Perimeter and Area | FIO

Question 21

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

Solution

Understand the Question

Using the exact same $7$ pieces means the total space covered (area) remains identical for both the square and the rectangle.
Perimeter is the total length around the outer boundary, which changes depending on how the pieces are arranged.
For a given fixed area, a square always has the smallest perimeter compared to any non-square rectangle.

Step 1 · Area Comparison of the Shapes

Since both the square and the rectangle are made from the exact same $7$ tangram pieces, their areas are equal: $\text{Area of Square} = \text{Area of Rectangle}$ Diagram 1

Step 2 · Compare the Perimeters

Perimeter is the total length of the outer boundary. Shapes with the same area can have different perimeters.

For example, consider an area of $36\text{ units}^2$ :

Square with side $6\text{ units}$ ( $6 \times 6 = 36$ ):

\begin{aligned} \text{Perimeter} &= 4 \times 6 \\[0.6em] &= 24\text{ units} \end{aligned}

Rectangle with dimensions $9\text{ units} \times 4\text{ units}$ ( $9 \times 4 = 36$ ):

\begin{aligned} \text{Perimeter} &= 2 \times (9 + 4) \\[0.6em] &= 2 \times 13 \\[0.6em] &= 26\text{ units} \end{aligned}

Among all rectangles of a given area, a square has the minimum perimeter. Therefore, the perimeters are different, and the square has a smaller perimeter than the rectangle.

Answer

The perimeters are different. The perimeter of the square is less than the perimeter of the rectangle.

Common Mistakes

Equating Area and Perimeter: Assuming that because both figures are made from the same pieces (equal area), their perimeters must also be equal.
Boundary Edge Changes: Overlooking that rearranging shapes exposes different edges on the outer perimeter.

More questions in FIO

Find the missing terms:

a. Perimeter of a rectangle = 14 cm; breadth = 2 cm; length = ?.

b. Perimeter of a square = 20 cm; side of a length = ?.

c. Perimeter of a rectangle = 12 m; length = 3 m; breadth = ?.

A rectangle having sidelengths $5\text{ cm}$ and $3\text{ cm}$ 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?

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.

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?

A piece of string is $36\text{ cm}$ long. What will be the length of each side, if it is used to form:

a. A square,

b. A triangle with all sides of equal length, and

c. A hexagon (a six sided closed figure) with sides of equal length?

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

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

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

Think and mark the positions as directed—

a. Mark 'A' at the point where Akshi will be after she ran 250 m. b. Mark 'B' at the point where Akshi will be after she ran 500 m. c. Now, Akshi ran 1000 m. How many full rounds has she finished running around her track? Mark her position as 'C'. d. Mark 'X' at the point where Toshi will be after she ran 250 m. e. Mark 'Y' at the point where Toshi will be after she ran 500 m. f. Now, Toshi ran 1000 m. How many full rounds has she finished running around her track? Mark her position as 'Z'.

Q10

The area of a rectangular garden $25 \text{ m}$ long is $300 \text{ sq m}$ . What is the width of the garden?

Q11

What is the cost of tiling a rectangular plot of land $500\text{ m}$ long and $200\text{ m}$ wide at the rate of $\text{₹}8$ per hundred $\text{sq m}$ ?

Q12

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?

Q13

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

a. b.

Q14

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

Q15

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

Q16

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

Q17

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

Q18

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

Hint: In the tangram pieces, by placing the shapes over each other, we can find out that Shapes A and B have the same area, Shapes C and E have the same area. You would have also figured out that Shape D can be exactly covered using Shapes C and E, which means Shape D has twice the area of Shape C or shape E, etc.

Q19

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

Q20

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.

Q21

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

Q22

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

Q23

Give the dimensions of a rectangle whose area is the sum of the areas of these two rectangles having measurements: $5\text{ m} \times 10\text{ m}$ and $2\text{ m} \times 7\text{ m}$ .

Q24

The area of a rectangular garden that is $50\text{ m}$ long is $1000\text{ sq m}$ . Find the width of the garden.

Q25

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

Q26

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

Q27

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.

Q28

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?

Q29

Draw a rectangle of size $12 \text{ units} \times 8 \text{ units}$ . Draw another rectangle inside it, without touching the outer rectangle that occupies exactly half the area.

Q30

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 $1\dfrac{1}{2}$ 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.