A trapezium with sides 10 cm and 21 cm, and height 8 cm is inscribed in a circle of radius 7 cm. The problem asks us to find the area of the region not covered by the trapezium. We are given $\pi = \frac{22}{7}$.

1. Problem Description

A trapezium with sides 10 cm and 21 cm, and height 8 cm is inscribed in a circle of radius 7 cm. The problem asks us to find the area of the region not covered by the trapezium. We are given $\pi = \frac{22}{7}$.

2. Solution Steps

First, we find the area of the circle.

The area of a circle is given by the formula:

$A_{circle} = \pi r^2$

Here, $r = 7$ cm, and $\pi = \frac{22}{7}$. So,

$A_{circle} = \frac{22}{7} \times (7)^2 = \frac{22}{7} \times 49 = 22 \times 7 = 154$ cm$^2$

Next, we find the area of the trapezium. The area of a trapezium is given by the formula:

$A_{trapezium} = \frac{1}{2} (a+b)h$

where $a$ and $b$ are the lengths of the parallel sides, and $h$ is the height. Here, $a = 10$ cm, $b = 21$ cm, and $h = 8$ cm.

$A_{trapezium} = \frac{1}{2} (10+21) \times 8 = \frac{1}{2} (31) \times 8 = 31 \times 4 = 124$ cm$^2$

Finally, to find the area of the region not covered by the trapezium, we subtract the area of the trapezium from the area of the circle:

$A_{uncovered} = A_{circle} - A_{trapezium} = 154 - 124 = 30$ cm$^2$

3. Final Answer

The area of the region not covered by the trapezium is 30 cm$^2$.

So the answer is C. 30 cm$^2$.