Area | Mathematics Grade 8 CBSE

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A formula for every shape

From flat polygons to the circle, then up to the surface and volume of solids — tap each shape to see the formula that measures it.

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The three big ideas

Parallelogram — area = base × height, where the height is the perpendicular distance between the base and the opposite side (not the slant side).
Triangle — area = ½ × base × height; a triangle is half a parallelogram on the same base.
Trapezium — area = ½ × (a + b) × h, the average of the two parallel sides times the height between them.
Rhombus — area = ½ × d₁ × d₂, half the product of the diagonals.

Worked example. Find the area of a trapezium with parallel sides 8 cm and 12 cm and a height of 5 cm.

1. Write the formula: area = ½ × (a + b) × h.

2. Add the parallel sides: a + b = 8 + 12 = 20.

3. Substitute: area = ½ × 20 × 5 = 10 × 5 = 50 cm².

Common mistake: using the slant side instead of the perpendicular height. Area always uses the height at right angles to the base, never the sloping edge.

Circumference (the distance around) = 2πr = πd, where r is the radius and d = 2r the diameter.
Area (the space inside) = πr².
The constant π ≈ 22/7 ≈ 3.14 — the same ratio of circumference to diameter for every circle.
Watch the powers: circumference grows with r, but area grows with r² — double the radius and the area becomes four times as big.

Worked example. A circle has radius 7 cm. Taking π = 22/7, find its circumference and area.

1. Circumference = 2πr = 2 × (22/7) × 7 = 2 × 22 = 44 cm.

2. Area = πr² = (22/7) × 7 × 7 = 22 × 7 = 154 cm².

Common mistake: mixing up the two formulas. Circumference = 2πr (a length, in cm); area = πr² (a space, in cm²). The squared one is the area.

Surface area is the total area of all the faces of a solid — the "wrapping paper" needed to cover it. Measured in square units (cm²).
Cuboid: surface area = 2(lb + bh + hl); volume = l × b × h.
Cube (edge a): surface area = 6a²; volume = a³.
Cylinder (radius r, height h): curved surface = 2πrh; volume = πr²h.
Volume measures the space inside, in cubic units (cm³). For liquids it becomes capacity: 1 litre = 1000 cm³.

Where you'll meet it

Measuring the world

Flooring & painting

Tiling a room or painting a wall needs the area. A 4 m by 3 m floor is 12 m² of tiles; a wall's paint is sold by how many square metres a tin covers — area decides how much to buy.

Fencing a field

The cost of fencing depends on the perimeter (or circumference for a round plot), while the crop depends on the area. A circular park of radius 14 m needs 88 m of fence — a circumference calculation in rupees.

Tanks & packaging

A cylindrical water tank's capacity is its volume, πr²h, converted to litres. The sheet metal to build it is the surface area. Designers balance both to hold more water with less material.

Check yourself

Competency quiz

Modelled on the competency-based pattern — MCQ, assertion–reason and a case study, testing whether you can use the ideas, not just recall them.

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