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Perimeter and Area — Class 6 Mathematics

Ganita Prakash · Chapter 6

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Chapter summary

This chapter studies perimeter and area of plane figures. The perimeter of any closed figure is the distance covered along its boundary in one round; for a polygon it is the sum of all side lengths. The perimeter of a rectangle is , of a square is , and of a triangle is the sum of its three sides; a regular polygon\u2019s perimeter is the number of sides times the length of a side. Area is the amount of region enclosed by a closed figure, measured in square units. The area of a rectangle is length breadth and the area of a square is side side. The chapter shows how to estimate areas of irregular shapes using squared or graph paper, counting full squares and half squares. A key result is that the area of a triangle is half the area of the rectangle that contains it, so the area of a right triangle is half of base height. Through tangram puzzles, house plans and area mazes, students see that two figures can have the same area but different perimeters, or the same perimeter but different areas.

Perimeter of PolygonsPerimeter of Rectangles, Squares and TrianglesArea and Square UnitsArea of Rectangle and SquareEstimating Area with GridsArea of a Triangle

Key ideas & terms

Perimeter: The total distance along the boundary of a closed figure, i.e. the sum of all its side lengths.
Area: The amount of region enclosed by a closed figure, measured in square units.
Perimeter of a rectangle: Twice the sum of length and breadth: .
Perimeter of a square: Four times the length of a side: .
Area of a rectangle: Length multiplied by breadth: .
Square unit: The unit of area, equal to the area of a square, such as a square centimetre or square metre.

Extra questions & answers

What is the perimeter of a closed figure?

It is the distance covered along the boundary in one complete round, equal to the sum of the lengths of all its sides.

What is the formula for the perimeter of a rectangle?

Perimeter .

What is the perimeter of a square of side ?

Perimeter , four times the length of one side.

What is area?

Area is the amount of region enclosed by a closed figure, measured in square units.

What is the area of a rectangle?

Area .

What is the area of a square of side ?

Area .

How is the area of a right triangle related to a rectangle?

A diagonal splits a rectangle into two equal triangles, so a right triangle\u2019s area is half the area of the rectangle, i.e. half of base height.

How can you estimate the area of an irregular shape?

Place it on squared or graph paper; count each full square as 1 unit, ignore parts less than half, count more than half as 1, and a half square as .

Can two figures with the same area have different perimeters?

Yes; for example different rectangles with area 24 square units can have very different perimeters.

Find the width of a rectangular garden 25 m long with area 300 sq m.

Width m.

Figure it Out

1. Find the missing terms: a. Perimeter of a rectangle = 14 cm; breadth = 2 cm; length = ?.

For a rectangle . So gives so .

1. Find the missing terms: b. Perimeter of a square = 20 cm; side of a length = ?.

For a square . So gives .

1. Find the missing terms: c. Perimeter of a rectangle = 12 m; length = 3 m; breadth = ?.

so and .

2. A rectangle having sidelengths 5 cm and 3 cm is made using a piece of wire. If the wire is straightened and then bent to form a square, what will be the length of a side of the square?

Length of wire = perimeter of rectangle = . This same length becomes perimeter of square, so and .

3. Find the length of the third side of a triangle having a perimeter of 55 cm and having two sides of length 20 cm and 14 cm, respectively.

Perimeter = sum of three sides. So third side = .

4. What would be the cost of fencing a rectangular park whose length is 150 m and breadth is 120 m, if the fence costs ₹40 per metre?

Perimeter = . Cost = so .

5. A piece of string is 36 cm long. What will be the length of each side, if it is used to form: a. A square,

For a square so .

5. A piece of string is 36 cm long. What will be the length of each side, if it is used to form: b. A triangle with all sides of equal length, and

For an equilateral triangle so .

5. A piece of string is 36 cm long. What will be the length of each side, if it is used to form: c. A hexagon (a six sided closed figure) with sides of equal length?

For a regular hexagon so .

6. A farmer has a rectangular field having length 230 m and breadth 160 m. He wants to fence it with 3 rounds of rope as shown. What is the total length of rope needed?

Perimeter of one round = . For 3 rounds = .

Figure it Out

1. Find out the total distance Akshi has covered in 5 rounds.

Outer track length = , breadth = . Perimeter = . In 5 rounds Akshi covers .

2. Find out the total distance Toshi has covered in 7 rounds. Who ran a longer distance?

Inner track length = , breadth = . Perimeter = . In 7 rounds Toshi covers . so Toshi ran the longer distance by .

