Example 1
Find the area of a rectangle 7 cm long and 4 cm wide.
- For a rectangle, area = length × width.
- Multiply 7 by 4.
- 7 × 4 = 28
So: The area is 28 cm².
Back to National 4
Geometry and Measures (National 4)
Perimeter, Area and Pythagoras
Geometry and area skills support practical design, access and construction decisions involving slopes, shapes, circles, scale factors and right-angled triangles.
Qualification focus: Gradient · Rectilinear, circular and composite perimeter · Triangle, quadrilateral, circle and composite area · Pythagoras · Linear scale factor
Topic explanation
Area is measured in square units such as cm² or m² because it counts squares covering a surface.
For triangles and parallelograms, use the perpendicular height, not a sloping side. For circles, check whether the labelled line is a radius or a diameter.
Pythagoras applies only to right-angled triangles. The hypotenuse is always the side opposite the right angle.
For a ramp or slope, compare rise with horizontal run. A rule such as no steeper than 1:12 means the horizontal distance must be at least 12 times the rise.
Quick methods
- Triangle
- Area = base × height divided by 2
- Quadrilaterals
- Use the supplied perpendicular height or diagonals for parallelograms, kites and rhombi.
- Circle
- Circumference = πd and area = πr²; distinguish radius from diameter
- Compound shapes
- Split into simple shapes, then add or subtract areas.
- Gradient
- Compare vertical height with horizontal distance and simplify the ratio.
- Pythagoras
- In a right-angled triangle, a² + b² = c² and c is opposite the right angle
- Scale factor
- Multiply each linear dimension by the supplied factor.
Worked examples
Example 2
Find the area of a triangle with base 10 cm and height 6 cm.
- For a triangle, area = base × height divided by 2.
- Multiply the base and height: 10 × 6 = 60
- Divide by 2: 60 divided by 2 = 30
So: The area is 30 cm².
Example 3
A compound shape is made from rectangles measuring 8 m by 3 m and 5 m by 4 m. Find the total area.
- Find the first rectangle area: 8 × 3 = 24 m².
- Find the second rectangle area: 5 × 4 = 20 m².
- Add the areas: 24 + 20 = 44
So: The total area is 44 m².