Area

 

What is Area?

Area is the amount of space enclosed within the perimeter of a 2D shape.

It is measured in square units such as $mm^2$, $cm^2$, $m^2$, or $km^2$.

Example: The area of a sports field or a painting.

How to Find the Area of a Shape on a Square Grid

1. Count the total number of whole squares inside the shape.

2. Pair up half squares to form whole squares.

3. Use the given scale to determine how much area one square represents.

4. Multiply the number of squares counted by this value to find the total area.

Key Area Formulae:

Area of a Rectangle: $$A = l \times w$$

where:

• l = length

• w = width

 

Area of a Triangle: $$A = \frac{1}{2}bh$$

where:

• b = base

• h = perpendicular height

 

Area of a Triangle Using Sine

To find the area of any triangle (not just right-angled), use:

$$\text{Area} = \frac{1}{2} ab \sin C$$

 

 

Where:

• $a$ and $b$ are two sides
• $C$ is the included angle

When to Use This Formula

You know two sides and the angle between them

It avoids having to use height directly

 

Worked Example: Area of a Triangle

In triangle DEF:

$DE = 5.2 m$

$EF = 7.1 m$

$\angle D = 64\degree$

 

 

Calculate the Area

$$\text{Area} = \frac{1}{2} \cdot 5.2 \cdot 7.1 \cdot \sin(64^\circ)$$

$$\text{Area} \approx 0.5 \cdot 5.2 \cdot 7.1 \cdot 0.8988 \approx 16.6 \text{ m}^2 \text{ (3 s.f.)}$$

 

 

Area of a Trapezium: $$A = \frac{1}{2}(a + b)h$$

where:

• a and b = lengths of the parallel sides

• h = perpendicular height

 

Area of a Parallelogram: $$A = bh$$

where:

• b = base

• h = perpendicular height (not the side length)

Examples

Example 1: Area of a Trapezium

• Given: a = 26 cm, b = 11 cm, h = 9 cm

 

 

$$A = \frac{1}{2}(26 + 11) \times 9$$

$$A = \frac{1}{2}(37) \times 9$$

$$A = 333 cm^2$$

 

Final Answer: 333 $cm^2$

 

Example 2: Area of a Parallelogram

• Given: b = 20 cm, h = 18 cm

 

 

$$A = 20 \times 18$$

$$A = 360 cm^2$$

 

Final Answer: 360 $cm^2$

 

Example 3: Area of a Isosceles triangle

• Given: b = 14 cm, h = 6 cm

 

 

$$A = \frac{1}{2} \times 13 \times 6$$

$$A = 39 cm^2$$

 

Final Answer: 39 $cm^2$