Mensuration — Class 8 IMO

Area of Trapezium & Rhombus

The area of a trapezium is ½ × (sum of the parallel sides) × height. The area of a rhombus is ½ × d₁ × d₂, where d₁ and d₂ are its diagonals.

Example 1: A trapezium has parallel sides 6 cm and 10 cm and height 4 cm. Find its area.

½ × (6 + 10) × 4 = ½ × 16 × 4 = 32 sq cm.

Example 2: A rhombus has diagonals 6 cm and 8 cm. Find its area.

½ × 6 × 8 = 24 sq cm.

Quick recap

Trapezium area = ½ × (sum of parallel sides) × height.
Rhombus area = ½ × d₁ × d₂.

✓ Quick check

A trapezium has parallel sides 5 cm and 7 cm and height 4 cm. Its area is ___ ?

½ × (5 + 7) × 4 = ½ × 12 × 4 = 24 sq cm.

A rhombus has diagonals 10 cm and 12 cm. Its area is ___ ?

½ × 10 × 12 = 60 sq cm.

Surface Area of Solids

What is a Cube? A cube is a three-dimensional solid shape with six square faces, all of equal size. All edges of a cube are equal in length.

What is a Cuboid? A cuboid is a three-dimensional solid shape with six rectangular faces. Opposite faces are identical. A cuboid has length (\(l\)), breadth (\(b\)), and height (\(h\)).

Nets of Solids: A net is a two-dimensional pattern that can be folded to form a three-dimensional solid. Nets help us understand how the faces of a solid are arranged.

Properties of Cubes and Cuboids:

Property	Cube	Cuboid
Faces	6 square faces	6 rectangular faces
Edges	12 equal edges	12 edges (4 length, 4 breadth, 4 height)
Vertices	8	8
Edge length	All edges = \(a\)	Length (\(l\)), Breadth (\(b\)), Height (\(h\))
Surface Area	\(6a^2\)	\(2(lb + bh + hl)\)
Volume	\(a^3\)	\(l \times b \times h\)

Formulas Summary:

Measurement	Cube (side = a)	Cuboid (l, b, h)
Lateral Surface Area (LSA)	\(4a^2\)	\(2h(l + b)\)
Total Surface Area (TSA)	\(6a^2\)	\(2(lb + bh + hl)\)
Volume	\(a^3\)	\(l \times b \times h\)
Diagonal	\(\sqrt{3}a\)	\(\sqrt{l^2 + b^2 + h^2}\)

Example 1: Find the surface area of a cube of side 3 cm.

6 × 3² = 6 × 9 = 54 sq cm.

Example 2: Find the surface area of a cuboid 2 × 3 × 4.

2(6 + 12 + 8) = 2 × 26 = 52 sq cm.

Example 3: Find the total surface area and volume of a cube with side 5 cm.

- TSA = \(6a^2 = 6 \times 5^2 = 6 \times 25 = 150\) cm² - Volume = \(a^3 = 5^3 = 125\) cm³ - **Answer:** TSA = 150 cm², Volume = 125 cm³ *Example 2: A cuboid has dimensions: length = 8 cm, breadth = 6 cm, height = 4 cm. Find its total surface area and volume. Solution: - TSA = \(2(lb + bh + hl) = 2(8×6 + 6×4 + 4×8)\) - \(= 2(48 + 24 + 32) = 2 \times 104 = 208\) cm² - Volume = \(l \times b \times h = 8 \times 6 \times 4 = 192\) cm³ - **Answer:** TSA = 208 cm², Volume = 192 cm³ *Example 3: A cuboid has a volume of 240 cm³. Its length is 10 cm and breadth is 6 cm. Find its height and total surface area. Solution: - Volume = \(l \times b \times h = 10 \times 6 \times h = 60h = 240\) - \(h = \frac{240}{60} = 4\) cm - TSA = \(2(lb + bh + hl) = 2(10×6 + 6×4 + 4×10)\) - \(= 2(60 + 24 + 40) = 2 \times 124 = 248\) cm² - **Answer:** Height = 4 cm, TSA = 248 cm²

Quick recap

Cube surface area = 6a².
Cuboid surface area = 2(lb + bh + hl).
Cube: all edges equal; 6 square faces; Volume = \(a^3\); TSA = \(6a^2\)
Cuboid: length, breadth, height; Volume = \(l \times b \times h\); TSA = \(2(lb + bh + hl)\)
Lateral surface area excludes top and bottom: Cube = \(4a^2\); Cuboid = \(2h(l+b)\)
A net is a 2D pattern that folds into a 3D shape
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✓ Quick check

What is the surface area of a cube of side 5 cm?

