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Mensuration: perimeter and area of plane figures — KCSE Mathematics

KCSE Mathematics · 104 practice questions · 4 syllabus objectives · 4 revision lessons

32 easy34 medium38 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.

Calculate the perimeter and area of triangles, parallelograms, trapeziums and compound shapes

Calculate the circumference and area of a circle; find arc length and area of a sector using the angle

Apply area formulae to solve problems involving land (in acres and hectares), material and design

Mensuration: perimeter and area of plane figures

Revision Notes

Concise lesson notes for Mensuration: perimeter and area of plane figures, written to the KCSE Mathematics marking standard. Read the first lesson free below.

Calculating Area and Perimeter of Shapes

To calculate the perimeter and area of various plane figures, you need to know the specific formulas for each shape:

  • Triangle:

    • Perimeter = sum of all sides.
    • Area = 1/2 × base × height.

  • Parallelogram:

    • Perimeter = 2 × (base + height).
    • Area = base × height.

  • Trapezium:

    • Perimeter = sum of all sides.
    • Area = 1/2 × (base1 + base2) × height.

  • Compound Shapes: Break down into simpler shapes, calculate area and perimeter separately, then sum them up.

For example, consider a triangle with a base of 6 cm and a height of 4 cm:

  • Area = 1/2 × 6 cm × 4 cm = 12 cm².

For a parallelogram with a base of 5 cm and a height of 3 cm:

  • Area = 5 cm × 3 cm = 15 cm².

Remember to always include the correct units in your answers!

Key points to remember

  • Use specific formulas for each shape.
  • Perimeter is the total length of the sides.
  • Area measures the space within a shape.
  • Break down complex shapes into simpler components.
  • Always include units in your final answers.

Worked example

Calculate the area and perimeter of a trapezium with bases 8 cm and 5 cm, height 4 cm, and sides 6 cm and 7 cm.

  • Area = 1/2 × (8 cm + 5 cm) × 4 cm = 26 cm².
  • Perimeter = 8 cm + 5 cm + 6 cm + 7 cm = 26 cm.

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More lessons in this topic

Lesson 2: Circumference and Area of a Circle

Objective: Calculate the circumference and area of a circle; find arc length and area of a sector using the angle

To calculate the circumference and area of a circle, we use the following formulas:

  • Circumference (C) = 2πr, where r is the radius.
  • Area (A) = πr².

When dealing with arcs and sectors, we also need to consider the angle in degrees.

  • Arc Length (L) = (θ/360) × C, where θ is the angle in degrees.
  • Area of a Sector (A_s) = (θ/360) × A.

For example, if a circle has a radius of 7 cm, we can calculate:

  • Circumference: C = 2 × π × 7 = 14π cm.
  • Area: A = π × (7)² = 49π cm².

If we want to find the arc length and area of a sector for an angle of 60°:

  • Arc Length: L = (60/360) × 14π = (1/6) × 14π = (7/3)π cm.
  • Area of Sector: A_s = (60/360) × 49π = (1/6) × 49π = (49/6)π cm².
  • Circumference formula: C = 2πr.
  • Area formula: A = πr².
  • Arc length depends on angle and circumference.
  • Sector area depends on angle and circle area.
  • Use degrees for angle calculations.

A circle has a radius of 10 cm. Find the circumference and area.

  • Circumference: C = 2π(10) = 20π cm.
  • Area: A = π(10)² = 100π cm².
Lesson 3: Calculating Area of Plane Figures

Objective: Apply area formulae to solve problems involving land (in acres and hectares), material and design

To solve problems involving land, material, and design, it is essential to apply the correct area formulae. Here are some common area formulae:

  • Rectangle: Area = Length × Width
  • Triangle: Area = 1/2 × Base × Height
  • Circle: Area = π × Radius² (use π ≈ 3.14)

When working with land, you might need to convert between acres and hectares:

  • 1 acre = 0.4047 hectares
  • 1 hectare = 2.471 acres

Example Problem: A rectangular plot of land measures 50 meters by 30 meters. Calculate the area in square meters and convert it to acres.