3. Think and mark the positions as directed— a. Mark ‘A’ at the point where Akshi will be after she ran 250 m.

Akshi's perimeter is . After she has completed one full round and is into the next round from her starting point, i.e. along the top side clockwise from the starting point.

3. Think and mark the positions as directed— b. Mark ‘B’ at the point where Akshi will be after she ran 500 m.

. After two full rounds , she is into the third round, i.e. almost at the first corner after the starting side, clockwise from the start.

3. Think and mark the positions as directed— c. Now, Akshi ran 1000 m. How many full rounds has she finished running around her track? Mark her position as ‘C’.

full rounds with remainder because . She has finished 4 full rounds and is into the 5th round, which puts her on the opposite side of the rectangle from the starting point.

3. Think and mark the positions as directed— d. Mark ‘X’ at the point where Toshi will be after she ran 250 m.

Toshi's perimeter is . After she has completed one round and is into the second round, i.e. beyond the first corner on the second side clockwise from her start.

3. Think and mark the positions as directed— e. Mark ‘Y’ at the point where Toshi will be after she ran 500 m.

. After two full rounds , she is into the third round, i.e. near the third side of the inner rectangle.

3. Think and mark the positions as directed— f. Now, Toshi ran 1000 m. How many full rounds has she finished running around her track? Mark her position as ‘Z’.

full rounds with remainder because . She has finished 5 full rounds and is into the 6th round, which puts her on the second side of her track about past the first corner.

Deep Dive:

In races, usually there is a common finish line for all the runners. Here are two square running tracks with the inner track of 100 m each side and outer track of 150 m each side. The common finishing line for both runners is shown by the flags in the figure which are in the center of one of the sides of the tracks. If the total race is of 350 m, then we have to find out where the starting positions of the two runners should be on these two tracks so that they both have a common finishing line after they run for 350 m. Mark the starting points of the runner on the inner track as ‘A’ and the runner on the outer track as ‘B’.

Inner square perimeter = . Outer square perimeter = . Flag is at the mid-point of a side. Going backwards from the flag gives the start. Inner: back from the flag is before completing a full lap, i.e. behind the flag on the track, which is half a side behind the flag. Outer: back from the flag is along the 600 m loop, which leaves ahead to the flag if going forward. On the diagram this places ‘A’ on the inner track one half-side before the common finish line and ‘B’ on the outer track 350 m upstream of the same finish line, as shown by counting sides: inner to the flag, outer to the flag.

Estimate and Verify

Take a rough sheet of paper or a sheet of newspaper. Make a few random shapes by cutting the paper in different ways. Estimate the total length of the boundaries of each shape then use a scale or measuring tape to measure and verify the perimeter for each shape.

This is a hands-on activity. First look at the shape and guess its perimeter by eye, for example guessing . Then place a thread along the boundary, straighten it and measure with a ruler or tape to get the actual perimeter, e.g. . Compare estimate and actual to see how close the guess was. Repeating for different irregular shapes shows that perimeter is always the total length around the shape.

Akshi says that the perimeter of this triangle shape is 9 units. Toshi says it can’t be 9 units and the perimeter will be more than 9 units. What do you think?

Toshi is correct. The triangle has two sides along the grid lines and one diagonal. A diagonal across a unit square is longer than a side (by Pythagoras about times). So counting the diagonal as 1 unit underestimates. The true perimeter is more than units, precisely is not because .

Write the perimeters of the figures below in terms of straight and diagonal units.

Counting straight segments (s) and diagonal segments (d) on the dot paper: the four figures have perimeters approximately for the first, for the second, for the third and for the fourth. In each case so the numerical value exceeds the count of unit edges.

Find various objects from your surroundings that have regular shapes and find their perimeters. Also, generalise your understanding for the perimeter of other regular polygons.

Examples: a square tile with side has perimeter ; an equilateral triangular coasters with side has perimeter . In general for a regular polygon with sides each of length , perimeter . So for a regular pentagon , for a regular hexagon and so on.

Find out the length of the boundary (i.e., the perimeter) of each of the other arrangements below.

Find out the length of the boundary (i.e., the perimeter) of each of the other arrangements below. b.

Arrangement b. is formed by joining the two pieces in an L-shape. Its outer boundary measures (the calculation is the sum of all outer edges, overlapping interior joint is not counted).

Find out the length of the boundary (i.e., the perimeter) of each of the other arrangements below. c.

Arrangement c. with the two pieces in a T-shape gives a perimeter of about (the central overlap reduces the total by twice the joining length).

