Geometry and Measurement - Perimeter and Area of 2D Shapes including Circles

Calculate the area and circumference of a circular garden with a radius of $14$ m. (Take $\pi \approx \frac{22}{7}$)

1. Identify the given radius: $r = 14$ m.
2. Calculate circumference using $C = 2\pi r$: $C = 2 \times \frac{22}{7} \times 14$ $C = 2 \times 22 \times 2 = 88$ m.
3. Calculate area using $A = \pi r^2$: $A = \frac{22}{7} \times 14^2$ $A = \frac{22}{7} \times 196$ $A = 22 \times 28 = 616$ $m^2$.

We substitute the radius into the standard circle formulas. Using the fraction $\frac{22}{7}$ for $\pi$ is helpful here because $14$ is a multiple of $7$, allowing for easy simplification.

A trapezium has parallel sides of length $10$ cm and $16$ cm. If the perpendicular height between them is $5$ cm, find its area.

1. Identify the parallel sides: $a = 10$ cm, $b = 16$ cm.
2. Identify the height: $h = 5$ cm.
3. Apply the trapezium area formula: $A = \frac{1}{2}(a + b)h$
4. Substitute the values: $A = \frac{1}{2}(10 + 16) \times 5$
5. Simplify the sum in parentheses: $A = \frac{1}{2}(26) \times 5$
6. Calculate the final result: $A = 13 \times 5 = 65$ $cm^2$.

The area is found by taking the sum of the parallel bases, dividing by $2$ to find the average length, and then multiplying by the vertical height.

Using the formula for the area of a triangle: $A = \frac{1}{2} \times b \times h$ $A = \frac{1}{2} \times 12 \times 7$ $A = 6 \times 7$ $A = 42 \text{ cm}^2$

To find the area of a triangle, multiply the base by the perpendicular height and then divide the result by 2.

Area of the track = Area of outer circle - Area of inner circle $A = \pi R^2 - \pi r^2$ $A = \pi(R^2 - r^2)$ $A = 3.14 \times (25^2 - 20^2)$ $A = 3.14 \times (625 - 400)$ $A = 3.14 \times 225$ $A = 706.5 \text{ m}^2$

The track is the region between two concentric circles. We calculate the area of the larger circle and subtract the area of the smaller circle to find the remaining path area.