Mensuration and trigonometry Study Guide

A trapezium is a quadrilateral (a four-sided shape) with at least one pair of parallel sides. These parallel sides are often denoted as 'a' and 'b'. The perpendicular distance between these parallel sides is called the 'height' (h). To calculate the area of a trapezium, we use a specific formula. Imagine taking two identical trapeziums and placing one upside down next to the other. This forms a parallelogram. The base of this parallelogram would be the sum of the two parallel sides of the trapezium (a + b), and its height would be 'h'. The area of this parallelogram would be (a + b) × h. Since this parallelogram is made up of two identical trapeziums, the area of a single trapezium is half of the area of the parallelogram. Therefore, the formula for the area of a trapezium is Area = ½ × (a + b) × h.

To apply this formula, first identify the two parallel sides (a and b) and the perpendicular height (h). Ensure all measurements are in the same units. Add the lengths of the two parallel sides, then multiply this sum by the height, and finally, divide by two (or multiply by ½). It's crucial that the height is the perpendicular distance, not a slanted side.

For example, if a = 5 cm, b = 7 cm, and h = 4 cm: Area = ½ × (5 + 7) × 4 Area = ½ × 12 × 4 Area = ½ × 48 Area = 24 cm².

The area of a circle refers to the total amount of space enclosed within its boundary, or circumference. To calculate this area, we use a fundamental formula that involves two key components: 'pi' (π) and the 'radius' (r) of the circle. Pi is a mathematical constant, approximately equal to 3.14159, representing the ratio of a circle's circumference to its diameter. The radius is the distance from the centre of the circle to any point on its circumference. The formula for the area of a circle is given by A = πr², where 'A' is the area and 'r' is the radius squared (r × r).

If you are given the diameter (d) of a circle instead of the radius, you first need to find the radius. The diameter is the distance across the circle passing through its centre, so the radius is simply half of the diameter (r = d/2). Once you have the radius, you can apply the area formula. For instance, if a circle has a radius of 5 cm, its area would be A = π × (5 cm)² = π × 25 cm² = 25π cm². This value can further be approximated by multiplying 25 by 3.14159, giving approximately 78.54 cm². Remember to always include the appropriate square units (e.g., cm², m²) when stating the area.