Find out the length of the boundary (i.e., the perimeter) of each of the other arrangements below. d.

Arrangement d. offset vertically by gives a perimeter of about . Different join positions change how much interior edge is hidden, so perimeters differ even though area stays .

Arrange the two pieces to form a figure with a perimeter of 22 cm.

To get the smallest perimeter, overlap the longest possible cut edges fully. Place the cut of one piece exactly on the cut of the other so that the two rectangles together form a rectangle again but with the cut internal. Then perimeter = if fully rejoined. To get leave a step, i.e. slide the pieces so of the cut is overlapped and remains exposed, giving perimeter .

Figure it Out

1. The area of a rectangular garden 25 m long is 300 sq m. What is the width of the garden?

Area = length width. So gives .

2. What is the cost of tiling a rectangular plot of land 500 m long and 200 m wide at the rate of ₹ 8 per hundred sq m?

Area = . Hundred sq m units = . Cost = .

3. A rectangular coconut grove is 100 m long and 50 m wide. If each coconut tree requires 25 sq m, what is the maximum number of trees that can be planted in this grove?

Grove area = . Number of trees = .

4. By splitting the following figures into rectangles, find their areas (all measures are given in metres). a.

Figure a can be split into three rectangles. Calculated total area = .

4. By splitting the following figures into rectangles, find their areas (all measures are given in metres). b.

Figure b is a large rectangle with a notch removed. Area = .

Figure it Out

1. Explore and figure out how many pieces have the same area.

In the 7-piece tangram, the two large triangles A and B have equal area, the two small triangles C and E have equal area, and the medium triangle D, the square F and the parallelogram G all relate by multiples of the smallest triangle area.

2. How many times bigger is Shape D as compared to Shape C? What is the relationship between Shapes C, D and E?

Shape D is exactly twice the area of Shape C. Since C and E are equal, D can be covered exactly by C + E, so .

3. Which shape has more area: Shape D or F? Give reasons for your answer.

They have the same area. In this tangram, Shape D (the medium triangle), Shape F (the square) and Shape G (the parallelogram) are each equal to twice the smallest triangle C, so Shape D and Shape F have equal area.

4. Which shape has more area: Shape F or G? Give reasons for your answer.

Shapes F (square) and G (parallelogram) have the same area, each equal to area of small triangle C, because each can be formed by two small triangles. So neither is larger.

5. What is the area of Shape A as compared to Shape G? Is it twice as big? Four times as big?

Shape A is the large triangle, with area area of small triangle C, while G is . So A is twice as big as G (and four times as big as C).

6. Can you now figure out the area of the big square formed with all seven pieces in terms of the area of Shape C?

Total area = A + B + D + C + E + F + G = . So the big square is 16 times the area of the smallest triangle C.

7. Arrange these 7 pieces to form a rectangle. What will be the area of this rectangle in terms of the area of Shape C now? Give reasons for your answer.

Rearranging the pieces does not change total area. The rectangle formed with all 7 pieces still has area , same as the square, because area is conserved under rearrangement.

8. Are the perimeters of the square and the rectangle formed from these 7 pieces different or the same? Give an explanation for your answer.

The perimeters are different. Although area is the same, the shape of the boundary changes. The square is more compact with smaller perimeter, while the rectangle is longer and thinner, exposing more edge, so its perimeter is larger.

Find the area of the following figures.

Using the rule count full squares as 1, more than half as 1, half as 0.5, the four sample shapes have areas about , , and respectively.

Let’s Explore! Why is area generally measured using squares?

Why is area generally measured using squares? Draw a circle on a graph sheet with diameter (breadth) of length 3. Count the squares and use them to estimate the area of the circular region. As you can see, circles can’t be packed tightly without gaps in between. So, it is difficult to get an accurate measurement of area using circles as units. Here, the same rectangle is packed in two different ways with circles—the first one has 42 circles and the second one has 44 circles. Try using different shapes (triangle and rectangle) to fill the given space (without overlaps and gaps) and find out the merits associated with using a square shape to find the area rather than another shape. List out the points that make a square the best shape to use to measure area.

Squares tile a plane perfectly without gaps or overlaps, have equal sides and right angles which makes counting easy, and allow easy subdivision into smaller squares. Circles leave gaps, triangles and other shapes may not tile as uniformly or require orientation changes. So the square with side giving area is the simplest consistent unit for area measurement.

1. Find the area (in square metres) of the floor outside of the corridor.

This is a practical measurement task. Measure the floor dimensions and corridor dimensions and subtract: Area outside = Area of whole floor − Area of corridor.

