What this chapter is about
This chapter develops your understanding of how to measure the boundary and the interior of two-dimensional shapes. Perimeter tells you the total length of the boundary, while area tells you the amount of flat surface a shape covers. You have met these ideas in earlier classes for simple figures; now you work with a wider variety of shapes, including composite figures formed by combining or removing standard shapes.
At the Class 9 level, you learn to calculate perimeter and area of triangles, quadrilaterals (including parallelograms, rhombuses, rectangles, squares and trapeziums), and circles. You also meet Heron's formula, which lets you find the area of any triangle when you know all three sides but not the height. Being confident with these calculations matters because they appear in everyday problems—flooring a room, fencing a plot, painting walls—and they lay the groundwork for surface area and volume work later in the year.
After studying this chapter, you should be able to choose the right formula for a given shape, apply Heron's formula correctly, and break a complicated figure into simpler parts to find its total perimeter or area.
Key ideas
- Perimeter is the sum of all side lengths of a closed plane figure; its unit is a length unit (metre, centimetre, etc.).
- Area is the measure of the region enclosed; its unit is a square of a length unit (m², cm², etc.).
- For a triangle with base b and corresponding height h, area = (1/2) × b × h.
- Heron's formula: if a triangle has sides a, b, c and semi-perimeter s = (a + b + c)/2, then area = √[s(s − a)(s − b)(s − c)].
- For a parallelogram, area = base × height; for a rectangle, area = length × breadth; for a square, area = side².
- For a rhombus with diagonals d₁ and d₂, area = (1/2) × d₁ × d₂.
- For a trapezium with parallel sides a and b and perpendicular distance h between them, area = (1/2) × (a + b) × h.
- For a circle with radius r, circumference = 2πr and area = πr².
Formulas and facts to remember
- Shape: Triangle (general) · Perimeter: a + b + c · Area: (1/2) × base × height, or use Heron's formula
- Shape: Rectangle · Perimeter: 2(l + b) · Area: l × b
- Shape: Square · Perimeter: 4 × side · Area: side²
- Shape: Parallelogram · Perimeter: 2(sum of adjacent sides) · Area: base × height
- Shape: Rhombus · Perimeter: 4 × side · Area: (1/2) × d₁ × d₂
- Shape: Trapezium · Perimeter: sum of all four sides · Area: (1/2) × (sum of parallel sides) × height
- Shape: Circle · Perimeter: 2πr (circumference) · Area: πr²
Semi-perimeter s = (a + b + c)/2 is the first step in Heron's formula.
Worked examples
### Example 1: Finding area using Heron's formula
A triangular garden has sides 13 m, 14 m and 15 m. Find its area.
Step 1: Calculate the semi-perimeter. s = (13 + 14 + 15)/2 = 42/2 = 21 m
Step 2: Apply Heron's formula. Area = √[s(s − a)(s − b)(s − c)] = √[21 × (21 − 13) × (21 − 14) × (21 − 15)] = √[21 × 8 × 7 × 6] = √7056 = 84 m²
The area of the garden is 84 m².
### Example 2: Area of a trapezium-shaped field
A plot of land is shaped like a trapezium. The two parallel sides measure 18 m and 12 m, and the perpendicular distance between them is 8 m. Find the area.
Area = (1/2) × (sum of parallel sides) × height = (1/2) × (18 + 12) × 8 = (1/2) × 30 × 8 = 120 m²
The plot covers 120 m².
### Example 3: Cost of tiling a room with a circular fountain
A rectangular hall is 10 m by 8 m. In the centre is a circular fountain of radius 1.4 m that cannot be tiled. Tiles cost ₹45 per m². Find the tiling cost. (Use π = 22/7.)
Step 1: Area of the rectangle = 10 × 8 = 80 m²
Step 2: Area of the circular fountain = πr² = (22/7) × (1.4)² = (22/7) × 1.96 = 6.16 m²
Step 3: Area to be tiled = 80 − 6.16 = 73.84 m²
Step 4: Cost = 73.84 × 45 = ₹3322.80
The tiling cost is ₹3322.80.
Common mistakes
- Forgetting to halve the perimeter when starting Heron's formula → always compute s = (a + b + c)/2 first.
- Using slant side instead of perpendicular height in parallelogram or trapezium area → the height must be perpendicular to the base.
- Mixing up diameter and radius in circle formulas → radius is half the diameter; double-check which is given.
- Leaving area in the wrong unit after converting lengths → if lengths are in cm, area is in cm², not m².
- Adding areas when a region is removed (like the fountain example) → subtract the removed part from the larger area.
Quick revision
- Perimeter = boundary length; Area = surface covered.
- Heron's formula needs only the three sides and gives area without needing height.
- Trapezium area = (1/2) × (sum of parallel sides) × height.
- Circle: circumference = 2πr, area = πr².
- For composite figures, add or subtract areas of simpler shapes.
- Always check that all measurements are in the same unit before calculating.