Mensuration | ICSE Class 6 Maths Notes

This note covers perimeter, area, rectangle and square formulas, missing measurements, shapes with equal perimeters or equal areas, area relationships in triangles, the idea of volume, conversions of measurement units, and practical problems involving boundaries and covered regions.

What does perimeter measure?

Mensuration means measuring geometrical figures. A plane figure is a flat shape. Its boundary is its enclosing outline. A figure is closed when its outline joins up and encloses a region.

Definition: The perimeter of a closed plane figure is the distance along its boundary when you go around it once.

A polygon is a closed plane figure made from straight line segments. These segments are its sides. To find a polygon's perimeter, add the lengths of all its sides. Count each boundary side once as you travel around the figure.

How are perimeter measurements written?

Use a unit of length, such as a centimetre, written cm, or a metre, written m. A measurement combines a number with its unit. A perimeter written without its unit does not tell the reader the length being described.

In the calculations below, = means equals, + means addition, × means multiplication, ÷ means division, and − means subtraction. Brackets group a calculation that must be completed together. Read the words and units alongside these signs.

Worked example 1. A triangle, a polygon with three sides, has side lengths 4 cm, 5 cm and 7 cm. Find its perimeter.

Answer: Add its three side lengths: 4 cm + 5 cm + 7 cm = 16 cm. The perimeter is 16 cm.

The calculation follows the boundary, rather than measuring the region inside it. This distinction matters when deciding whether a problem about a shape needs addition of side lengths or a calculation of the space it covers.

How is the perimeter of a rectangle calculated?

A rectangle has four sides and four right angles. A right angle is the square corner made by two perpendicular lines, which meet at that angle. Opposite sides of a rectangle have equal lengths. Adjacent sides meet at a corner.

The two adjacent measurements are called length and breadth; width is another name for breadth. Let P stand for perimeter, l for length and b for breadth. These letters represent measurements, rather than particular fixed numbers.

Result: Perimeter of a rectangle

1. Travel around the four sides in order: length, breadth, length, breadth.
2. Add them to obtain length + breadth + length + breadth.
3. There are two lengths and two breadths, so the sum is twice length plus twice breadth.
4. Equivalently, add one length and one breadth, then multiply their sum by two.

P = 2 × (l + b)

Worked example 2. Find the perimeter of a rectangle with length 12 cm and breadth 8 cm.

Answer: P = 2 × (12 + 8) cm = 2 × 20 cm = 40 cm. Adding the sides separately gives 12 + 8 + 12 + 8 = 40, confirming the result.

The corners are labelled A, B, C and D, with A at the upper left. AB denotes the side joining A to B. The upper side AB is labelled 12 cm and the left side AD is labelled 8 cm.

Reference: NCERT Class 6, page 129

The letters at the corners identify positions; they are different from the letters used for measurements in the formula. A single length plus a single breadth covers only two adjacent sides. Doubling that sum accounts for the opposite pair as well.

How do squares and equal-sided shapes simplify perimeter calculations?

A square is a rectangle whose four sides are equal. Since each side contributes the same length to its boundary, finding its perimeter requires the length of just one side. Let s represent the length of a side.

Result: Perimeter of a square

P = 4 × s

This result comes from adding s + s + s + s. It also follows from the rectangle rule: when length and breadth are both s, twice their sum is four times s. The two methods describe the same boundary.

Worked example 3. A square photo frame has side 1 m. What length of coloured tape is needed all around it?

Answer: The tape follows the boundary. Perimeter = 4 × 1 m = 4 m, so the required length of tape is 4 m.

How are repeated rounds counted?

Worked example 4. Usha takes three complete rounds of a square park of side 75 m. Find the distance she travels.

Answer: One round is the perimeter: 4 × 75 m = 300 m. Three rounds cover 3 × 300 m = 900 m.

A regular polygon has equal sides and equal angles. For such a polygon, multiply the number of sides by the length of one side. An equilateral triangle has three equal sides and three equal angles, so its perimeter is three times a side.