2. Find the area (in square metres) occupied by your school playground.

Measure length and breadth of the playground, e.g. . Exact value depends on your school.

Let’s Explore! On a squared grid paper (1 square = 1 square unit), make as many rectangles as you can whose lengths and widths are a whole number of units such that the area of the rectangle is 24 square units. a. Which rectangle has the greatest perimeter? b. Which rectangle has the least perimeter?

Let’s Explore! On a squared grid paper (1 square = 1 square unit), make as many rectangles as you can whose lengths and widths are a whole number of units such that the area of the rectangle is 24 square units. a. Which rectangle has the greatest perimeter?

Factor pairs of 24: , , , . Perimeters are , , , units. The rectangle has the greatest perimeter .

Let’s Explore! On a squared grid paper (1 square = 1 square unit), make as many rectangles as you can whose lengths and widths are a whole number of units such that the area of the rectangle is 24 square units. b. Which rectangle has the least perimeter?

From the same list the rectangle (closest to a square) has the least perimeter .

Let’s Explore! On a squared grid paper (1 square = 1 square unit), make as many rectangles as you can whose lengths and widths are a whole number of units such that the area of the rectangle is 24 square units. c. If you take a rectangle of area 32 sq cm, what will your answers be? Given any area, is it possible to predict the shape of the rectangle with the greatest perimeter as well as the least perimeter? Give examples and reasons for your answer.

For area : pairs perimeter , perimeter , perimeter . Greatest is , least is (most square-like). In general for a fixed area the rectangle that is most elongated () gives the largest perimeter and the rectangle closest to a square gives the smallest perimeter.

Area of a Triangle

Draw a rectangle on a piece of paper and draw one of its diagonals. Cut the rectangle along that diagonal and get two triangles. Check! whether the two triangles overlap each other exactly. Do they have the same area?

Yes, the two triangles exactly coincide when placed one over the other, so they have equal area, each is half the area of the original rectangle.

Is the area of the blue rectangle more or less than the area of the yellow triangle? Or is it the same? Why? Can you see some relationship between the blue rectangle and the yellow triangle and their areas? Write the relationship here.

The yellow triangle is exactly half of the blue rectangle that encloses it. So the triangle's area is one-half the rectangle's area.

1. Find the area of blue triangle BAD.

Triangle BAD is half of rectangle ABCD, so its area = area of rectangle ABCD.

2. Find the area of red triangle ABE.

Triangle ABE is also half of rectangle ABCD, formed by two halves of two smaller rectangles. So its area = area of rectangle ABCD, same as the blue triangle despite different shape.

Conclusion

Any triangle that shares the same base and has its opposite vertex on the line parallel to the base within the same rectangle has area equal to half the rectangle's area.

Figure it Out

1. Find the areas of the figures below by dividing them into rectangles and triangles. a.

By dividing into rectangles and triangles, area a = .

1. Find the areas of the figures below by dividing them into rectangles and triangles. b.

Area b = .

1. Find the areas of the figures below by dividing them into rectangles and triangles. c.

Area c = .

1. Find the areas of the figures below by dividing them into rectangles and triangles. d.

Area d = .

1. Find the areas of the figures below by dividing them into rectangles and triangles. e.

Area e = .

Making it ‘More’ or ‘Less’

Using 9 unit squares, solve the following. 1. What is the smallest perimeter possible?

The smallest perimeter is obtained when the 9 squares form a square. Perimeter = .

Using 9 unit squares, solve the following. 2. What is the largest perimeter possible?

The largest perimeter is obtained when all 9 squares are in a straight row . Perimeter = .

Using 9 unit squares, solve the following. 3. Make a figure with a perimeter of 18 units.

A shape with perimeter can be made, for example a block plus three extra squares attached to make a stepped shape, or a square with one corner square moved to create two extra protrusions, giving perimeter .

Using 9 unit squares, solve the following. 4. Can you make other shaped figures for each of the above three perimeters, or is there only one shape with that perimeter? What is your reasoning?

For perimeter there is only one arrangement (the square) up to rotation. For larger perimeters such as and many different polyominoes give the same perimeter because indentations can be rearranged without changing the boundary length, so multiple shapes share the same perimeter.

We have a figure below having perimeter 24 units. Without calculating all over again, observe, think and find out what will be the change in the perimeter if a new square is attached as shown on the right.

If a new square is attached flush along a full side interior to the figure, one side of the new square and one side of the old figure become interior and are removed from the perimeter. So change = units. If attached at a corner touching only at a point or along a half edge the change differs.

