面積與體積

A solid iron sphere of radius $r$ is melted and recast into a circular cone and a circular cylinder. If both of them have the same height $h$ and the same base radius $r$, find $h$ in terms of $r$.

A cylindrical hole of radius $r$ is drilled through a solid cylinder, base radius $2r$ and height $r$, as shown in the figure. The percentage increase in the total surface area is

The figure shows the circumscribed circle $P$ and the inscribed circle $Q$ of the square $ABCD$. Find area of $P$ : area of $Q$.

$ABCD$ is a square of side $2\text{ cm}$. $O$ is the mid-point of $AD$. A sector with centre $O$ is inscribed in the square as shown in the figure. What is the area of the sector?

In the figure, $ABCD$ is a G-shaped curve, where $ABC$ is an arc of a circle and $DC$ is a radius. If the length of the curve $ABCD$ is the same as that of the complete circle, find, in radians, the angle subtended by the arc $ABC$ at the centre.

$ABCD$ is a trapezium in which $AB\parallel DC$, $AB=8\text{ cm}$, $DC=18\text{ cm}$, $AD=BC=13\text{ cm}$. Find the area of the trapezium.

A solid rectangular iron block, $4\text{ cm}\times2\text{ cm}\times1\text{ cm}$, is melted and recast into a cube. The decrease in the total surface area is

Figure a shows a circular measuring cylinder $4\text{ cm}$ in diameter containing water. Three iron balls, each of diameter $2\text{ cm}$, are dropped into the cylinder as shown in Figure b. What is the rise in the water level?

The circumference of a circle is $6\pi\text{ cm}$. The length of an arc of the circle which subtends an angle of $\dfrac{1}{3}$ radian at the centre is

In the figure, $O$ is the centre of the circle of radius 4. The area of the shaded region is