Equal side lengths permit repeated addition to become multiplication. For a triangle whose sides differ, use all three given lengths instead. For repeated rounds, first establish the boundary length of one round, then multiply by the number of complete rounds.

How can a missing length be found from a perimeter?

A perimeter can be known even when one measurement is missing. Work backwards from the rule that links the known measurements to the unknown one. The missing quantity is a length, so its answer needs a length unit.

How is the rectangle rule reversed?

Half the perimeter of a rectangle equals length plus breadth. Divide the given perimeter by two, then subtract the known adjacent side. Do not subtract just one side directly from the whole perimeter, because the whole perimeter includes both pairs of opposite sides.

Worked example 5. A rectangle has perimeter 14 cm and breadth 2 cm. Find its length.

Answer: Length + breadth = 14 ÷ 2 = 7 cm. Length = 7 − 2 = 5 cm. Check: 2 × (5 + 2) cm = 14 cm, matching the given perimeter.

For a square, divide the perimeter by four because the four sides are equal. A square with perimeter 20 cm therefore has side 20 ÷ 4 = 5 cm. Multiplying this side length by four returns the given perimeter.

How is a missing triangle side found?

Worked example 6. A triangle has perimeter 55 cm. Two sides measure 20 cm and 14 cm. Find the third side.

Answer: The two known sides total 20 + 14 = 34 cm. The third side is 55 − 34 = 21 cm. Check: 20 + 14 + 21 = 55 cm.

Checking by substitution means putting the calculated value back into the original rule. It tests whether the proposed answer reproduces the supplied total. The check should use the original perimeter, rather than just repeating the subtraction that produced the missing side.

What is area, and why are square units used?

Definition: Area measures the region enclosed by a closed figure. It describes how much flat space the figure covers.

A unit square is a square with each side one unit long. Its area is one square unit. Covering a region with equal unit squares, without gaps or overlaps, lets us measure area by counting those squares.

How do area units differ from length units?

A square centimetre, written cm², is the area of a square of side 1 cm. A square metre, written m², is the area of a square of side 1 m. Here the raised ² indicates a square unit.

Area is generally measured in square units. A boundary length and an enclosed area describe different quantities. Measuring the distance around a floor is different from measuring the region to be covered by a carpet.

How can a square grid estimate area?

A square grid is a pattern of equal squares. Trace a shape onto transparent paper and place it over the grid. When the boundary cuts through squares, the following counting conventions give an estimate, meaning an approximate measurement.

1. Count each complete small square as one square unit.
2. Ignore a portion smaller than half a square.
3. Count a portion larger than half a square as one square unit.
4. Count a portion exactly half a square as half a square unit.

These conventions do not make every estimate exact. They provide a consistent way to handle partly covered squares. A region that can be divided completely into unit squares can have its area determined by an exact count.

Why do the rectangle and square area formulas work?

Let A represent area. Imagine a rectangle divided into equal unit squares. The number of squares along its length tells how many squares are in each row. The number along its breadth tells how many such rows cover the rectangle.

Result: Area of a rectangle

Multiplying the number in each row by the number of rows counts all the unit squares. This gives the rule A = l × b. Use length and breadth in the same length unit so the answer is in the corresponding square unit.

The rule is therefore a compact way to count the enclosed region. It is not a boundary calculation. Adding length and breadth does not count all the squares in the rows, and doubling their sum calculates the perimeter instead.

Result: Area of a square

A square has equal length and breadth. Replacing both measurements by its side length gives A = s × s, also written A = s². The expression s² means s multiplied by itself.

Worked example 7. A rectangular floor is 5 m long and 4 m wide. A square carpet of side 3 m is laid on it. Find the uncovered area.

Answer: Floor area = 5 × 4 = 20 m². Carpet area = 3 × 3 = 9 m². Uncovered area = 20 − 9 = 11 m².

The floor provides the whole region, and the carpet covers part of it. Both areas have the same unit, so subtraction gives the remaining area directly. The side of the carpet is a length; it must first be converted into an area by multiplication.