Experiment placing this new square at different places and think what the change in perimeter will be. Can you place the square so that the perimeter: a) increases; b) decreases; c) stays the same?

a) Perimeter increases if the square is attached sharing only one side on an outer edge: new perimeter = old + 2 units. b) Perimeter decreases if the square is placed into a notch filling a concavity sharing three sides: new perimeter = old − 2 units. c) Perimeter stays the same if the square fills a notch sharing two sides: new perimeter = old. So all three are possible by choosing placement.

House plan of Charan

Below is the house plan of Charan. It is in a rectangular plot. Look at the plan. What do you notice? Some of the measurements are given. a. Find the missing measurements.

Plot is . Master Bedroom , Kitchen , Toilet . Then Small Bedroom , Utility , Hall , Parking , Garden . These fill the remaining widths and heights.

Below is the house plan of Charan. It is in a rectangular plot. Look at the plan. What do you notice? b. Find out the area of his house.

Area of whole plot/house = . This equals sum of all room areas: .

House plan of Sharan

Now, find out the missing dimensions and area of Sharan’s home. Below is the plan: Some of the measurements are given. a. Find the missing measurements.

Plot is . Master Bedroom , Kitchen , Hall , Small Bedroom , Entrance , Utility , Toilet .

Now, find out the missing dimensions and area of Sharan’s home. Below is the plan: Some of the measurements are given. b. Find out the area of his house.

Area = , same as Charan’s house. Perimeter of Charan’s house = , perimeter of Sharan’s house = . Areas equal but Sharan’s more elongated rectangle has larger perimeter.

Area Maze Puzzles

In each figure, find the missing value of either the length of a side or the area of a region. a.

Figure a: large rectangle divided vertically. Top left , top right , bottom left implies scale factor 2, so missing bottom right = .

In each figure, find the missing value of either the length of a side or the area of a region. b.

Figure b: widths proportional to areas. Left columns 60 and 42 with common height give bottom missing area = after solving.

In each figure, find the missing value of either the length of a side or the area of a region. c.

Figure c: with total height split, missing upper area = .

In each figure, find the missing value of either the length of a side or the area of a region. d.

Figure d: given areas and sharing widths, missing side = .

Figure it Out

1. Give the dimensions of a rectangle whose area is the sum of the areas of these two rectangles having measurements: 5 m × 10 m and 2 m × 7 m.

Area1 = , Area2 = , sum = . Any factor pair of 64 works, e.g. or or .

2. The area of a rectangular garden that is 50 m long is 1000 sq m. Find the width of the garden.

3. The floor of a room is 5 m long and 4 m wide. A square carpet whose sides are 3 m in length is laid on the floor. Find the area that is not carpeted.

Floor area = . Carpet area = . Not carpeted = .

4. Four flower beds having sides 2 m long and 1 m wide are dug at the four corners of a garden that is 15 m long and 12 m wide. How much area is now available for laying down a lawn?

Garden area = . Each bed area = . Four beds = . Remaining lawn area = .

5. Shape A has an area of 18 square units and Shape B has an area of 20 square units. Shape A has a longer perimeter than Shape B. Draw two such shapes satisfying the given conditions.

This demonstrates area does not determine perimeter. For example Shape A could be rectangle perimeter units area , Shape B could be rectangle perimeter units area . So despite smaller area, Shape A has longer perimeter. Many polyomino examples also work.

6. On a page in your book, draw a rectangular border that is 1 cm from the top and bottom and 1.5 cm from the left and right sides. What is the perimeter of the border?

If page size is e.g. (A4), border dimensions become . Perimeter = . General formula: perimeter = where are page width and height.

7. Draw a rectangle of size 12 units × 8 units. Draw another rectangle inside it, without touching the outer rectangle that occupies exactly half the area.

Outer area = . Half is . Choose inner rectangle e.g. or centered inside with a gap of at least one unit from the outer boundary.

8. A square piece of paper is folded in half. The square is then cut into two rectangles along the fold. Regardless of the size of the square, one of the following statements is always true. Which statement is true here? a. The area of each rectangle is larger than the area of the square. b. The perimeter of the square is greater than the perimeters of both the rectangles added together. c. The perimeters of both the rectangles added together is always 1 1/2 times the perimeter of the square. d. The area of the square is always three times as large as the areas of both rectangles added together.

Only statement c is true. Let square side = , perimeter = . Each small rectangle is with perimeter . Sum of both = . Areas halve, so a, b, d are false.