In this guide
Draw a 3 × 3 square on squared paper and ask a Class 4 child for its perimeter. Quite often the answer is 8: the child has counted the squares along the border instead of the unit lengths around it. Ask for the area of a 6 × 2 rectangle and someone will add the sides. Mensuration questions in CTET test the formulas, but the pedagogy questions test whether you understand why children mix these ideas up.
Paper I touches perimeter and area through counting squares, and measures capacity. Paper II covers perimeter and area of plane figures, circles, and the surface area and volume of cubes, cuboids and cylinders at the Class 6–8 level. This guide covers both.
Three different questions
- Perimeter: how far is it around the edge? A length, in cm or m.
- Area: how much surface does it cover? Counted in unit squares, in cm² or m².
- Volume: how much space does it fill? Counted in unit cubes, in cm³ or m³. Capacity is how much a container holds, in mL or L.
If a child can say which question is being asked, most formula errors disappear.
Plane figures
| Figure | Perimeter | Area |
|---|---|---|
| Rectangle (length l, breadth b) | 2(l + b) | l × b |
| Square (side a) | 4a | a² |
| Triangle (base b, height h) | Sum of the three sides | ½ × b × h |
| Parallelogram (base b, height h) | Sum of the four sides | b × h |
| Trapezium (parallel sides a and b, height h) | Sum of the four sides | ½ × (a + b) × h |
| Rhombus (diagonals d₁ and d₂) | 4 × side | ½ × d₁ × d₂ |
| Circle (radius r) | Circumference 2πr | πr² |
The height is always perpendicular to the base. Using the slanting side instead is one of the most common errors in triangle and parallelogram questions. Take π = 22/7 unless told otherwise.
Solids
| Solid | Volume | Surface area |
|---|---|---|
| Cube (side a) | a³ | Total 6a²; lateral 4a² |
| Cuboid (l, b, h) | l × b × h | Total 2(lb + bh + hl); lateral 2h(l + b) |
| Cylinder (radius r, height h) | πr²h | Curved 2πrh; total 2πr(r + h) |
Units and conversions
| Converting | Rule |
|---|---|
| Area | 1 cm² = 100 mm²; 1 m² = 10,000 cm² |
| Land | 1 hectare = 10,000 m²; 1 km² = 100 hectares |
| Volume and capacity | 1 cm³ = 1 mL; 1,000 cm³ = 1 L; 1 m³ = 1,000 L |
A length conversion factor gets squared for area and cubed for volume: 1 m = 100 cm, so 1 m² = 100 × 100 cm².
Worked examples
1. Rectangle. A rectangle is 8 cm by 5 cm. Perimeter = 2(8 + 5) = 26 cm. Area = 8 × 5 = 40 cm².
2. From perimeter to area. A square has perimeter 36 cm. Side = 36 ÷ 4 = 9 cm, so area = 81 cm².
3. A path round a garden. A garden is 20 m by 15 m, with a path 2 m wide all round it on the outside. The outer rectangle is (20 + 4) by (15 + 4) = 24 m by 19 m, with area 456 m². The garden is 300 m². Path = 456 − 300 = 156 m². The usual slip is adding 2 m instead of 4 m to each side.
4. Circle. Radius 7 cm. Circumference = 2 × 22/7 × 7 = 44 cm. Area = 22/7 × 7 × 7 = 154 cm².
5. A wheel. A wheel has diameter 70 cm, so radius 35 cm. One turn covers 2 × 22/7 × 35 = 220 cm. In 500 turns it covers 1,10,000 cm = 1.1 km.
6. Cuboid. A box is 5 cm × 4 cm × 3 cm. Volume = 60 cm³. Total surface area = 2(20 + 12 + 15) = 2 × 47 = 94 cm².
7. Water tank. A tank is 2 m × 1.5 m × 1 m. Volume = 3 m³ = 3,000 litres.
8. Cylinder. Radius 7 cm, height 10 cm. Volume = 22/7 × 49 × 10 = 1,540 cm³. Curved surface = 2 × 22/7 × 7 × 10 = 440 cm².
9. Scaling. Double the side of a square and the perimeter doubles, but the area becomes 4 times. Double the side of a cube and the volume becomes 8 times.
10. Cost. A floor 5 m × 4 m is tiled at ₹50 per m²: 20 × 50 = ₹1,000. Fencing a field 20 m × 15 m at ₹10 per metre: perimeter 70 m, so ₹700. Tiling uses area; fencing uses perimeter.