Explaining which region each multiplication measures makes the solution easier to follow. Write “floor area” and “carpet area” before combining the answers, rather than presenting a sequence of numbers without identifying the quantities.

How are areas combined and missing dimensions found?

A dimension is a measurement such as a shape's length or breadth. Knowing a rectangular region's area and one dimension lets us find the other. Since length multiplied by breadth gives area, division reverses that multiplication.

Worked example 8. A rectangular garden has area 300 m² and length 25 m. Find its width.

Answer: Width = area ÷ length = 300 ÷ 25 = 12 m. Check by multiplying: 25 m × 12 m gives 300 m², the required garden area.

How do we subtract several covered regions?

Decomposing a region means dividing it into simpler parts. If parts fill the region without overlapping, add their areas to find the whole. If known parts are removed or reserved, subtract their combined area from the whole area.

Worked example 9. Four square flower beds, each of side 4 m, occupy the four corners of land measuring 12 m by 10 m. Find the remaining area.

Answer: Whole area = 12 × 10 = 120 m². One bed has area 4 × 4 = 16 m². Four beds occupy 4 × 16 = 64 m². Remaining area = 120 − 64 = 56 m².

The number of flower beds and the side length of each bed have different jobs. The side length determines one bed's area. The number of beds tells how many copies of that area must be subtracted.

1. Identify the full region whose area is needed first.
2. Find the area of each smaller rectangular or square part.
3. Multiply by the number of equal parts where appropriate.
4. Add or subtract the areas according to the region requested.

Check that the final answer names the requested region. In the flower-bed problem, 64 m² measures the beds, while 56 m² measures the remaining land. A correct intermediate calculation does not by itself answer the final question.

Can shapes have the same perimeter but different areas?

Yes. Equal perimeters mean equal total boundary lengths. They do not require the enclosed regions to have equal areas. A piece of wire can form different closed shapes while keeping the same length of wire around their boundaries.

Worked example 10. A wire forms a rectangle with sides 5 cm and 3 cm. It is straightened and bent into a square. Find the square's side.

Answer: Wire length = rectangle perimeter = 2 × (5 + 3) = 16 cm. The square has the same perimeter. Its side = 16 ÷ 4 = 4 cm.

The rectangle's area is 5 × 3 = 15 cm². The square's area is 4 × 4 = 16 cm². The boundary length is unchanged, but the areas differ. This provides a counterexample: a case that disproves a general claim.

Can equal areas also have different perimeters?

Yes. Moving the same unit squares into a new arrangement can keep the total area unchanged while changing the outer boundary. Equal area tells us how much region is covered, but it does not fully describe the shape of its outline.

One figure is a square arranged in three rows of three unit squares. The other is a C-shaped arrangement of nine unit squares. Their areas are both nine square units, but their perimeters are 12 units and 20 units respectively.

Reference: NCERT Class 6, page 145

When counting a boundary in such arrangements, follow the exposed outline. A side shared by two adjoining squares lies inside the figure and is not part of its perimeter. Counting every side of every small square would include these internal joins.

Note: Do not use an area comparison to decide a perimeter comparison. Calculate or trace the relevant boundary separately, even when the shapes contain the same number of unit squares.

How is a triangle's area related to a rectangle's area?

A diagonal is a line segment joining two non-adjacent corners of a polygon. Drawing a diagonal across a rectangle divides it into two triangles. Cutting along it gives two triangles that fit exactly over one another.

Because the two triangles cover the rectangle together and have equal areas, each occupies half the rectangle's area. This connects a triangle's area to the familiar rectangle rule, rather than introducing an unrelated calculation to memorise.

How do base and height describe the comparison?

A triangle's base is the side chosen for its area calculation. The corresponding height is the perpendicular distance from the opposite corner to the line containing that base. The height measures straight across at a right angle to the base.

The area of a triangle is half the product of its base and corresponding height. In words: triangle area = half × base × height. The rectangle with that base and height has twice the triangle's area.

