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International Baccalaureate · MYP 1

Area

⌚ ~82 min ▢ 8 blocks ✓ 34 exercises ➣ Prerequisites : Length and Perimeter, Area
I Units of Area, the Are and the Hectare
Method — Converting Using Multiplication or Division
  • Use multiplication to go from a larger unit to a smaller one (like square metres to square centimetres).
  • Use division to go from a smaller unit to a larger one (like square centimetres to square metres).
Figure
Definition — Units of Area
  • Square kilometre \(\left(\mathrm{km}^2\right)\): A very large unit of area, the area of a square with sides of \(1\) km: about \(140\) football pitches.
    Figure
  • Square metre \(\left(\mathrm{m}^2\right)\): An everyday unit of area, the area of a square with sides of \(1\) m: about the size of a small table top.
    Figure
  • Square centimetre \(\left(\mathrm{cm}^2\right)\): A small unit of area, about the size of a fingernail.
    Figure
  • Square millimetre \(\left(\mathrm{mm}^2\right)\): A very small unit of area, about the size of a tiny dot made by a pencil.
    Figure
For land and fields, two more units are common: the are (\(\mathrm{a}\)) and the hectare (\(\mathrm{ha}\)), with \(1\,\mathrm{a}=100\,\mathrm{m}^2\) and \(1\,\mathrm{ha}=100\,\mathrm{a}=10\,000\,\mathrm{m}^2\).
Skills to practice
  • Converting with the Are and the Hectare
II The Area Formulas of the Square and the Rectangle
Proposition — The Area Formulas of the Square and the Rectangle
  • Square: the area of a square is equal to its side multiplied by itself. In words: \(\text{area} = \text{side} \times \text{side}\). In letters, with \(s\) the side: $$A = s \times s.$$
    Figure
  • Rectangle: the area of a rectangle is equal to the product of its length and its width. In words: \(\text{area} = \text{length} \times \text{width}\). In letters, with \(l\) the length and \(w\) the width: $$A = l \times w.$$
    Figure
The area is expressed in the unit of area that matches the lengths: sides in cm give an area in \(\text{cm}^2\), sides in m give an area in \(\text{m}^2\). The perimeter formulas use the same letters: \(P = 4 \times s\) for the square and \(P = 2 \times (l + w)\) for the rectangle.
Method — Using a Formula: Replacing the Letters
For a rectangle of length \(l = 4.6\) m and width \(w = 2.5\) m:
  1. Write the formula with its letters: \(A = l \times w\).
  2. Replace each letter by its value, all the lengths being in the same unit (convert first if needed): \(A = 4.6 \times 2.5\).
  3. Calculate: \(A = 11.5\).
  4. Write the unit, the unit of area that matches the lengths: \(A = 11.5 \, \text{m}^2\).
The order of the two lengths does not matter: \(l \times w = w \times l\), the rectangle turned on its side has the same area.
Example
  1. A rectangle is \(4.6\) m long and \(2.5\) m wide. Calculate its area and its perimeter.
  2. A square has a side of \(6.5\) cm. Calculate its area.
  3. A rectangle is \(7.5\) m long and \(120\) cm wide. Calculate its area in \(\text{m}^2\).
  4. A rectangle is \(5\) m long and \(\dfrac{3}{4}\) m wide. Calculate its area.

  1. \(A = l \times w = 4.6 \times 2.5 = 11.5 \, \text{m}^2\) and \(P = 2 \times (l + w) = 2 \times (4.6 + 2.5) = 2 \times 7.1 = 14.2\) m.
  2. \(A = s \times s = 6.5 \times 6.5 = 42.25 \, \text{cm}^2\).
  3. The two lengths must be in the same unit: \(120\) cm \(= 1.2\) m. Then \(A = 7.5 \times 1.2 = 9 \, \text{m}^2\) (check in cm: \(750 \times 120 = 90\,000 \, \text{cm}^2 = 9 \, \text{m}^2\)).
  4. \(A = 5 \times \dfrac{3}{4} = \dfrac{15}{4} \, \text{m}^2\), and \(\dfrac{15}{4} = 15 \div 4 = 3.75\): the area is \(3.75 \, \text{m}^2\).

Skills to practice
  • Calculating with the Formula
  • Finding Area and Perimeter with the Formulas
  • Converting Before Calculating
III Cutting and Completing
Method — Finding the Area of a Composite Figure
To find the area of a composite figure, follow these steps:
  1. Divide the figure into simple, non-overlapping shapes whose area can be calculated with a formula (rectangles, squares, triangles, …).
  2. Find the area of each simpler shape using the appropriate formula.
  3. Add the areas together to find the total area of the composite figure, or, when a part is missing from a rectangle, subtract its area from the area of the rectangle.
Proposition — Area of a Triangle
The area of a triangle is found by multiplying the base by the height and dividing by 2: $$\text{Area of a triangle} = \dfrac{\text{base} \times \text{height}}{2}$$ $$A = \dfrac{b \times h}{2}$$ where \(b\) is the length of the base and \(h\) is the corresponding height. The height can fall outside the triangle: the base is then extended with a dotted line, and the height meets this extension at a right angle (figure on the right).
Figure  Figure
Example
This L-shaped figure is made of rectangles. Find its area in two different ways.
Figure

Figure
First way: cut the figure into two rectangles, \(6\) cm by \(2\) cm and \(2\) cm by \(2\) cm: \(A = 6 \times 2 + 2 \times 2 = 12 + 4 = 16 \, \text{cm}^2\). Second way: complete the figure into a \(6\) cm by \(4\) cm rectangle and remove the missing \(4\) cm by \(2\) cm rectangle: \(A = 6 \times 4 - 4 \times 2 = 24 - 8 = 16 \, \text{cm}^2\). Both cuttings give the same area.

Skills to practice
  • Finding Areas by Cutting or Completing
IV Going further
Skills to practice
  • Using the Formulas in New Ways
  • Using the Formulas to Solve Problems
  • Investigating Area Patterns
  • Quiz