6 × 25 = 150 sq cm.

What is the surface area of a cuboid 5 × 4 × 3?

2(20 + 12 + 15) = 2 × 47 = 94 sq cm.

Volume & Capacity

What is a Cylinder? A cylinder is a three-dimensional solid with two parallel circular bases connected by a curved surface. Examples: a soda can, a pipe, a candle.

Parts of a Cylinder:

Radius (\(r\)): distance from center to edge of circular base
Height (\(h\)): perpendicular distance between the two bases

Formulas for a Cylinder:

Measurement	Formula
Curved Surface Area (CSA)	\(2\pi r h\)
Total Surface Area (TSA)	\(2\pi r h + 2\pi r^2 = 2\pi r (r + h)\)
Volume	\(\pi r^2 h\)

Understanding the Formulas:

Curved surface area = circumference of base × height = \((2\pi r) \times h\)
Volume = area of base × height = \((\pi r^2) \times h\)

Comparison of Solids:

Solid	TSA Formula	Volume Formula
Cube	\(6a^2\)	\(a^3\)
Cuboid	\(2(lb + bh + hl)\)	\(l \times b \times h\)
Cylinder	\(2\pi r(r + h)\)	\(\pi r^2 h\)

Example 1: Find the volume of a cylinder with radius 7 cm and height 10 cm (π = 22/7).

(22/7) × 7² × 10 = 22 × 7 × 10 = 1540 cm³.

Example 2: How many litres is 1000 cm³?

1 L.

Example 3: Find the curved surface area and total surface area of a cylinder with radius 7 cm and height 10 cm. (Use \(\pi = \frac{22}{7}\))

- CSA = \(2\pi r h = 2 \times \frac{22}{7} \times 7 \times 10 = 2 \times 22 \times 10 = 440\) cm² - TSA = \(2\pi r (r + h) = 2 \times \frac{22}{7} \times 7 \times (7 + 10)\) - \(= 2 \times 22 \times 17 = 748\) cm² - **Answer:** CSA = 440 cm², TSA = 748 cm² *Example 2: Find the volume of a cylinder with radius 3.5 cm and height 12 cm. (Use \(\pi = \frac{22}{7}\)) Solution: - Volume = \(\pi r^2 h = \frac{22}{7} \times (3.5)^2 \times 12\) - \((3.5)^2 = 12.25 = \frac{49}{4}\) - Volume = \(\frac{22}{7} \times \frac{49}{4} \times 12 = 22 \times 7 \times 3 = 462\) cm³ - **Answer:** 462 cm³ *Example 3: A cylindrical water tank has a height of 10 m and a radius of 4 m. Find the cost of painting its curved surface at ₹50 per square meter. (Use \(\pi = 3.14\)) Solution: - CSA = \(2\pi r h = 2 \times 3.14 \times 4 \times 10 = 2 \times 3.14 \times 40 = 251.2\) m² - Cost = \(251.2 \times 50 = ₹12,560\) - **Answer:** ₹12,560

Quick recap

Cube a³; cuboid l×b×h; cylinder πr²h.
1 cm³ = 1 mL; 1000 cm³ = 1 L.
Cylinder has two circular bases and one curved surface
Curved Surface Area (CSA) = \(2\pi r h\)
Total Surface Area (TSA) = \(2\pi r (r + h)\)
Volume = \(\pi r^2 h\)
CSA is the area of the curved surface only; TSA includes both circular ends
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✓ Quick check

What is the volume of a cube of side 4 cm?

4³ = 64 cm³.

2000 cm³ equals how many litres?

2000 ÷ 1000 = 2 L.