Mensuration and trigonometry Study Guide

The standard formula for the area of a triangle, Area = ½ × base × height, requires the perpendicular height, which is not always directly given. For these situations, especially when dealing with non-right-angled triangles, a more advanced formula involving trigonometry can be used. This formula is Area = ½ab sin C, where 'a' and 'b' are the lengths of two sides of the triangle, and 'C' is the included angle between those two sides. The 'included angle' is the angle formed by the two sides whose lengths you are using in the formula. This formula works for any triangle. To use it, you need to know the lengths of two sides and the measure of the angle located between those two sides (the included angle). If you are given three sides and no angles, you would first need to find an angle using the cosine rule before applying this area formula. Similarly, if you are given two angles and one side, you might need to use the sine rule to find a second side, and then calculate the third angle to ensure you have two sides and their included angle. This method is particularly useful in higher-level geometry and problem-solving contexts where direct height measurement is impractical or impossible.

A parallelogram is a four-sided shape where opposite sides are parallel and equal in length. Unlike a rectangle, its internal angles are not necessarily 90 degrees. To find the area of a parallelogram, we use a simple formula: Area = base × perpendicular height. The 'base' is any one of the sides of the parallelogram. The 'perpendicular height' (or simply 'height') is the shortest distance from the base to the opposite side, measured at a right angle (90 degrees) to the base. It is crucial to use the perpendicular height, not the length of the slanted side. Imagine transforming a parallelogram into a rectangle: you can cut off a right-angled triangle from one end and attach it to the other end. This forms a rectangle with the same base and the same perpendicular height as the original parallelogram. This visual transformation helps to understand why the formula Area = base × height works. Always ensure that the base and height are in the same units before calculating the area, and the final area will be in square units (e.g., cm², m²).