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Mensuration - Perimeter of Rectangles and Regular Polygons

Grade 6ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The perimeter of a closed plane figure is the total length of its boundary. For a rectangle, it is the sum of all four sides, where opposite sides are equal.

Rectangle diagram showing length and breadth labels on all sides.
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A regular polygon has all sides of equal length and all interior angles of equal measure. The perimeter is found by multiplying the number of sides (nn) by the length of one side (ss).

Regular pentagon with all five sides labeled 's'.
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For a square, which is a regular polygon with four sides, the formula simplifies to P=4×sP = 4 \times s.

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To find the side of a regular polygon when the perimeter is known, we divide the perimeter by the number of sides: s=Pns = \frac{P}{n}.

Regular hexagon illustrating the relationship between perimeter and side length.

📐Formulae

Perimeter of a Rectangle=2×(l+b)\text{Perimeter of a Rectangle} = 2 \times (l + b) where ll is length and bb is breadth

Perimeter of a Square=4×s\text{Perimeter of a Square} = 4 \times s where ss is the length of a side

Perimeter of an Equilateral Triangle=3×s\text{Perimeter of an Equilateral Triangle} = 3 \times s

Perimeter of a Regular Pentagon=5×s\text{Perimeter of a Regular Pentagon} = 5 \times s

Perimeter of a Regular Hexagon=6×s\text{Perimeter of a Regular Hexagon} = 6 \times s

Perimeter of a Regular Polygon=n×s\text{Perimeter of a Regular Polygon} = n \times s where nn is the number of sides

Side of a Regular Polygon=Perimetern\text{Side of a Regular Polygon} = \frac{\text{Perimeter}}{n}

💡Examples

Problem 1:

Calculate the perimeter of a rectangular park whose length is 45 m45\text{ m} and breadth is 32 m32\text{ m}.

Solution:

Given: Length (ll) = 45 m45\text{ m} Breadth (bb) = 32 m32\text{ m}

Using the formula: Perimeter=2×(l+b)\text{Perimeter} = 2 \times (l + b) Perimeter=2×(45+32)\text{Perimeter} = 2 \times (45 + 32) Perimeter=2×77\text{Perimeter} = 2 \times 77 Perimeter=154 m\text{Perimeter} = 154\text{ m}

Explanation:

To find the perimeter of a rectangle, we first add the length and the breadth together to find the sum of two adjacent sides. Then, we multiply that sum by 2 to include the other two equal opposite sides.

Problem 2:

The perimeter of a regular pentagon is 65 cm65\text{ cm}. Find the length of each side.

Solution:

Given: Perimeter (PP) = 65 cm65\text{ cm} Number of sides (nn) for a pentagon = 55

Using the formula: Side(s)=Perimetern\text{Side} (s) = \frac{\text{Perimeter}}{n} s=655s = \frac{65}{5} s=13 cms = 13\text{ cm}

Explanation:

Since a regular pentagon has 5 sides of equal length, we divide the total perimeter by 5 to find the length of one individual side.

Problem 3:

A wire in the shape of a square of side 12 cm12\text{ cm} is reshaped into a regular hexagon. Find the length of each side of the hexagon.

A square of side 12cm being converted into a regular hexagon.

Solution:

  1. First, find the total length of the wire by calculating the perimeter of the square: Perimeter of Square=4×side\text{Perimeter of Square} = 4 \times \text{side} Perimeter=4×12 cm=48 cm\text{Perimeter} = 4 \times 12\text{ cm} = 48\text{ cm}

  2. Since the same wire is used to make a regular hexagon, the perimeter of the hexagon is also 48 cm48\text{ cm}.

  3. A regular hexagon has n=6n = 6 sides. Calculate the side length: Side of Hexagon=Perimeter6\text{Side of Hexagon} = \frac{\text{Perimeter}}{6} Side=486=8 cm\text{Side} = \frac{48}{6} = 8\text{ cm}

Explanation:

Because the same wire is reshaped, its total length (perimeter) remains constant. We calculate the total length using the square's properties and then distribute that length equally among the 6 sides of the hexagon.

Problem 4:

Find the cost of fencing a rectangular field of length 120 m120\text{ m} and breadth 80 m80\text{ m} at the rate of Rs 1515 per metre.

Rectangular field with dimensions 120m by 80m and an outer line representing fencing.

Solution:

  1. Calculate the perimeter of the rectangular field: Perimeter=2×(l+b)\text{Perimeter} = 2 \times (l + b) Perimeter=2×(120+80)\text{Perimeter} = 2 \times (120 + 80) Perimeter=2×200=400 m\text{Perimeter} = 2 \times 200 = 400\text{ m}

  2. Calculate the total cost of fencing: Cost=Perimeter×Rate\text{Cost} = \text{Perimeter} \times \text{Rate} Cost=400×15=6000\text{Cost} = 400 \times 15 = 6000

The total cost of fencing is Rs 6000.

Explanation:

Fencing is done along the boundary of the field, so we calculate the perimeter. The total cost is the product of the total boundary length and the cost per unit length.