Rectangle ABCD has A and B at the bottom. D and C are above them. E lies on the upper side and F lies directly below E on AB. Lines DB, AE, BE and EF mark the triangles and the two smaller rectangles.

Reference: NCERT Class 6, page 143

Triangle BAD is half of rectangle ABCD. Triangle ABE can be split into triangles AEF and BEF. Each is half of its corresponding smaller rectangle. Adding these halves gives half of the whole rectangle ABCD.

Thus differently shaped triangles can have the same area. In this diagram, both triangles use the same bottom side and reach the same upper side. Their equal areas follow from the rectangle comparison, even though their outlines look different.

How are units of mass, time, money and capacity converted?

Unit conversion expresses the same quantity in a different unit. A smaller unit needs a larger numerical count for the same quantity. A larger unit needs a smaller count. The quantity itself remains unchanged during conversion.

Mass describes how much matter an object contains. The gram, written g, and kilogram, written kg, are mass units. Time measures duration, using units such as seconds, minutes and hours. A rupee and a paisa are money units; paise is the plural of paisa.

Volume measures the space occupied by an object. Capacity describes how much a container can hold. The litre, written L, and millilitre, written mL, are capacity units. The symbol ₹ denotes rupees.

Which conversion facts are needed?

Worked example 11. Express 254 g in kilograms, using 1 kg = 1000 g.

Answer: Divide by 1000 because kilograms are the larger unit. Thus 254 g = 254 ÷ 1000 kg = 0.254 kg. Multiplying 0.254 by 1000 converts it back to 254 g.

Worked example 12. Express 75 paise in rupees, using 1 rupee = 100 paise.

Answer: Divide the number of paise by 100. Thus 75 paise = 75 ÷ 100 rupee = ₹0.75. Multiplying 0.75 by 100 returns 75 paise.

Use the conversion factor, meaning the number of smaller units in one larger unit, for the particular pair. Time conversions here use 60, money uses 100, and kilogram-to-gram and litre-to-millilitre conversions use 1000.

Length units also need attention in mensuration: 1 m = 100 cm. Before adding side lengths or multiplying a rectangle's dimensions, express them in a common length unit. Keep the distinction between length units and square units when writing the result.

How can practical mensuration problems be solved and checked?

Begin by identifying what the problem measures. Fencing, edging and travelling around a closed boundary involve perimeter. Covering a floor or finding available land involves area. The operation should follow the meaning of the required measurement.

How is fencing cost calculated?

Worked example 13. A rectangular park is 150 m long and 120 m broad. Fencing costs ₹40 per metre. Find the cost of fencing its boundary.

Answer: Perimeter = 2 × (150 + 120) = 540 m. Cost = 540 × ₹40 = ₹21,600.

A rate expresses an amount for each unit. Here the rate is a cost for each metre of boundary. Multiply by the perimeter in metres. Multiplying by the park's area would apply a length-based rate to the wrong measurement.

Which steps make a solution clear?

1. List the supplied dimensions and identify the quantity to be found.
2. Check the units and convert measurements where necessary.
3. Write the perimeter or area rule that matches the question.
4. Insert the given values and calculate in a clear order.
5. Include any extra step, such as multiplying by rounds or a cost rate.
6. Write the final quantity with its unit and check it against the question.

A useful check relates the answer to the original information. Add sides again to check a perimeter, multiply a missing width by the known length to check area, or add covered and uncovered areas to recover a whole floor.

Keep calculations for different quantities separate. A boundary in metres, a region in square metres and a cost in rupees answer different questions. Naming each intermediate result helps prevent a correct calculation from being used in the wrong place.

