Introduction to Chapter 2

In ‎Chapter 1, we discovered the geometric method as a highly effective approach for calculating the area of irregular polygons. Whether you are a roofer estimating the cost of a roof with polygonal sections, a skilled mason meticulously selecting edgy granite stones to fit along a retaining wall, or a horticulturist needing to determine the area of a polygon-shaped field for precise fertilizer spraying, this method is both reliable and easily applicable mathematically. However, as you may have realised from the investigation in Chapter 1 Practice Problems (Question 7), the geometric method has its limits when the contour of the shape is curvilinear. Take, for example, the heart symbol or a typical leaf (Figure ‎2.1), unless we use multiple vertices and straight lines to outline its perimeter, the area produced by the geometric method will be inaccurate.

‎Chapter 2 of this mini-course introduces an analytical approach to calculating the area of irregular spaces called the offset method. This method, based on calculus, divides a shape into a series of trapezoids to estimate its area. Calculus is the branch of mathematics that focuses on change and motion. Integration, which is part of calculus, involves finding the area under a curve. In the case of the offset method, we gather width data evenly across the entire length of an irregular shape and depict it on a graph. The resulting curve corresponds to the shape of the space, which can be divided into numerous small trapezoids. Integration allows us to add up the areas of all these trapezoids to find the total area. So, by applying the offset method and using integration, tradespeople who depend on accurate area measurements can confidently determine the area of irregular spaces.