Areas and Volumes：練習問題

The radius of a right circular cone is 3 cm and its vertical height is 4 cm. Find the volume of the cone in terms of \(\pi\).

The heights of two similar cones are in the ratio \(2:3\). If the total surface area of the smaller cone is \(16\pi \text{ cm}^2\), find the total surface area of the larger cone in terms of \(\pi\).

A solid toy is formed by joining a right circular cone and a hemisphere. The cone has a base radius of \(3 \text{ cm}\) and a height of \(4 \text{ cm}\). The base of the cone is attached to the flat face of a hemisphere with the same radius. Calculate the total volume of the solid in terms of \(\pi\).

A solid right circular cylinder has a base radius of 4 cm and a height of 5 cm. Find its curved surface area in terms of \(\pi\).

Calculate the total surface area of a solid hemisphere with a radius of 6 cm in terms of \(\pi\).

Find the surface area of a sphere with a radius of \(7 \text{ cm}\) in terms of \(\pi\).

The heights of two similar pyramids are in the ratio \(3:4\). If the volume of the larger pyramid is \(128 \text{ cm}^3\), find the volume of the smaller pyramid.

In the figure, a right prism of height \(20 \text{ cm}\) has a base sector \(OAB\) with a radius of \(10 \text{ cm}\) and \(\angle AOB = 90^\circ\). Calculate the total surface area of the prism in terms of \(\pi\).

A right circular cone has a base radius of \(9 \text{ cm}\) and a slant height of \(15 \text{ cm}\).

(a) Find the vertical height of the cone.
(b) Calculate the volume of the cone in terms of \(\pi\).

In the figure, \(VABCD\) is a right pyramid with a square base \(ABCD\) of side length \(12 \text{ cm}\). The vertical height of the pyramid is \(8 \text{ cm}\).

(a) Find the volume of the pyramid.
(b) Find the length of the slant height of a triangular face.
(c) Find the total surface area of the pyramid.