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Geometry: quadrilaterals and polygons — KCSE Mathematics

KCSE Mathematics · 97 practice questions · 4 syllabus objectives · 4 revision lessons

36 easy34 medium27 hard

Last updated · Aligned to the KNEC KCSE syllabus

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

Identify and state the properties of quadrilaterals (parallelogram, rectangle, rhombus, square, trapezium, kite)

Calculate the sum of interior and exterior angles of any polygon and the size of each angle of a regular polygon

Solve problems involving the area of quadrilaterals and polygons using appropriate formulae

Geometry: quadrilaterals and polygons

Revision Notes

Concise lesson notes for Geometry: quadrilaterals and polygons, written to the KCSE Mathematics marking standard. Read the first lesson free below.

Properties of Quadrilaterals

Quadrilaterals are four-sided polygons with various properties. Here are the key types:

  • Parallelogram: Opposite sides are equal and parallel; opposite angles are equal; diagonals bisect each other.
  • Rectangle: All properties of a parallelogram; all angles are right angles; diagonals are equal.
  • Rhombus: All properties of a parallelogram; all sides are equal; diagonals bisect at right angles.
  • Square: All properties of a rectangle and rhombus; all sides are equal; all angles are right angles.
  • Trapezium: At least one pair of parallel sides; angles on the same side are supplementary.
  • Kite: Two pairs of adjacent sides are equal; one pair of opposite angles are equal; diagonals intersect at right angles.

Understanding these properties helps in solving problems related to quadrilaterals. Remember to identify the type of quadrilateral before stating its properties.

Key points to remember

  • Parallelograms have opposite sides equal and parallel.
  • Rectangles have right angles and equal diagonals.
  • Rhombuses have equal sides and diagonals bisect at right angles.
  • Squares combine properties of rectangles and rhombuses.
  • Kites have two pairs of equal adjacent sides.

Worked example

Identify the properties of a rectangle:

  • All angles are right angles.
  • Opposite sides are equal and parallel.
  • Diagonals are equal in length.

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Lesson 2: Angles in Polygons

Objective: Calculate the sum of interior and exterior angles of any polygon and the size of each angle of a regular polygon

To calculate the sum of the interior angles of a polygon, use the formula: (n - 2) × 180°, where n is the number of sides. For example, a hexagon (6 sides) has a sum of interior angles calculated as follows:

  • (6 - 2) × 180° = 4 × 180° = 720°.

For the exterior angles, the sum of the exterior angles of any polygon is always 360°.

To find the size of each angle in a regular polygon, divide the sum of the interior angles by the number of sides. For a regular pentagon (5 sides):

  • Sum of interior angles = (5 - 2) × 180° = 540°.
  • Each interior angle = 540° ÷ 5 = 108°.

For exterior angles of a regular polygon, each exterior angle can be found by dividing the total exterior angle sum by the number of sides. For the same pentagon:

  • Each exterior angle = 360° ÷ 5 = 72°.
  • Sum of interior angles = (n - 2) × 180°.
  • Sum of exterior angles = 360° for any polygon.
  • Each angle of a regular polygon = Sum of angles ÷ n.
  • Regular polygon angles are equal in size.
  • Use the number of sides to calculate angles.

Calculate the sum of interior angles and each angle of a regular octagon.

  • Sum = (8 - 2) × 180° = 1080°.
  • Each angle = 1080° ÷ 8 = 135°.
Lesson 3: Calculating Area of Quadrilaterals and Polygons

Objective: Solve problems involving the area of quadrilaterals and polygons using appropriate formulae

To solve problems involving the area of quadrilaterals and polygons, it's essential to use the correct formulae. Common quadrilaterals include:

  • Rectangle: Area = length × width
  • Square: Area = side²
  • Parallelogram: Area = base × height
  • Trapezium: Area = 1/2 × (base1 + base2) × height

For polygons, especially regular ones (where all sides and angles are equal), you can use the formula:

  • Regular Polygon: Area = (Perimeter × Apothem) / 2

Example Problem: Calculate the area of a rectangle with a length of 10 cm and a width of 5 cm.

  • Solution: Area = length × width = 10 cm × 5 cm = 50 cm².

Another Example Problem: Find the area of a trapezium with bases of 8 cm and 6 cm, and a height of 4 cm.

