上图显示了一个几何体。 该几何体的横截面是一个半径为 5 cm，圆心角为 30° 的扇形。 水平截面 OST 和 RUV 相距 7 cm。 S、T、U 和 V 位于该几何体的曲面上。 线段 OR、TU 和 SV 是垂直的。

1. Find the arc length of the sector forming the curved edge of the surface.

Arc length
$=\frac{30}{360}\times 2\pi(5)$
$=\frac{5\pi}{6}\text{ cm}$

2. Multiply the arc length by the vertical distance to get the curved surface area.

Curved surface area
$=\frac{5\pi}{6}\times 7$
$=\frac{35\pi}{6}\text{ cm}^2$

1. Find the straight-line length $ST$ of the chord in the sector.

In the horizontal cross-section $OST$, $OS=OT=5$ cm and $\angle SOT=30^\circ$.

Using the cosine rule in $\triangle OST$,
$ST^2=5^2+5^2-2(5)(5)\cos30^\circ$
$ST=\sqrt{50-50\cos30^\circ}=2.588\ldots\text{ cm}$

2. Use right-angled trigonometry in triangle $STU$ to find $\angle UST$.

Since $TU$ is vertical and $ST$ lies in the horizontal plane, $TU\perp ST$.

In right-angled $\triangle STU$, $TU=7$ cm.
$\tan \angle UST=\frac{TU}{ST}=\frac{7}{2.588\ldots}$
$\angle UST=69.7^\circ\text{ }(3\text{ s.f.})$

1. Cube the ratio of corresponding radii to obtain the volume ratio.

Smaller radius : larger radius
$=3:5$
For similar solids,
$\text{volume ratio}=(\text{length ratio})^3$
So
$\text{smaller volume : larger volume}=3^3:5^3=27:125.$