Perimeter and Area — Measuring Boundaries and Spaces
"'How much fencing do I need for my garden? How much carpet for this room?' Perimeter and area answer these everyday questions."
1. Perimeter — Distance Around a Shape
Perimeter is the TOTAL LENGTH of the boundary of a closed figure. Measured in LINEAR units (cm, m, km).
Perimeter Formulas
| Shape | Formula | Variables |
|---|---|---|
| Square | P = 4s | s = side length |
| Rectangle | P = 2(l + b) | l = length, b = breadth |
| Triangle | P = a + b + c | a, b, c = side lengths |
| Parallelogram | P = 2(a + b) | a, b = adjacent sides |
| Circle (Circumference) | C = 2πr | r = radius |
Worked Example: 'A rectangular AP farm plot is 80 m long and 50 m wide. Find the cost of fencing at Rs 15 per metre.' Perimeter = 2(80 + 50) = 2(130) = 260 m. Cost = 260 × 15 = Rs 3900.
2. Area — Space Inside a Shape
Area is the measure of the SURFACE enclosed by a shape. Measured in SQUARE units (cm², m², km², hectare).
Area Formulas
| Shape | Formula | Variables |
|---|---|---|
| Square | A = s² | s = side |
| Rectangle | A = l × b | l = length, b = breadth |
| Triangle | A = ½ × b × h | b = base, h = height |
| Parallelogram | A = b × h | b = base, h = height (perpendicular) |
| Circle | A = πr² | r = radius |
3. Squares and Rectangles — In Detail
Converting Units
'Area calculations require UNIFORM units. Always convert before multiplying.' 1 m = 100 cm. 1 m² = 100 × 100 = 10,000 cm². 1 km² = 1000 × 1000 = 1,000,000 m². 1 hectare = 10,000 m².
| Conversion | Value |
|---|---|
| 1 m² | 10,000 cm² |
| 1 km² | 100 hectares |
| 1 hectare | 10,000 m² |
Worked Example
'A rectangular park is 120 m long and 80 m wide. Find its area in hectares.' Area = 120 × 80 = 9600 m². In hectares: 9600/10000 = 0.96 hectares.
4. Triangles — Area In Detail
Formula: Area = ½ × base × height. The height (altitude) must be PERPENDICULAR to the base.
Right Triangle
Base = one leg, Height = other leg. Area = ½ × (product of legs).
Example: A right triangle with legs 6 cm and 8 cm. Area = ½ × 6 × 8 = 24 cm².
Any Triangle
'Choose any side as the base. Drop a perpendicular from the opposite vertex. That is the height.'
Example: Triangle with base = 10 cm and height = 7 cm. Area = ½ × 10 × 7 = 35 cm².
5. Parallelograms — Area In Detail
Formula: Area = base × height. Height = perpendicular distance between the base and the opposite side.
Shape Comparison: 'A parallelogram with the SAME base and height as a rectangle has the SAME area. But the parallelogram's perimeter is calculated differently.'
| Shape | Base | Height | Area | Perimeter |
|---|---|---|---|---|
| Rectangle 6×4 | 6 | 4 | 24 | 20 |
| Parallelogram (same b, h) | 6 | 4 | 24 | 2(6 + slanted side) |
6. Circles — Area and Circumference
Key constant: π (pi) ≈ 22/7 ≈ 3.14.
| Measurement | Formula | Example (r = 7 cm) |
|---|---|---|
| Circumference | C = 2πr | 2 × (22/7) × 7 = 44 cm |
| Diameter | d = 2r | 14 cm |
| Area | A = πr² | (22/7) × 7 × 7 = 154 cm² |
Worked Example: 'A circular AP well has a radius of 3.5 m. Find the cost of tiling its floor at Rs 200 per m².' Area = (22/7) × 3.5 × 3.5 = (22/7) × 12.25 = 38.5 m². Cost = 38.5 × 200 = Rs 7700.