Solution:

  1. Calculate area in square meters:
    Area = Length × Width = 50 m × 30 m = 1500 m²
  2. Convert to acres:
    Area in acres = 1500 m² × (1 acre / 4047 m²) ≈ 0.37 acres

Thus, the area of the plot is 1500 m² or approximately 0.37 acres.

  • Use appropriate area formulae for different shapes.
  • Convert units accurately between acres and hectares.
  • Show all steps clearly in calculations.

A triangular garden has a base of 12 m and a height of 5 m. Calculate the area.
Area = 1/2 × Base × Height = 1/2 × 12 m × 5 m = 30 m².

Lesson 4: Understanding Perimeter and Area of Plane Figures

Objective: Mensuration: perimeter and area of plane figures

In mensuration, it is crucial to understand the concepts of perimeter and area of plane figures.

  • Perimeter is the total distance around a figure. For example:

    • Rectangle: Perimeter = 2(length + width)
    • Triangle: Perimeter = side1 + side2 + side3

  • Area measures the space enclosed within a figure. For instance:

    • Rectangle: Area = length × width
    • Triangle: Area = 1/2 × base × height

To solve problems, always identify the shape and use the appropriate formulas. Remember to express your final answers in square units for area and linear units for perimeter.

  • Perimeter is the total distance around a plane figure.
  • Area measures the space within a plane figure.
  • Use specific formulas for different shapes.
  • Express area in square units, perimeter in linear units.
  • Always identify the shape before applying formulas.

Calculate the perimeter and area of a rectangle with length 5 cm and width 3 cm.

  • Perimeter = 2(5 + 3) = 16 cm
  • Area = 5 × 3 = 15 cm²

Sample Questions

Read 3 questions and answers free. Sign up to access all 104 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER4 marks

A circular flower bed has a radius of 2 m. Explain how to find: (a) the area of the flower bed in square metres. (b) the circumference of the flower bed in metres. (4 marks)

Answer & marking scheme

Part (a) — 2 marks
Area = π × radius² = π × (2 m)² = 4π m² (1 mk)
Area ≈ 12.57 m² (using π ≈ 3.14) (1 mk)
Part (b) — 2 marks
Circumference = 2π × radius = 2π × 2 m = 4π m (1 mk)
Circumference ≈ 12.57 m (using π ≈ 3.14) (1 mk)
2
easySHORT ANSWER3 marks

A rectangular garden measures 30 m by 40 m. Calculate the area of the garden in square metres and convert this area to hectares. (3 marks)

Answer & marking scheme

Part (a) — 1 mark
Area = length × width = 30 m × 40 m = 1200 m² (1 mk)
Part (b) — 2 marks
Convert area to hectares: 1200 m² ÷ 10000 = 0.12 hectares (2 mks)
3
easySHORT ANSWER3 marks

A sector of a circle has a radius of 10 cm and a central angle of 60°. Calculate the area of the sector. (Take π = 22/7) (3 marks)

Answer & marking scheme

Part (a) — 3 marks
Area formula: A = (angle/360) × πr² (1 mk)
A = (60/360) × 22/7 × 10² (1 mk)
A = (1/6) × 22/7 × 100 = 35.33 cm² (1 mk)
4

A trapezium has parallel sides measuring 10 cm and 6 cm, with a height of 4 cm. State the area of the trapezium. (2 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Mensuration: perimeter and area of plane figures" cover?

Mensuration: perimeter and area of plane figures covers Calculate the perimeter and area of triangles, parallelograms, trapeziums and compound shapes; Calculate the circumference and area of a circle; find arc length and area of a sector using the angle; Apply area formulae to solve problems involving land (in acres and hectares), material and design, and more, all aligned to the official KNEC KCSE Mathematics syllabus.

How many practice questions are available for Mensuration: perimeter and area of plane figures?

HighMarks has 104 Mensuration: perimeter and area of plane figures 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 Mensuration: perimeter and area of plane figures for the KCSE exam?

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