Glossary

• Mensuration — The measurement of geometrical figures, including their boundary lengths and enclosed regions.
• Perimeter — The distance travelled along a closed figure's boundary when going around it once.
• Polygon — A closed plane figure whose boundary consists of straight line segments called sides.
• Rectangle — A four-sided figure with four right angles and equal opposite sides.
• Square — A rectangle whose four sides all have the same length.
• Area — The measure of the region enclosed by a closed plane figure.
• Unit square — A square with each side one unit long, covering one square unit.
• Regular polygon — A polygon in which all sides and all angles are equal.
• Diagonal — A line segment joining two corners that are not adjacent in a polygon.
• Height of a triangle — The perpendicular distance from the opposite corner to the line containing the chosen base.
• Counterexample — A particular case that shows a proposed general statement is false.
• Volume — The measure of the space occupied by an object in three dimensions.
• Capacity — The amount a container can hold, expressed using units such as litres.
• Conversion factor — The number of smaller units equivalent to one specified larger unit.

Common errors and misconceptions

• Misconception: Length plus breadth is a rectangle's perimeter. Correct: This covers only two adjacent sides. Double their sum to include all four boundary sides.
• Misconception: A square's perimeter is its side multiplied by itself. Correct: That calculates its area. Perimeter is four times the side length.
• Misconception: Perimeter and area have the same units. Correct: Perimeter uses length units; area is generally measured in square units, such as square centimetres.
• Misconception: Shapes with equal perimeters must have equal areas. Correct: The 5 cm by 3 cm rectangle and the square of side 4 cm both have perimeter 16 cm, but their areas differ.
• Misconception: Every side of every unit square counts towards an arrangement's perimeter. Correct: Shared internal sides do not belong to the outer boundary. Follow the exposed outline.
• Misconception: Subtracting the area of one flower bed accounts for four equal beds. Correct: First multiply one bed's area by four, then subtract the combined area.
• Misconception: Converting to a larger unit requires multiplication. Correct: Divide by the number of smaller units in that larger unit; the physical quantity stays unchanged.
• Misconception: Counting partly covered grid squares gives an exact area in every case. Correct: The stated rounding conventions estimate area; they do not guarantee an exact measurement.

Exam-style questions with model answers

Q1. Define perimeter and find the perimeter of a triangle with sides 4 cm, 5 cm and 7 cm. [2 marks]

1. Perimeter is the distance around a closed figure's boundary once.
2. The triangle's perimeter is the sum of its sides: 4 + 5 + 7 = 16 cm.

Q2. A rectangle has perimeter 14 cm and breadth 2 cm. Find its length and check your answer. [3 marks]

1. The rectangle's perimeter is twice the sum of its length and breadth. Therefore, length plus breadth equals half of 14 cm, which is 7 cm.
2. Subtract the known breadth from this sum. The length is 7 − 2 = 5 cm.
3. Check using both dimensions: 2 × (5 + 2) = 14 cm. This matches the supplied perimeter.

Q3. Usha walks three complete rounds of a square park whose side is 75 m. Find her total distance, showing the distance for one round. [3 marks]

1. The park is square, so its boundary has four equal sides. Its perimeter is four times the given side length.
2. One complete round covers the perimeter: 4 × 75 = 300 m. This is the distance for one circuit of the park.
3. Three complete rounds cover three times that distance: 3 × 300 = 900 m. Usha walks 900 m altogether.

Q4. A 5 cm by 3 cm rectangular wire frame is reshaped into a square using the same wire length. Find the square's side and compare the two areas to test whether equal perimeters imply equal areas. [5 marks]

1. The rectangle's perimeter is twice the sum of its adjacent sides: 2 × (5 + 3) = 16 cm. This is the wire's total length.
2. Using the same wire gives the square a perimeter of 16 cm. Its four equal sides therefore each measure 16 ÷ 4 = 4 cm.
3. The rectangle's area is length multiplied by breadth: 5 × 3 = 15 cm². This measures its enclosed region.
4. The square's area is its side multiplied by itself: 4 × 4 = 16 cm². Its enclosed region is larger.
5. The shapes have equal perimeters of 16 cm but different areas. Thus equal perimeters do not imply equal areas.

