Quadrilaterals – JSS2 Basic Technology Lesson Note (Second Term)

A quadrilateral is a closed figure with four straight sides, and its four angles add up to 360°.

Behavioural objectives

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

1. define a quadrilateral and state that the sum of its angles is 360°;
2. name the five types of quadrilaterals covered in the lesson;
3. state the properties of a parallelogram, rectangle, square, rhombus and trapezium;
4. write the formulas for the area and perimeter of each type;
5. use the comparison table to tell the quadrilaterals apart.

What is a quadrilateral?

A quadrilateral is a polygon with four sides, four corners and four angles.

• Polygon: a closed figure made of straight sides, such as a triangle or a quadrilateral.
• Sides: a quadrilateral has four straight sides that join end to end.
• Vertices: the four corners where two sides meet.
• Diagonals: the two lines that join opposite corners.
• Angle sum: the four angles add up to 360° in every quadrilateral.
• Reason for 360°: one diagonal splits the figure into two triangles, and each triangle has angles that add up to 180°.

What do the key words mean?

These words are used to describe the properties of every quadrilateral in this note.

What are the types of quadrilaterals?

There are five types of quadrilaterals in this lesson: the parallelogram, rectangle, square, rhombus and trapezium.

• Parallelogram: both pairs of opposite sides are parallel.
• Rectangle: a parallelogram with four right angles, like a classroom door or a blackboard.
• Square: four equal sides and four right angles, like a floor tile.
• Rhombus: four equal sides with opposite sides parallel, like the shape of a kite.
• Trapezium: only one pair of parallel sides.

• Angle sum: the angles of each type add up to 360°.

What is a parallelogram?

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

What are the properties of a parallelogram?

• Opposite sides: parallel and congruent.
• Opposite angles: congruent.
• Adjacent angles: supplementary.
• Diagonals: they bisect each other, and each diagonal divides the parallelogram into two congruent triangles.
• Right angle test: if one angle is a right angle, all the other angles are right angles and the figure is a rectangle.

What are the formulas for a parallelogram?

• Area: L × H, where L is the base and H is the perpendicular height.
• Perimeter: 2(L + B), where L is the base and B is the slanting side.

What is a rectangle?

A rectangle is a parallelogram in which all four angles are right angles.

What are the properties of a rectangle?

• Opposite sides: parallel and congruent.
• Angles: all four are right angles.
• Diagonals: congruent, and they bisect each other (divide each other equally).
• Angles at the meeting point: the opposite angles formed where the diagonals cross are congruent.
• Link to parallelogram: a rectangle is a special parallelogram with right angles.

What are the formulas for a rectangle?

If the length is L and the breadth is B, then:

• Length of diagonal: √(L² + B²).
• Area: L × B.
• Perimeter: 2(L + B).

What is a square?

A square is a quadrilateral with four equal sides and four right angles.

What are the properties of a square?

• Sides and angles: all sides are congruent and all angles are congruent.
• Opposite sides: parallel to each other.
• Diagonals: congruent.
• Diagonals meeting: perpendicular to each other, and they bisect each other.
• Link to parallelogram: a square is a special parallelogram with equal angles and equal sides.
• Square test: a parallelogram is a square when its diagonals are equal and are right bisectors of each other.

What are the formulas for a square?

If L is the length of a side of a square, then:

• Length of diagonal: L√2.
• Area: L².
• Perimeter: 4L.

What is a rhombus?

A rhombus is a parallelogram with four equal sides.

What are the properties of a rhombus?

• Sides: all four sides are congruent.
• Opposite angles: congruent.
• Diagonals: perpendicular to each other, and they bisect each other.
• Adjacent angles: supplementary (for example, ∠A + ∠B = 180°).
• Link to parallelogram: a rhombus is a parallelogram whose diagonals are perpendicular to each other.

What are the formulas for a rhombus?

If a and b are the lengths of the diagonals and L is the length of one side, then:

• Area: (a × b) / 2.
• Perimeter: 4L.

What is a trapezium?

A trapezium is a quadrilateral with one pair of parallel sides, called the bases.

What are the properties of a trapezium?

• Bases: the two bases are parallel to each other (MN ⫽ OP in the diagram).
• General trapezium: no sides, angles or diagonals are congruent when the slanting sides are unequal.

What are the formulas for a trapezium?

Let L and L2 be the parallel sides, L1 and L3 the slanting sides, and h the perpendicular height.

• Area: (1/2) × h × (L + L2).
• Perimeter: L + L1 + L2 + L3.

How are the quadrilaterals related?

Some quadrilaterals are special cases of others.

• Rectangle: a parallelogram with right angles.
• Rhombus: a parallelogram with four equal sides.
• Square: a rectangle with four equal sides, and also a rhombus with right angles.
• Trapezium: only one pair of parallel sides, so it is not a parallelogram.

How do the properties of the quadrilaterals compare?

The table compares the parallelogram, rectangle, rhombus and square. A tick (✓) means the property is always true.

How do we use the formulas? (worked examples)

Example 1: A rectangle is 8 cm long and 6 cm broad. Find its area, perimeter and diagonal.

1. Area = L × B = 8 × 6 = 48 cm².
2. Perimeter = 2(L + B) = 2(8 + 6) = 28 cm.
3. Diagonal = √(8² + 6²) = √100 = 10 cm.

Example 2: A square has a side of 7 cm. Find its area and perimeter.

1. Area = L² = 7 × 7 = 49 cm².
2. Perimeter = 4L = 4 × 7 = 28 cm.

Example 3: A rhombus has diagonals of 10 cm and 24 cm. Find its area.

1. Area = (a × b) / 2.
2. Area = (10 × 24) / 2 = 240 / 2 = 120 cm².

Example 4: A parallelogram has a base of 8 cm and a perpendicular height of 5 cm. Find its area.

1. Area = L × H.
2. Area = 8 × 5 = 40 cm².

Example 5: A trapezium has parallel sides of 6 cm and 10 cm, and a height of 4 cm. Find its area.

1. Area = (1/2) × h × (L + L2).
2. Area = (1/2) × 4 × (6 + 10) = 2 × 16 = 32 cm².

Evaluation

1. State the sum of the angles of a quadrilateral.
2. List the five types of quadrilaterals covered in this lesson.
3. Write the formulas for the area and perimeter of a rectangle.
4. State two properties of the diagonals of a rhombus.
5. Which of the quadrilaterals in the table have congruent diagonals?

Assignment

Answer both parts.

1. Define a polygon and a quadrilateral.
2. A rectangle is 12 cm long and 5 cm broad. Find its area, perimeter and the length of its diagonal.