Area
This chapter studies how much flat space a shape covers. We open with rectangles and squares, where the area is just the product of two sides, and then pause on a deeper question: why do we count unit squares to measure area instead of relying on the boundary length? Starting from a rectangle, we slice, fold and shuffle pieces until a single elegant result emerges, the area of a triangle, and we notice that every polygon can be carved into triangles. From there we assemble tidy formulas for the parallelogram, the rhombus and the trapezium, each by reshaping the figure into a rectangle of matching area (a technique called dissection, well known to the ancient Indian Śulba-Sūtra geometers). We close with everyday units of area, square inches, square feet, acres and square kilometres.
Learning Outcomes
By the end of this chapter, you will be able to:
- explain why area is counted in unit squares, and why perimeter is not a measure of area;
- find the area of a rectangle and a square, and of composite figures built from rectangles;
- derive and apply area of a triangle = \tfrac12 \times \text{base} \times \text{height} for triangles of every shape;
- split any polygon into triangles to compute its area;
- find the area of a parallelogram, a rhombus and a trapezium by dissection;
- convert between units of area, \text{cm}^2, \text{in}^2, \text{ft}^2, acres and \text{km}^2.