What this chapter is about
This chapter builds on what you already know about finding areas of simple shapes like rectangles and squares. In Class 8, you move forward to learn how to calculate the area of more complex figures — quadrilaterals such as parallelograms, rhombuses, trapeziums and also triangles. You will see how these formulas connect to each other and to the basic rectangle formula you learned earlier.
Understanding area is useful in daily life. When your family wants to tile a floor, paint a wall, buy cloth for a tablecloth or calculate how much land a farmer owns, you need to find area. These shapes appear everywhere — in farms, plots of land, designs and art.
After studying this chapter, you should be able to find the area of any parallelogram, triangle, rhombus or trapezium when given the right measurements. You will also understand why each formula works, not just memorise it.
Key ideas
- The area of a rectangle is length × breadth. This simple idea is the foundation for all other area formulas in this chapter.
- A parallelogram can be thought of as a slanted rectangle. Its area equals base × height, where the height is the perpendicular distance between the two parallel sides, not the slant side.
- A triangle is exactly half of a parallelogram with the same base and height. So area of a triangle = (1/2) × base × height.
- A rhombus is a special parallelogram with all sides equal. Its area can be found using its two diagonals: area = (1/2) × d₁ × d₂, where d₁ and d₂ are the lengths of the diagonals.
- A trapezium has one pair of parallel sides. Its area = (1/2) × (sum of parallel sides) × height.
- The height in any of these formulas is always measured perpendicular (at right angles) to the base, never along a slant.
- You can find the area of irregular shapes by dividing them into triangles, rectangles or other known figures, then adding the areas.
Formulas and facts to remember
- Area of rectangle = length × breadth
- Area of square = side × side = side²
- Area of parallelogram = base × height (height is perpendicular to the base)
- Area of triangle = (1/2) × base × height
- Area of rhombus = (1/2) × d₁ × d₂ (d₁ and d₂ are the two diagonals)
- Area of trapezium = (1/2) × (a + b) × h (a and b are the parallel sides, h is the height between them)
- All area measurements are in square units (cm², m², etc.)
Worked examples
Example 1: Area of a parallelogram
A parallelogram-shaped plot of land has a base of 24 metres and a perpendicular height of 15 metres. Find its area.
Solution: Area of parallelogram = base × height Area = 24 × 15 = 360 m²
The area of the plot is 360 square metres.
Example 2: Area of a triangle
Priya is making a triangular banner for her school sports day. The base of the banner is 80 cm and the height from the base to the top point is 50 cm. How much cloth does she need?
Solution: Area of triangle = (1/2) × base × height Area = (1/2) × 80 × 50 Area = (1/2) × 4000 = 2000 cm²
Priya needs 2000 square centimetres of cloth.
Example 3: Area of a trapezium
A farmer owns a field shaped like a trapezium. The two parallel sides measure 40 m and 60 m, and the perpendicular distance between them is 25 m. Find the area of the field.
Solution: Area of trapezium = (1/2) × (sum of parallel sides) × height Area = (1/2) × (40 + 60) × 25 Area = (1/2) × 100 × 25 Area = (1/2) × 2500 = 1250 m²
The field has an area of 1250 square metres.
Common mistakes
- Using the slant side instead of the perpendicular height in parallelogram or triangle formulas → always measure height at right angles to the base.
- Forgetting to multiply by (1/2) when finding the area of a triangle, rhombus or trapezium → check which formulas have the half in them.
- Mixing up the diagonals of a rhombus with its sides → the diagonal formula uses the two lines that cross inside, not the four equal edges.
- Adding all four sides of a trapezium instead of only the two parallel sides → only the parallel pair goes into the formula.
- Writing the answer without square units → area is always in cm², m² or other squared units.
Quick revision
- Parallelogram area = base × perpendicular height.
- Triangle area = half of base × height.
- Rhombus area = half the product of its two diagonals.
- Trapezium area = half the sum of parallel sides × height.
- Height is always perpendicular to the base, never slanted.
- Always state area in square units.