  • Solution: Area = 1/2 × (base1 + base2) × height = 1/2 × (8 cm + 6 cm) × 4 cm = 28 cm².
  • Use correct formulae for different quadrilaterals.
  • Area of rectangle: length × width.
  • Area of trapezium: 1/2 × (base1 + base2) × height.
  • Regular polygon area: (Perimeter × Apothem) / 2.
  • Show all steps clearly for full marks.

Calculate the area of a square with a side of 4 cm.

  • Area = side² = 4 cm × 4 cm = 16 cm².
Lesson 4: Understanding Quadrilaterals and Polygons

Objective: Geometry: quadrilaterals and polygons

Quadrilaterals are four-sided polygons with various properties. The main types of quadrilaterals include:

  • Square: All sides equal, all angles 90°.
  • Rectangle: Opposite sides equal, all angles 90°.
  • Rhombus: All sides equal, opposite angles equal.
  • Trapezium: At least one pair of parallel sides.

To classify a polygon, remember:

  • A polygon is a closed figure with straight sides.
  • The sum of interior angles of a polygon can be calculated using the formula: (n-2) × 180°, where n is the number of sides.

For example, a pentagon (5 sides) has an interior angle sum of:

  • (5-2) × 180° = 3 × 180° = 540°.

Understanding these properties helps in solving problems related to area, perimeter, and angle measurements in quadrilaterals and polygons. Always label your diagrams and show your working to maximize your marks in exams.

  • Quadrilaterals have four sides with specific properties.
  • Types include squares, rectangles, rhombuses, and trapeziums.
  • Polygons are closed figures with straight sides.
  • Interior angle sum formula: (n-2) × 180°.
  • Label diagrams and show working for full marks.

Calculate the sum of interior angles of a hexagon.

  • A hexagon has 6 sides.
  • Using the formula: (6-2) × 180° = 720°.

Sample Questions

Read 3 questions and answers free. Sign up to access all 97 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER2 marks

A rhombus has diagonals measuring 8 cm and 6 cm. Find the area of the rhombus. (2 marks)

Answer & marking scheme

Part (a) — 2 marks
Area = ½ × diagonal1 × diagonal2 = ½ × 8 cm × 6 cm (1 mk)
State the area = 24 cm² (1 mk)
2
easySHORT ANSWER2 marks

In a rectangle, the length is 10 cm and the width is 5 cm. Calculate the area of the rectangle. (2 marks)

Answer & marking scheme

Part (a) — 2 marks
Area = length × width = 10 cm × 5 cm (1 mk)
State the area = 50 cm² (1 mk)
3
easySHORT ANSWER4 marks

Each exterior angle of a regular polygon measures 45°. (a) Find the number of sides of the polygon. (b) Calculate the sum of all interior angles of the polygon. (4 marks)

Answer & marking scheme

Part (a) — 2 marks
Use the formula n = 360/45 (1 mk)
State n = 8 (1 mk)
Part (b) — 2 marks
Sum of interior angles = (n − 2) × 180 = (8 − 2) × 180 (1 mk)
Sum of interior angles = 1080° (1 mk)
4

A regular polygon has a sum of interior angles equal to 720°. (a) Determine the number of sides of the polygon. (b) Calculate the measure of each interior angle. (4 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Geometry: quadrilaterals and polygons" cover?

Geometry: quadrilaterals and polygons covers Identify and state the properties of quadrilaterals (parallelogram, rectangle, rhombus, square, trapezium, kite); Calculate the sum of interior and exterior angles of any polygon and the size of each angle of a regular polygon; Solve problems involving the area of quadrilaterals and polygons using appropriate formulae, and more, all aligned to the official KNEC KCSE Mathematics syllabus.

How many practice questions are available for Geometry: quadrilaterals and polygons?

HighMarks has 97 Geometry: quadrilaterals and polygons practice questions for KCSE Mathematics, each with a full marking scheme. The first 3 are free; sign up to access the rest, plus all KCSE mock exams and past papers.

Are these aligned with the KNEC KCSE syllabus?

Yes. Every objective on this page is taken directly from the official KNEC KCSE Mathematics syllabus. Practice questions match the KCSE exam format and are graded against the standard KNEC marking scheme.

How should I revise Geometry: quadrilaterals and polygons for the KCSE exam?

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Why Practise Geometry: quadrilaterals and polygons?

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