Perimeter and Area | IT

Question 14

Let's do something tricky now! 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.

Question diagram 1
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Solution
Understand the Question
  • Perimeter is the total distance around the outside boundary of a closed figure.
  • When a new square is inserted into an inner corner of the cross shape:
    • It covers 22 existing outer edges, removing them from the boundary.
    • It exposes 22 of its own outer edges, adding them to the boundary.
  • Since the number of edges gained equals the number of edges lost, the overall perimeter does not change.

Step 1 · Find the Side Length of Each Square

The original cross shape is formed by 55 squares with a total of 1212 equal boundary edges.

Side length of one square=24 units12=2 units\text{Side length of one square} = \dfrac{24\text{ units}}{12} = 2\text{ units}

Step 2 · Analyze the Change in Boundary Edges

Diagram 1

When the new square is placed in the corner:

  • Edges covered (removed from perimeter): 22 sides Decrease=2×2 units=4 units\text{Decrease} = 2 \times 2\text{ units} = 4\text{ units}

  • New exposed edges (added to perimeter): 22 sides Increase=2×2 units=4 units\text{Increase} = 2 \times 2\text{ units} = 4\text{ units}

Step 3 · Calculate the Net Change in Perimeter

Total change=Increase−Decrease=4 units−4 units=0 units\begin{aligned} \text{Total change} &= \text{Increase} - \text{Decrease} \\[0.6em] &= 4\text{ units} - 4\text{ units} \\[0.6em] &= 0\text{ units} \end{aligned}

Therefore, the perimeter remains the same.

Answer

There is no change in the perimeter (it stays the same at 24 units24\text{ units}).

Common Mistakes
  • Adding All Four Sides: Assuming that adding a square increases the perimeter by 44 sides, forgetting that two sides are glued inside and cover existing boundary edges.
  • Ignoring Covered Boundary: Forgetting to subtract the original outer edges that become internal edges when the new square is attached.

More questions in IT

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Q3

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Q4

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A rectangular paper chit of dimension 6 cm×4 cm6\text{ cm} \times 4\text{ cm} is cut as shown into two equal pieces. These two pieces are joined in different ways.

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

Q5

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Q6

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Q7

Look at the figures below and guess which one of them has a larger area.

Q8

Find the area of the following figures.

Q9

Now, see the figures below. 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.
Q10

Use your understanding from previous grades to calculate the area of any closed figure using grid paper and—

  1. Find the area of blue triangle BAD.
Q11

Use your understanding from previous grades to calculate the area of any closed figure using grid paper and—

  1. Find the area of red triangle ABE\text{ABE}.
Q12

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Q13

Using 9 unit squares, solve the following.

  1. What is the smallest perimeter possible?
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  3. Make a figure with a perimeter of 18 units.
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Q14

Let's do something tricky now! 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.

Q15

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?

Q16

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.

b. Find out the area of his house.

Q17

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.

b. Find out the area of his house.

What are the dimensions of all the different rooms in Sharan's house? Compare the areas and perimeters of Sharan's house and Charan's house.

Q18

Area Maze Puzzles

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

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