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Chapter 14: Area

How much paint do you need to colour a rectangle? How much grass covers a triangular plot? Both questions are answered by area , the count of unit squares (squares of side 11) that fit inside a region.

This chapter takes a careful tour of area. We start where it all begins: rectangles and squares, where area is simply length×width\text{length} \times \text{width}. We then ask a sharp question , is perimeter a good measure of area? (Spoiler: no , two regions can have the same perimeter but very different areas.) Next we develop the area formula for triangles, 12×base×height\dfrac{1}{2} \times \text{base} \times \text{height}, and see why the same formula works for every triangle, including stretched-out obtuse ones. Finally we tackle real-life composite figures , paths around parks, crosspaths through plots, and spirals , and a beautiful family of identities involving triangles between two parallel lines.

By the end you will see area not as a memorised formula but as a flexible idea: count the squares. Once you can rearrange and decompose a region, you can find its area, no matter how irregular it looks.

What's inside

  1. Rectangle and squares , the foundation of area.
  2. Perimeter is not area , why the boundary length can mislead you.
  3. Area of triangles , the half-rectangle formula.
  4. Paths and composite figures , break, subtract, or rearrange.
  5. Triangles between parallel lines , same base, same height, same area.

Key results

ShapeArea
Rectanglelength×width\text{length} \times \text{width}
Square (side aa)a2a^2
Right triangle12×base×height\dfrac{1}{2} \times \text{base} \times \text{height}
Any triangle12×base×height\dfrac{1}{2} \times \text{base} \times \text{height}

Key idea , rearrangement. Cutting a region into pieces and reassembling them does not change its area.

Triangles on the same base between two parallels have equal areas , because they share the same base and the same perpendicular height.

How to read this chapter

Always check the units. Area is reported in square units , square centimetres, square metres , never in plain centimetres. And whenever you meet a strange region, ask: can I split it into rectangles and triangles I already know?

Sub-topics

5 pages
01
Rectangles and squares
Counting unit squares gives the simplest area formula of all , and the special case for squares falls out instantly.
02
Perimeter is not area
Two regions can have the same perimeter but very different areas , and vice versa. Why the boundary length is not a substitute for the area inside.
03
Area of triangles
Every triangle fits inside a rectangle whose area it shares exactly half , and that single fact gives the universal formula.
04
Paths and composite figures
Break a complicated shape into rectangles and triangles you know, add or subtract , and the area falls out.
05
Triangles between parallel lines
Slide a triangle's apex along a line parallel to its base , its area stays exactly the same, even as its shape changes.

Practice quiz

Answer the questions; explanations appear after each.

Quiz
Chapter 14 : Mixed practice: Area
8 questions · pick the best answer
Q1

Area of a rectangle 1212 cm ×\times 77 cm

Q2

Area of a square with side 99 m

Q3

Area of a triangle with base 1010 cm, height 66 cm

Q4

Two rectangles with same perimeter 2020 cm : which has more area?

Q5

A diagonal of an 8×68 \times 6 rectangle divides it into two triangles, each of area

Q6

11 m2^2 equals

Q7

Triangles between two parallel lines, on the same base, have

Q8

Garden 2020 m ×\times 1212 m with a 22-m-wide path outside on all sides. Path area