Class 8 — Mathematics (NCERT)

Mensuration: Area, Surface Area and Volume

Class 8

✓By the end of this lesson students will be able to calculate the area of trapeziums and general polygons.
✓By the end of this lesson students will be able to determine the total surface area and volume of cuboids and cubes.
✓By the end of this lesson students will be able to find the curved surface area, total surface area, and volume of cylinders.
✓By the end of this lesson students will be able to apply mensuration formulas to solve real-life problems involving these shapes.
✓By the end of this lesson students will be able to understand and use appropriate units for area and volume.

Key concepts

Area of a Trapezium

A trapezium is a quadrilateral with one pair of parallel sides. The area of a trapezium is found by taking half the sum of the lengths of its parallel sides multiplied by the perpendicular distance (height) between them.

Area = 1/2 × (sum of parallel sides) × height = 1/2 × (a + b) × h

Area of a General Polygon

To find the area of a general polygon, we divide it into simpler shapes whose areas we already know how to calculate, such as triangles and trapeziums. The total area of the polygon is the sum of the areas of these constituent shapes.

Surface Area of a Cuboid

A cuboid is a three-dimensional shape with six rectangular faces. The total surface area of a cuboid is the sum of the areas of all its six faces. If 'l' is length, 'b' is breadth, and 'h' is height, then there are three pairs of identical faces.

Total Surface Area (TSA) = 2(lb + bh + hl)

Volume of a Cuboid

The volume of a cuboid is the amount of space it occupies. It is calculated by multiplying its length, breadth, and height.

Volume (V) = l × b × h

Surface Area of a Cube

A cube is a special type of cuboid where all its edges are of equal length, meaning all its six faces are identical squares. If 'a' is the length of an edge, then the area of one face is a². The total surface area is the sum of the areas of all six square faces.

Total Surface Area (TSA) = 6a²

Volume of a Cube

The volume of a cube is the amount of space it occupies. It is calculated by cubing the length of its edge.

Volume (V) = a³

Surface Area of a Cylinder

A cylinder has two circular bases and a curved surface. The curved surface area (C.S.A.) is the area of the curved part. The total surface area (T.S.A.) is the sum of the curved surface area and the areas of the two circular bases. If 'r' is the radius of the base and 'h' is the height.

Curved Surface Area (C.S.A.) = 2πrh\nTotal Surface Area (T.S.A.) = 2πr(r + h)

Volume of a Cylinder

The volume of a cylinder is the amount of space it occupies. It is calculated by multiplying the area of its base (a circle) by its height.

Volume (V) = πr²h

Key facts to remember

1Area is the measure of the surface enclosed by a closed figure, expressed in square units (e.g., cm², m²).
2Volume is the measure of the space occupied by a three-dimensional object, expressed in cubic units (e.g., cm³, m³).
31 m³ = 1000 litres.
41 litre = 1000 cm³.
5The value of π (pi) is approximately 22/7 or 3.14.
6To find the area of an irregular polygon, divide it into simpler known shapes like triangles and trapeziums.
7The total surface area of a solid is the sum of the areas of all its faces.

Worked examples

Example 1

The parallel sides of a trapezium are 15 cm and 9 cm, and the perpendicular distance between them is 7 cm. Find its area.

IGiven: Length of parallel side a = 15 cm

IILength of parallel side b = 9 cm

IIIHeight h = 7 cm

IVFormula for the area of a trapezium: Area = 1/2 × (a + b) × h

VSubstitute the given values into the formula:

VIArea = 1/2 × (15 + 9) × 7

VIIArea = 1/2 × (24) × 7

VIIIArea = 12 × 7

9Area = 84 cm²

Answer

The area of the trapezium is 84 cm².

Example 2

A cuboidal box is 75 cm long, 50 cm wide, and 40 cm high. Find its total surface area and volume.

IGiven: Length (l) = 75 cm

IIBreadth (b) = 50 cm

IIIHeight (h) = 40 cm

IVTo find Total Surface Area (TSA):

VFormula: TSA = 2(lb + bh + hl)

VITSA = 2((75 × 50) + (50 × 40) + (40 × 75))

VIITSA = 2(3750 + 2000 + 3000)

VIIITSA = 2(8750)

9TSA = 17500 cm²

10To find Volume (V):

11Formula: V = l × b × h

12V = 75 × 50 × 40

13V = 3750 × 40

14V = 150000 cm³

Answer

The total surface area of the cuboidal box is 17500 cm² and its volume is 150000 cm³.

Example 3

A cylindrical pillar has a radius of 1.4 m and a height of 5 m. Find its volume and curved surface area. (Use π = 22/7)

IGiven: Radius (r) = 1.4 m

IIHeight (h) = 5 m

IIIValue of π = 22/7

IVTo find Volume (V):

VFormula: V = πr²h

VIV = (22/7) × (1.4)² × 5

VIIV = (22/7) × (1.4 × 1.4) × 5

VIIIV = (22/7) × 1.96 × 5

9V = 22 × 0.28 × 5

10V = 22 × 1.4

11V = 30.8 m³

12To find Curved Surface Area (C.S.A.):

13Formula: C.S.A. = 2πrh

14C.S.A. = 2 × (22/7) × 1.4 × 5

15C.S.A. = 2 × 22 × 0.2 × 5

16C.S.A. = 44 × 1

17C.S.A. = 44 m²

Answer

The volume of the cylindrical pillar is 30.8 m³ and its curved surface area is 44 m².

Remember to use the correct units for volume (cubic units) and area (square units).

Common mistakes

✗Confusing the formulas for surface area and volume, or using them interchangeably.
✗Using diameter instead of radius (or vice-versa) in formulas for cylindrical shapes.
✗Incorrectly applying units (e.g., using cm for area instead of cm² or cm³ for volume).
✗Making calculation errors, especially when dealing with fractions or the approximate value of π.
✗Forgetting to include all faces when calculating the total surface area of a cuboid or cube.

Exam tips

★Always draw a neat diagram of the given shape, labelling the dimensions clearly.
★Write down the correct formula before substituting the values to ensure you use the right one.
★Pay close attention to the units of measurement and ensure consistency throughout the problem. Convert units if necessary.
★Show all steps of your calculation clearly and systematically, as this helps in getting partial marks even if the final answer is incorrect.
★Double-check your calculations and the final answer for accuracy, especially when dealing with decimal numbers or fractions.

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