text{ cm}\) cut from each corner. Find the area of the remaining card.

A solid is formed by a cone of height 12 cm and base radius 5 cm, from which a smaller cone of height 4 cm has been removed from the top. The top surface of the frustum is parallel to the base.

(a) Calculate the radius of the top surface of the frustum.

(b) Calculate the volume of the original (large) cone.

(c) Calculate the volume of the smaller cone that was removed.

(d) Calculate the volume of the frustum, leaving your answer in terms of \(\pi\).

(e) Calculate the curved surface area of the frustum, leaving your answer in terms of \(\pi\).

A running track consists of a rectangle with semi-circles at each end. The length of the rectangular part is 100 m and the width is 60 m (which is also the diameter of the semi-circles).

(a) Calculate the total length of the inner perimeter of the track.

(b) Calculate the total area enclosed by the inner perimeter of the track.

(c) The track itself is 10 m wide. Calculate the area of the track (the shaded region between the inner and outer perimeter).