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Mensuration: surface area of solids — KCSE Mathematics

KCSE Mathematics · 95 practice questions · 3 syllabus objectives · 3 revision lessons

33 easy35 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.

Calculate the total surface area of a cuboid, cylinder, cone and sphere

Calculate the surface area of composite solids formed by combining two or more basic shapes

Mensuration: surface area of solids

Revision Notes

Concise lesson notes for Mensuration: surface area of solids, written to the KCSE Mathematics marking standard. Read the first lesson free below.

Calculating Total Surface Area of Solids

To calculate the total surface area of various solids, use the following formulas:

  • Cuboid: Total Surface Area (TSA) = 2(lw + lh + wh)
  • Cylinder: TSA = 2πr(h + r)
  • Cone: TSA = πr(r + l), where l is the slant height
  • Sphere: TSA = 4πr²

Example Calculations:

  1. Cuboid: For a cuboid with length = 5 cm, width = 3 cm, and height = 4 cm:

    • TSA = 2(53 + 54 + 3*4) = 2(15 + 20 + 12) = 2(47) = 94 cm²

  2. Cylinder: For a cylinder with radius = 3 cm and height = 7 cm:

    • TSA = 2π(3)(7 + 3) = 2π(3)(10) = 60π cm² ≈ 188.4 cm²

Remember to use the correct units and round off your answers appropriately. This ensures clarity and precision in your calculations.

Key points to remember

  • Cuboid TSA formula involves length, width, and height.
  • Cylinder TSA includes base area and lateral surface area.
  • Cone TSA requires slant height for accurate calculation.
  • Sphere TSA is based solely on the radius.
  • Always express final answers with correct units.

Worked example

Calculate the TSA of a cone with radius 4 cm and slant height 5 cm.

  • TSA = π(4)(4 + 5) = π(4)(9) = 36π cm² ≈ 113.1 cm².

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Lesson 2: Calculating Surface Area of Composite Solids

Objective: Calculate the surface area of composite solids formed by combining two or more basic shapes

To calculate the surface area of composite solids, we first identify the basic shapes that make up the solid. Then, we calculate the surface area of each basic shape and sum them up, adjusting for any overlapping areas. Steps to follow:

  1. Identify basic shapes: Break down the composite solid into recognizable shapes (e.g., cylinders, cones, cubes).
  2. Calculate individual areas: Use the formulas for surface area of each basic shape:
    • Cylinder: SA = 2πr(h + r)
    • Cone: SA = πr(l + r) where l is the slant height
    • Cube: SA = 6a²
  3. Sum the areas: Add the surface areas calculated, making sure to subtract any areas that are not exposed.

Example: Calculate the surface area of a cylinder with radius 3 cm and height 5 cm combined with a hemisphere of radius 3 cm.

  • Cylinder SA = 2π(3)(5 + 3) = 2π(3)(8) = 48π cm²
  • Hemisphere SA = 2π(3)² = 18π cm²
  • Total SA = 48π + 18π = 66π cm².

Thus, the total surface area of the composite solid is 66π cm².

  • Identify basic shapes in the composite solid.
  • Calculate surface area for each basic shape.
  • Sum the surface areas, adjusting for overlaps.

Calculate the surface area of a cube with side length 4 cm and a cylinder with radius 2 cm and height 6 cm.

  • Cube SA = 6(4²) = 96 cm².
  • Cylinder SA = 2π(2)(6 + 2) = 32π cm².
  • Total SA = 96 + 32π cm².
Lesson 3: Understanding Surface Area of Solids

Objective: Mensuration: surface area of solids

In mensuration, the surface area of solids is crucial for various applications. To calculate the surface area, one must understand the formulas for different shapes:

  • Cuboid: Surface Area = 2(lw + lh + wh)
  • Cube: Surface Area = 6a² (where a is the side length)
  • Cylinder: Surface Area = 2πr(h + r) (where r is the radius and h is the height)
  • Sphere: Surface Area = 4πr² (where r is the radius)

To solve problems, identify the solid shape, use the correct formula, and substitute the values correctly. Always include units in your final answer.

For example, to find the surface area of a cuboid with length 5 cm, width 3 cm, and height 4 cm:

  1. Calculate: 2(53 + 54 + 3*4) = 2(15 + 20 + 12) = 2(47) = 94 cm².

Thus, the surface area of the cuboid is 94 cm².

  • Identify the solid shape before calculation.
  • Use correct formulas for each type of solid.
  • Substitute values accurately into the formulas.
  • Include units in your final answers.
  • Check calculations for accuracy.

Calculate the surface area of a cylinder with radius 3 cm and height 5 cm. Surface Area = 2πr(h + r) = 2π(3)(5 + 3) = 2π(3)(8) = 48π cm².

Sample Questions

Read 3 questions and answers free. Sign up to access all 95 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER4 marks

A cone has a base radius of 4 cm and a height of 6 cm. Determine its total surface area. (Use π = 22/7) (4 marks)

Answer & marking scheme

Part (a) — 4 marks
Calculate the slant height using Pythagoras' theorem: l = √(r² + h²) (1 mk)
Curved surface area = πrl = 22/7 × 4 × l (1 mk)
Area of circular base = πr² = 22/7 × 4² (1 mk)
Total surface area = Curved surface area + Area of circular base (1 mk)
2
easySHORT ANSWER3 marks

A cylindrical storage tank has a radius of 3 m and a height of 5 m. Calculate its total surface area. (Take π = 3.142) (3 marks)

Answer & marking scheme

Part (a) — 3 marks
Curved surface area = 2πrh = 2 × 3.142 × 3 × 5 (1 mk)
Area of circular top and bottom = 2 × πr² = 2 × 3.142 × 3² (1 mk)
Total surface area = Curved surface area + Area of circular top and bottom (1 mk)
3
easySHORT ANSWER4 marks

A cylinder has a radius of 4 cm and a height of 10 cm. Calculate its total surface area. (4 marks)

Answer & marking scheme

Part (a) — 4 marks
Total surface area = 2πr(h + r) (1 mk)
Substituting values: = 2 * 22/7 * 4 * (10 + 4) (1 mk)
Calculate the height and radius: = 2 * 22/7 * 4 * 14 (1 mk)
Final calculation = 2 * 22/7 * 56 = 704/7 ≈ 100.57 cm² (1 mk)
4

Calculate the total surface area of a cuboid with a length of 10 cm, a width of 5 cm, and a height of 3 cm. (3 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Mensuration: surface area of solids" cover?

Mensuration: surface area of solids covers Calculate the total surface area of a cuboid, cylinder, cone and sphere; Calculate the surface area of composite solids formed by combining two or more basic shapes; Mensuration: surface area of solids, all aligned to the official KNEC KCSE Mathematics syllabus.

How many practice questions are available for Mensuration: surface area of solids?

HighMarks has 95 Mensuration: surface area of solids 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: surface area of solids for the KCSE exam?

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