The area-perimeter confusion
Children often believe that shapes with the same perimeter have the same area. Give them a 16 cm loop of string and ask them to make rectangles:
| Rectangle | Perimeter | Area |
|---|---|---|
| 7 × 1 | 16 cm | 7 cm² |
| 6 × 2 | 16 cm | 12 cm² |
| 5 × 3 | 16 cm | 15 cm² |
| 4 × 4 | 16 cm | 16 cm² |
The perimeter stays fixed while the area changes, and the square gives the largest area. The reverse works too: rectangles of area 12 cm² (12 × 1, 6 × 2, 4 × 3) have perimeters of 26, 16 and 14 cm.
Children's misconceptions
| Misconception | Example | Teaching response |
|---|---|---|
| Same perimeter means same area | 6 × 2 and 4 × 4 "must be equal" | The string activity above |
| Counting squares for perimeter | A 3 × 3 square has "perimeter 8" | Trace the boundary with a finger and count unit lengths: 12 |
| Wrong units | Area written as 40 cm | Ask "40 of what?" Area counts squares, so cm² |
| Mixed units | 2 m × 50 cm = 100 | Convert first: 200 × 50 = 10,000 cm² = 1 m² |
| Slant side used as height | Area of a parallelogram = base × slant side | Cut and rearrange the parallelogram into a rectangle |
| Diameter used as radius | Area of a circle of diameter 14 as π × 14² | Draw and label r before calculating |
| Taller container holds more | A tall, thin glass "has more water" | Pour between containers and compare |
| Area changes when a shape is cut and rearranged | Tangram pieces "make a bigger shape" | Rearrange the same pieces into different shapes |
Teaching mensuration so formulas make sense
- Area as covering. Cover a notebook with identical cards, then with square tiles. Children discover why squares are the best unit.
- Rows × columns. Counting squares in a rectangle row by row leads to l × b without any rule being given.
- Derive, don't dictate. Cut a triangle off one end of a paper parallelogram and fit it on the other end: it becomes a rectangle with the same base and height. Two identical triangles make a parallelogram, so a triangle's area is half.
- Circle area. Cut a paper circle into many equal sectors and arrange them alternately. They form a near-rectangle whose length is half the circumference, πr, and whose height is r, so the area is πr × r = πr².
- Perimeter as walking around the playground or tracing with a string.
- Volume with unit cubes. Build cuboids and count layers; fill boxes with sand or water for capacity.
- Estimate first. "Is the classroom floor nearer 5 m² or 50 m²?" builds a sense of size that catches unit errors.
For the Paper I side of measuring length, weight, capacity and time, see measurement. For shapes themselves, see geometry and shapes.
Practice set
- A square has perimeter 48 cm. What is its area?
- A rectangle has area 96 m² and length 12 m. Find its perimeter.
- Find the circumference of a circle of radius 14 cm (π = 22/7).
- Find the area of a circle of diameter 14 cm.
- How many 2 cm cubes fit into a box 10 cm × 8 cm × 6 cm?
- The side of a square is increased by 50%. By what percentage does its area increase?
- A trapezium has parallel sides 12 cm and 8 cm and height 5 cm. Find its area.
- A child says a 6 × 2 rectangle and a 4 × 4 square have the same area "because they both have perimeter 16". What is the best teaching response?
Answers
- Side 12 cm, area 144 cm².
- Breadth = 96 ÷ 12 = 8 m, perimeter = 2(12 + 8) = 40 m.
- 2 × 22/7 × 14 = 88 cm.
- Radius 7 cm, area = 154 cm².
- 5 × 4 × 3 = 60 (check: 480 cm³ ÷ 8 cm³ = 60).
- New area = 1.5 × 1.5 = 2.25 times, so an increase of 125%.
- ½ × (12 + 8) × 5 = 50 cm².
- Let the child build both on squared paper and count the squares (12 and 16), or make rectangles from a 16 cm loop of string. The contradiction does the teaching.
What to do next
- Write the plane-figure and solid tables from memory, including units.
- Solve five "path around a field" and "cost of tiling or fencing" problems, labelling each as area or perimeter first.
- Do the parallelogram and circle paper-cutting activities yourself, so you can describe them in one line.
- Continue with measurement.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Central Board of Secondary Education website .
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