Perimeter and Area | A

Question 3

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.

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Solution
Understand the Question
  • A regular polygon is a closed geometric shape where all sides are of equal length and all interior angles are equal.
  • The perimeter is the total distance around the boundary of a closed figure, found by adding the lengths of all its sides.
  • Since all sides of a regular polygon are equal, adding the same length nn times is equivalent to multiplying the length of one side by the number of sides nn.

Step 1 · Perimeter of a Square Table

A square table top has 44 equal sides. Let each side length be 1 m1\text{ m}.Diagram 1

Perimeter=Side length+Side length+Side length+Side length=1 m+1 m+1 m+1 m=4×1 m=4 m\begin{aligned} \text{Perimeter} &= \text{Side length} + \text{Side length} + \text{Side length} + \text{Side length} \\[0.6em] &= 1\text{ m} + 1\text{ m} + 1\text{ m} + 1\text{ m} \\[0.6em] &= 4 \times 1\text{ m} = 4\text{ m} \end{aligned}

Step 2 · Perimeter of an Equilateral Triangular Clock

An equilateral triangular clock has 33 equal sides. Let each side length be 30 cm30\text{ cm}.Diagram 2

Perimeter=Side length+Side length+Side length=30 cm+30 cm+30 cm=3×30 cm=90 cm\begin{aligned} \text{Perimeter} &= \text{Side length} + \text{Side length} + \text{Side length} \\[0.6em] &= 30\text{ cm} + 30\text{ cm} + 30\text{ cm} \\[0.6em] &= 3 \times 30\text{ cm} = 90\text{ cm} \end{aligned}

Step 3 · Perimeter of a Regular Hexagonal Tile

A regular hexagonal tile has 66 equal sides. Let each side length be 10 cm10\text{ cm}.$$ \begin{aligned} \text{Perimeter} &= \text{Side length} + \text{Side length} + \text{Side length} + \text{Side length} + \text{Side length} + \text{Side length} \[0.6em] &= 10\text{ cm} + 10\text{ cm} + 10\text{ cm} + 10\text{ cm} + 10\text{ cm} + 10\text{ cm} \[0.6em] &= 6 \times 10\text{ cm} = 60\text{ cm} \end{aligned}

Step 4 · Generalisation for Regular Polygons

In each regular polygon, the perimeter is obtained by multiplying the number of sides by the length of one side.

Let nn be the number of sides and ss be the length of each side:

Perimeter of a Regular Polygon=Number of sides×Side length=n×s\text{Perimeter of a Regular Polygon} = \text{Number of sides} \times \text{Side length} = n \times s

Answer

Perimeter of a Regular Polygon=(Number of sides)×(Side length)\text{Perimeter of a Regular Polygon} = (\text{Number of sides}) \times (\text{Side length})

Common Mistakes
  • Applying to Irregular Shapes: The formula Perimeter=n×s\text{Perimeter} = n \times s only holds for regular polygons where all sides are equal. For irregular polygons, each side length must be added individually.
  • Omitting Units: Forgetting to write the appropriate unit of measurement (such as m\text{m} or cm\text{cm}) alongside the numerical value of the perimeter.

More questions in A

Q1

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 m100\text{ m} each side and outer track of 150 m150\text{ 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 m350\text{ 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 m350\text{ m}. Mark the starting points of the runner on the inner track as 'A' and the runner on the outer track as 'B'.

Q2

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.

Q3

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.

Q4

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.

Q5

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.

Q6

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

Q7

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

Q8

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?

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.

Q9

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?

Try this with more rectangles having different dimensions. You can check this for a square as well.

  • Can you draw any inferences from this exercise? Please write it here.
Q10

Draw suitable triangles on grid paper to verify your inferences and relationships observed in the above exercises.

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