Area of Plane Figures – JSS2 Basic Technology Lesson Note (Second Term)

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

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 formulas for the triangle, rectangle, trapezium, square, parallelogram and circle;
3. calculate the area of a rectangle, a circle and a triangle from given measurements;
4. break a combined shape into simpler parts and find its total area.

What is area?

Area is the size of a surface, found by measuring how much space lies inside the edges of a plane figure.

• Plane figure: a flat shape such as a square, triangle or circle.
• Unit of area: a square unit, such as square centimetres (cm²) or square metres (m²).
• One square metre: the area of a square with sides of 1 m.
• Measurements used: base, height, width, radius or side length, depending on the shape.
• Everyday use: finding the size of a floor to tile, a farm plot to clear or a wall to paint.

What are the area formulas at a glance?

Each plane figure has its own area formula, shown in the table below.

How do you find the area of each plane figure?

What is the area of a triangle?

A triangle is a plane figure with three straight sides, and its area is half of base times vertical height.

• Formula: Area = ½ × b × h
• b: base.
• h: vertical height.

What is the area of a rectangle?

A rectangle is a four-sided figure with four right angles and equal opposite sides, and its area is width times height.

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

What is the area of a trapezium?

A trapezium (trapezoid in US English) is a four-sided figure with one pair of parallel sides.

• Formula: Area = ½(a + b) × h
• a and b: the two parallel sides.
• h: vertical height between the parallel sides.

What is the area of an ellipse?

An ellipse is an oval-shaped closed curve, and its area is π times a times b.

• Formula: Area = πab
• a and b: half of the longest width and half of the shortest width.

What is the area of a square?

A square is a four-sided figure with four equal sides and four right angles, and its area is the side multiplied by itself.

• Formula: Area = a²
• a: length of side.

What is the area of a parallelogram?

A parallelogram is a four-sided figure with two pairs of parallel sides, and its area is base times vertical height.

• Formula: Area = b × h
• b: base.
• h: vertical height.

What is the area of a circle?

A circle is a round plane figure in which every point on the edge is the same distance from the centre.

• Area formula: Area = π × r²
• Circumference formula: Circumference = 2 × π × r
• r: radius, the distance from the centre to the edge.
• Value of π: about 3.14159, often rounded to 3.14.

What is the area of a sector?

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

• Formula (radians): Area = ½ × r² × θ
• Formula (degrees): Area = (θ ÷ 360) × π × r²
• r: radius.
• θ: the angle at the centre. Use radians in the first formula and degrees in the second.

How are the height and base related?

The vertical height is the straight distance from the base to the top of the figure, measured at right angles to the base.

• Right angle: h is at right angles to b.
• Not the slanted side: never use a sloping side as the height.
• Used in: the triangle, parallelogram and trapezium.

How do you calculate the area of a plane figure?

Follow these steps to find the area of any plane figure.

1. Identify the shape.
2. Write down the formula for that shape.
3. Write the given measurements next to the letters in the formula.
4. Substitute the numbers into the formula.
5. Work out the answer.
6. Write the answer with square units.

Worked examples

Example 1: What is the area of this rectangle?

1. Formula: Area = w × h, where w = width and h = height.
2. Given: w = 5 and h = 3.
3. Area = 5 × 3 = 15 square units

Example 2: What is the area of this circle?

Radius = r = 3

• Area = π × r²
• = π × 3²
• = π × (3 × 3)
• = 3.14159… × 9
• = 28.27 square units (to 2 decimal places)

Example 3: What is the area of this triangle?

1. Height = h = 12 and base = b = 20.
2. Area = ½ × b × h
3. = ½ × 20 × 12
4. = 120 square units

Example 4 (a harder example): How much does Sam earn cutting grass?

Sam cuts grass at $0.10 per square metre, and the area is made of a square and a triangle.

1. Break the area into two parts, A and B.
2. Part A is a square:

   

   Area of A = a² = 20 m × 20 m = 400 m²

3. Part B is a triangle. Viewed sideways, it has a base of 20 m and a height of 14 m.
   Area of B = ½ × b × h = ½ × 20 m × 14 m = 140 m²
4. Add the two parts: Area = 400 m² + 140 m² = 540 m²
5. Multiply by the rate: Sam earns $0.10 × 540 = $54

What mistakes should students avoid?

Most errors in area come from a few common slips.

• Wrong height: using a slanted side instead of the vertical height.
• Forgetting the half: the triangle formula needs ½.
• Squaring wrongly: r² means r × r, not r × 2.
• Missing units: area must carry square units such as m².
• Mixed units: change all lengths to the same unit before multiplying.

Evaluation

1. What is area?
2. Write the formula for the area of a triangle.
3. Find the area of a rectangle with width 7 and height 4.
4. Find the area of a square with sides of length 6.
5. Find the area of a circle with radius 2. Give your answer to 2 decimal places.

Assignment

Answer all parts of this task, showing the formula and your working each time.

1. Find the area of a triangle with base 14 and vertical height 10.
2. Find the area of a parallelogram with base 9 and vertical height 5.
3. Write the formulas for the area and the circumference of a circle.