Q5. Four non-overlapping square flower beds, each of side 4 m, occupy the four corners of rectangular land measuring 12 m by 10 m. Find the remaining area and check your result. [5 marks]

1. Calculate the whole rectangular area using length multiplied by breadth: 12 × 10 = 120 m². This includes the beds and the remaining land.
2. Each bed is a square of side 4 m. The area of one bed is therefore 4 × 4 = 16 m².
3. There are four equal beds, so their combined area is 4 × 16 = 64 m². The beds do not overlap.
4. Subtract the combined bed area from the whole area: 120 − 64 = 56 m². This is the remaining land.
5. Check by adding the two regions: 56 + 64 = 120 m². The result recovers the full rectangular area.

Q6. A rectangular park is 150 m long and 120 m broad. A fence is to run once around the complete boundary at ₹40 per metre. Find its length and cost. [4 marks]

1. Fencing follows the boundary, so the required measurement is the park's perimeter. For a rectangle, double the sum of length and breadth.
2. The sum of the adjacent sides is 150 + 120 = 270 m. The perimeter is 2 × 270 = 540 m.
3. The fence costs ₹40 for each metre, so multiply the required length by that rate: 540 × 40 = 21,600.
4. The required fence length is 540 m and its cost is ₹21,600. These answer the length and cost parts separately.

Q7. Convert 254 g into kilograms and 75 paise into rupees. Use 1 kg = 1000 g and 1 rupee = 100 paise, and explain each operation. [4 marks]

1. A kilogram is the larger mass unit. Since 1000 grams make one kilogram, divide the number of grams by 1000.
2. Therefore, 254 g = 254 ÷ 1000 kg = 0.254 kg. The mass stays the same; its numerical expression changes.
3. A rupee is the larger money unit. Since 100 paise make one rupee, divide the number of paise by 100.
4. Therefore, 75 paise = 75 ÷ 100 rupee = ₹0.75. Multiplication by 100 would convert this rupee amount back to paise.

Q8. A rectangular floor measures 5 m by 4 m. A square carpet of side 3 m lies completely inside it. Find the uncovered area. [3 marks]

1. The rectangular floor's area is its length multiplied by its breadth: 5 × 4 = 20 m². This is the whole region under consideration.
2. The carpet is square, so its area is its side multiplied by itself: 3 × 3 = 9 m².
3. Subtract the carpeted area from the floor area: 20 − 9 = 11 m². The uncovered region has area 11 m².

Key takeaways

• Perimeter measures a complete boundary once; for a polygon, add the lengths of all its sides.
• A rectangle's perimeter is twice length plus breadth taken together; a square's perimeter is four times its side.
• Area measures the enclosed region and is generally expressed in square units, rather than length units.
• A rectangle's area is length multiplied by breadth; a square's area is its side multiplied by itself.
• Find remaining area by subtracting the combined areas of covered or reserved parts from the whole region.
• Equal perimeters can enclose different areas, and equal areas can belong to shapes with different perimeters.
• A triangle's area is half the product of its chosen base and the corresponding perpendicular height.
• Convert units using the correct factor, and match the final unit to the quantity requested in the problem.

Test yourself

What is the difference between perimeter and area?

Perimeter measures the boundary length; area measures the region enclosed by that boundary.

Why is the sum of a rectangle's length and breadth doubled for its perimeter?

The boundary contains two equal lengths and two equal breadths, so one adjacent pair must be counted twice.

A square has perimeter 20 cm. What is its side length?

Its four sides are equal, so each side is 20 ÷ 4 = 5 cm.

Why does multiplying a rectangle's length by breadth give its area?

It counts the unit squares in each row multiplied by the number of rows covering the rectangle.

A rectangle has area 300 m² and length 25 m. What is its width?

Width equals area divided by length, so the width is 300 ÷ 25 = 12 m.

Why are shared sides between adjoining unit squares excluded from the combined perimeter?

They lie inside the combined figure, while perimeter follows its exposed outer boundary.

Using 1 L = 1000 mL, how do you convert litres into millilitres and back?

Multiply litres by 1000 to get millilitres. Divide millilitres by 1000 to recover litres.

Using 1 hour = 60 minutes, how do you convert hours into minutes and back?

Multiply hours by 60 to get minutes. Divide minutes by 60 to express that duration in hours.