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

The area of a plane figure is the amount of flat surface it covers, measured in square units.

Behavioural objectives

1. State the formulas for the area of a rectangle, a square, a triangle, a circle and a sector.
2. Calculate the area of a rectangle, a triangle and a circle from given measurements.
3. Calculate the area of a sector using the radius and the angle.
4. Split a compound shape into simpler figures.
5. Find the total area of a compound shape and use it to solve a money problem.

What are the formulas for the area of plane figures?

Each plane figure has its own area formula, and the answer is always in square units.

• Square units: area is written as cm², m² or km², depending on the unit of length used.
• Compound shape: a shape made from two or more simpler figures joined together.
• Value of π: use 3.14159… or the value your teacher gives (for example 3.14 or 22/7).

What is the area of a sector?

A sector is the part of a circle enclosed by two radii and the arc between them.

Area = ½ × r² × θ

• r: the radius of the circle.
• θ: the angle at the centre, in radians.

Sector with the angle in degrees

• Formula: Area = (θ ÷ 360) × π × r², with θ in degrees.
• Link between units: 180° = π radians.

Worked example: angle in radians

Find the area of a sector with radius 4 cm and angle 1.5 radians.

1. Write the formula: Area = ½ × r² × θ.
2. Substitute: Area = ½ × 4² × 1.5.
3. Work out: ½ × 16 × 1.5 = 12 cm².

Worked example: angle in degrees

Find the area of a sector with radius 6 cm and angle 60°.

1. Write the formula: Area = (θ ÷ 360) × π × r².
2. Substitute: Area = (60 ÷ 360) × π × 6² = ⅙ × π × 36.
3. Work out: 6π = 18.85 cm² (to 2 decimal places).

How do you find the area of a rectangle?

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

Example: What is the area of this rectangle?

• Formula: Area = w × h.
• w: width.
• h: height.

1. Read the measurements: w = 5 and h = 3.
2. Substitute: Area = 5 × 3.
3. Work out: Area = 15 square units.

How do you find the area of a circle?

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

Example: What is the area of this circle?

• Radius: r = 3.
• Diameter: if only the diameter is given, halve it to get the radius first.

How do you find the area of a triangle?

The area of a triangle is half of its base multiplied by its perpendicular height.

Example: What is the area of this triangle?

• Height: h = 12.
• Base: b = 20.
• Note: the height is measured at right angles to the base, not along a slanting side.

1. Write the formula: Area = ½ × b × h.
2. Substitute: Area = ½ × 20 × 12.
3. Work out: Area = 120 square units.

How do you find the area of a compound shape?

To find the area of a compound shape, split it into simple figures, find each area and add them.

Example: Sam cuts grass at $0.10 per square metre. How much does Sam earn cutting this area?

Steps

1. Break the shape into two parts, as shown below.
2. Find the area of each part.
3. Add the areas to get the total.
4. Multiply the total area by the price per square metre.

Part A: the square

• Formula: Area of A = a².
• Working: 20 m × 20 m = 400 m².

Part B: the triangle

• Measurements: viewed sideways, it has a base of 20 m and a height of 14 m.
• Working: Area of B = ½ × b × h = ½ × 20 m × 14 m = 140 m².

Total area and money earned

• Total area: Area of A + Area of B = 400 m² + 140 m² = 540 m².
• Earnings: $0.10 × 540 = $54.

What mistakes should you avoid when finding area?

Most errors come from wrong units, wrong radius or a missing half.

• Forgetting the half: a triangle needs ½ × b × h, not b × h.
• Squaring wrongly: r² means r × r, not r × 2.
• Using the diameter: the circle formula needs the radius.
• Mixing units: change all lengths to the same unit before multiplying.
• Wrong angle unit: use radians in ½ × r² × θ, and degrees in (θ ÷ 360) × π × r².

Evaluation

1. Write the formulas for the area of a circle and the area of a sector.
2. Find the area of a rectangle with width 6 and height 4.
3. Find the area of a triangle with base 10 and height 8.
4. Find the area of a circle of radius 2. Give your answer to 2 decimal places.
5. Explain how you would find the area of a shape made up of a square and a triangle.

Assignment

1. A triangle has base 12 cm and height 8 cm. Find its area. A rectangle with height 6 cm has the same area as the triangle. Find the width of the rectangle.