Areas (Edexcel GCSE Maths): Revision Notes

Areas

Understanding how to calculate areas is essential for GCSE mathematics. This topic covers everything from basic triangles to complex composite shapes and circles. Let's work through the key concepts and formulas you need to master.

Areas of triangles and quadrilaterals

When calculating areas of triangles and quadrilaterals, there's one fundamental principle that applies to all shapes in this category.

Triangle area

The area of any triangle can be found using the base and the perpendicular height drawn to that base.

The formula for triangle area is:

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{vertical height}$

Parallelogram area

A parallelogram's area is calculated by multiplying the base by the vertical height. Notice this is similar to a rectangle, but we must use the perpendicular height, not the slanted side.

$\text{Area of parallelogram} = \text{base} \times \text{vertical height}$

Trapezium area

A trapezium (or trapezoid) has two parallel sides. To find its area, we use the average of the parallel sides multiplied by the distance between them.

$\text{Area of trapezium} = \frac{1}{2}(a + b) \times \text{vertical height}$

Here, '$a$' and '$b$' are the lengths of the parallel sides, and the vertical height is the perpendicular distance between them.

Working with composite shapes

Many real-world problems involve composite shapes - these are made up of different basic shapes joined together. The key strategy is to break them down into simpler shapes that you can calculate separately.

For example, if you have a house-shaped wall, you might split it into a rectangle for the main wall and a triangle for the roof section. Calculate each area using the appropriate formula, then add them together.

Areas and circumference of circles

Circle calculations involve the mathematical constant π (pi). Understanding when and how to use π is crucial for both calculator and non-calculator questions.

Circle area

The area of a circle depends on its radius (remember that radius is half the diameter):

$\text{Area of circle} = \pi r^2$

Circumference

The circumference is the distance around the outside of the circle:

$\text{Circumference} = \pi D = 2\pi r$

where $D$ is the diameter and $r$ is the radius.

Areas of sectors and segments

Working with parts of circles requires understanding sectors, arcs, and segments. These calculations are based on the proportion of the full circle.

Sectors and arcs

A sector is like a slice of pie - it's a portion of the circle bounded by two radii and an arc. The formulas for sectors are based on the fraction of the full circle:

$\text{Area of sector} = \frac{x}{360} \times \text{Area of full circle}$

$\text{Length of arc} = \frac{x}{360} \times \text{Circumference of full circle}$

where $x$ is the angle at the centre in degrees.

Segments

A segment is the area between a chord and the arc it cuts off.

This process involves using trigonometry for the triangle area, typically using the formula: $\text{Area of triangle} = \frac{1}{2}r^2\sin x$, where $r$ is the radius and $x$ is the angle.

Important reminders about perimeters

For a semicircle, the perimeter includes the curved part ($\frac{1}{2} \times \text{circumference}$) plus the diameter.