Difference Between Circumference and Area
'Circumference is a LINEAR measure — the distance AROUND the circle. Area is the SPACE INSIDE. Do NOT confuse them!'
7. Area Between Two Rectangles (Paths)
'When a rectangular path runs INSIDE or BETWEEN two rectangles, find the area of the path by SUBTRACTING the inner rectangle from the outer rectangle.'
Type 1: Path Around a Rectangle (Outside)
'A rectangular garden 30 m × 20 m has a 2 m wide path around it OUTSIDE. Find the area of the path.'
Outer dimensions: Length = 30 + 2 + 2 = 34 m. Width = 20 + 2 + 2 = 24 m. Outer area = 34 × 24 = 816 m². Inner area = 30 × 20 = 600 m². Path area = 816 − 600 = 216 m².
Type 2: Path Inside a Rectangle
'A park 50 m × 40 m has a 3 m wide path INSIDE along its boundary. Find the area of the path.'
Inner dimensions: Length = 50 − 3 − 3 = 44 m. Width = 40 − 3 − 3 = 34 m. Outer area = 50 × 40 = 2000 m². Inner area = 44 × 34 = 1496 m². Path area = 2000 − 1496 = 504 m².
Type 3: Cross Paths — Two Paths Intersecting
'A rectangular field 80 m × 60 m has two paths: one 3 m wide parallel to length, another 2 m wide parallel to width, intersecting in the middle. Find total path area.'
Area of lengthwise path = 80 × 3 = 240 m². Area of widthwise path = 60 × 2 = 120 m². Intersection (counted twice) = 3 × 2 = 6 m². Total path area = 240 + 120 − 6 = 354 m².
'This is the KEY formula for cross paths — add the individual path areas, then SUBTRACT the intersection once.'
8. Common Mistakes and Fixes
| Mistake | Why It Is Wrong | Correct Approach |
|---|---|---|
| Area of triangle = base × height | Forgot the ½ factor | Area = ½ × base × height |
| Using slanted side as height for parallelogram/ triangle | Height must be PERPENDICULAR | Height = perpendicular distance |
| π = 22/7 always | 22/7 is an APPROXIMATION — use 3.14 for non-multiples of 7 | Use 22/7 when radius is multiple of 7; else use 3.14 |
| Perimeter and circumference confused | Circle uses circumference, not perimeter | Perimeter for polygons, circumference for circles |
| In path problems, adding width only once | Width is added on BOTH sides | Add twice the width for outside paths |
9. AP SSC Exam Focus
| Topic | Marks | Question Type |
|---|---|---|
| Perimeter of rectangles and squares | 2-3 | Direct computation |
| Area of triangles and parallelograms | 2-3 | Computation |
| Circle area and circumference | 3-4 | Word problems |
| Path problems (area between rectangles) | 3-4 | Application |
| Unit conversion (m², hectare, km²) | 1-2 | Conversion problems |
Quick Self-Test
Q1. Find the perimeter of a rectangle with length 12 cm and breadth 8 cm. A1. P = 2(12 + 8) = 2(20) = 40 cm.
Q2. Find the area of a triangle with base 14 cm and height 10 cm. A2. A = ½ × 14 × 10 = 70 cm².
Q3. A circle has radius 21 cm. Find its circumference and area. A3. C = 2 × (22/7) × 21 = 132 cm. A = (22/7) × 21 × 21 = 1386 cm².
Q4. A rectangular field 40 m × 30 m has a 2.5 m wide path outside. Find the area of the path. A4. Outer = (40+5)(30+5) = 45×35 = 1575 m². Inner = 40×30 = 1200 m². Path = 375 m².
Q5. Convert 5 hectares to m². A5. 5 hectares = 5 × 10000 = 50,000 m².
Q6. A parallelogram has base 12 cm and area 84 cm². Find its height. A6. 84 = 12 × h → h = 7 cm.
Q7. Which has more area: a square of side 10 m or a rectangle 12 m × 8 m? A7. Square area = 100 m². Rectangle area = 96 m². The square has more area.
