Area of Plane Figures – JSS3 Mathematics Lesson Note (Second Term)

Area is the amount of surface a plane figure covers, measured in square units such as cm² and m².

Behavioural objectives

By the end of the lesson, students should be able to:

1. State what area is and give its units.
2. Write the area formula for a triangle, rectangle, trapezium, ellipse, square, parallelogram and circle.
3. Say what each letter in these formulas stands for.
4. Write the formula for the circumference of a circle.
5. Use the formulas to find the area of simple plane figures.

What is area?

Area is the size of a flat surface, found by measuring the space inside its boundary.

• Plane figure: a flat shape with length and width only, such as a square or a circle.
• Measurements used: base, vertical height, width, side length or radius, depending on the shape.
• Units of area: square units, written with a small 2, for example cm² and m².
• Everyday example: the floor of a classroom, measured in m², to know how much tile is needed.

What are the area formulas for plane figures?

Each plane figure has its own area formula, and the table below lists all seven.

Triangle

The area of a triangle is half the base multiplied by the vertical height.

Area = ½ × b × h

• b: base.
• h: vertical height.

Rectangle

The area of a rectangle is its width multiplied by its height.

Area = w × h

• w: width.
• h: height.

Trapezium

A trapezium is a four-sided figure with one pair of parallel sides, called a trapezoid in the US.

Area = ½(a + b) × h

• a and b: the lengths of the two parallel sides.
• h: vertical height.

Ellipse

An ellipse is an oval shape, and its area is π multiplied by a and b.

Area = πab

• a and b: the semi-major and semi-minor axes (half the longest and shortest widths).

Square

The area of a square is the length of one side multiplied by itself.

Area = a²

• a: length of side.

Parallelogram

The area of a parallelogram is its base multiplied by its vertical height.

Area = b × h

• b: base.
• h: vertical height.

Circle

The area of a circle is π multiplied by the radius squared.

Area = π × r²

Circumference = 2 × π × r

• r: radius, the distance from the centre to the edge.
• Circumference: the distance round the circle.
• Diameter: twice the radius, so r = diameter ÷ 2.

How are the base and height measured?

The height is measured as the vertical height, at right angles to the base.

Note: h is at right angles to b.

• Vertical height: the straight distance from the base to the top, not along a sloping side.
• Sloping side: never use it as h in the triangle, trapezium or parallelogram formulas.

How do we use the formulas? (worked examples)

To find an area, write the formula, substitute the values, calculate and add the unit.

Steps

1. Write down the formula for the figure.
2. Replace each letter with its given value.
3. Work out the answer.
4. Write the answer with the correct square unit.

Example 1: Triangle

Find the area of a triangle with base 8 cm and vertical height 5 cm.

• Area = ½ × b × h
• Area = ½ × 8 × 5
• Area = 20 cm²

Example 2: Trapezium

Find the area of a trapezium with parallel sides 6 cm and 10 cm and vertical height 4 cm.

• Area = ½(a + b) × h
• Area = ½(6 + 10) × 4 = ½ × 16 × 4
• Area = 32 cm²

Example 3: Circle

A circle has radius 5 cm. Leave π in your answers.

• Area = π × r² = π × 5 × 5 = 25π cm²
• Circumference = 2 × π × r = 2 × π × 5 = 10π cm

What mistakes should students avoid?

Most errors come from using the wrong measurement or forgetting a step.

• Forgetting the half: the triangle and trapezium formulas both contain ½.
• Using the slanting side: always use the vertical height, at right angles to the base.
• Mixing up area and circumference: area uses r², circumference uses 2 × π × r.
• Using the diameter as the radius: halve the diameter first.
• Wrong units: area is in square units, circumference is in ordinary length units.

Evaluation

1. Write the formula for the area of a triangle.
2. Find the area of a triangle with base 10 cm and vertical height 6 cm.
3. Find the area of a square whose side is 7 cm.
4. Find the area of a trapezium with parallel sides 5 cm and 9 cm and vertical height 4 cm.
5. A circle has a radius of 7 cm. Leave π in your answers and find (a) the area and (b) the circumference.

Assignment

1. Answer all three parts: (a) find the area of a parallelogram with base 8 cm and vertical height 5 cm; (b) find the area of a rectangle with width 6 cm and height 9 cm; (c) a circle has a radius of 3 cm, so leave π in your answers and find its area and its circumference.