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Perimeter and Area

Perimeter and Area

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Study Action Plan (Sorted by Difficulty)
Area Estimation Techniques
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Area of Irregular Shapes
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Area of Rectangles and Squares
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Area of Triangles
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Comparing Perimeters and Areas
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Perimeter and Area Relationship
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Perimeter of Polygons
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Practical Applications of Perimeter and Area
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Regular Polygons
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Learning Objectives

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  • Understand the concept of perimeter and area of various geometric shapes.
  • Calculate the perimeter of rectangles, squares, and triangles using appropriate formulas.
  • Determine the area of rectangles and squares using the formula: Area = length × width (for rectangles) and Area = side × side (for squares).
  • Analyze and compare the perimeters and areas of different shapes to identify relationships.
  • Solve real-life problems involving perimeter and area, including fencing and tiling scenarios.
  • Explore the relationship between perimeter and area through practical exercises and examples.

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Revision Notes & Summary

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Chapter 6 - Perimeter and Area

6.1 Perimeter

  • Definition: The perimeter of any closed plane figure is the distance covered along its boundary when you go around it once.
  • Formula for Polygon: The perimeter of a polygon = the sum of the lengths of all its sides.

Examples:

  1. Rectangle:
    • Perimeter = 2 × (length + breadth)
    • Example: For a rectangle with length 12 cm and breadth 8 cm:
      • Perimeter = 2 × (12 cm + 8 cm) = 40 cm
  2. Square:
    • Perimeter = 4 × length of a side
    • Example: For a square with side 1 m:
      • Perimeter = 4 × 1 m = 4 m

6.2 Area

  • Definition: The area of a closed figure is the measure of the region enclosed by the figure.
  • Units: Area is generally measured in square units.
  • Formulas:
    • Area of a rectangle = length × width
    • Area of a square = length of a side × length of a side

Examples:

  1. Rectangle:
    • Example: A floor is 5 m long and 4 m wide:
      • Area = 5 m × 4 m = 20 sq m
  2. Square:
    • Example: A square carpet of sides 3 m:
      • Area = 3 m × 3 m = 9 sq m

6.3 Area of a Triangle

  • Observation Exercise: Cut a rectangle along its diagonal to form two triangles. Check if they overlap and if they have the same area.
  • Relationship: Explore the relationship between the areas of rectangles and triangles.

Important Notes

  • Two closed figures can have the same area with different perimeters or the same perimeter with different areas.
  • Areas can be estimated by breaking them into unit squares or other shapes whose areas can be calculated.
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Practice Test – MCQs, True/False

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Multiple Choice Questions

A.

25 m

B.

50 m

C.

30 m

D.

40 m
Correct Answer: B

Solution:

The perimeter of a rectangle is calculated as 2×(length+width)2 \times (\text{length} + \text{width}). Therefore, 2×(15+10)=502 \times (15 + 10) = 50 m.

Chapter Concept:

Practical Applications of Perimeter and Area

A.

34s2\frac{\sqrt{3}}{4} s^2

B.

s2s^2

C.

12s2\frac{1}{2} s^2

D.

34s2\frac{3}{4} s^2
Correct Answer: A

Solution:

The area of an equilateral triangle is calculated using the formula 34s2\frac{\sqrt{3}}{4} s^2, where ss is the side length.

Chapter Concept:

Area of Triangles

A.

6 cm and 6 cm

B.

8 cm and 4 cm

C.

10 cm and 2 cm

D.

12 cm and 0 cm
Correct Answer: B

Solution:

The perimeter of the triangle is 7 cm + 8 cm + 9 cm = 24 cm. For the rectangle to have the same perimeter, 2(l+b)=242(l + b) = 24. Solving for possible integer values, l+b=12l + b = 12. Possible dimensions are 8 cm and 4 cm.

Chapter Concept:

Area of Triangles

A.

5 cm

B.

6 cm

C.

4 cm

D.

3 cm
Correct Answer: A

Solution:

Using the formula for the perimeter of a rectangle, 2×(length+breadth)=142 \times (\text{length} + \text{breadth}) = 14. Solving for length gives 2×(length+2)=142 \times (\text{length} + 2) = 14, so length=5\text{length} = 5 cm.

Chapter Concept:

Comparing Perimeters and Areas

A.

10 square meters

B.

15 square meters

C.

20 square meters

D.

25 square meters
Correct Answer: D

Solution:

The area of a square is calculated by squaring the length of one of its sides. Therefore, 5×5=255 \times 5 = 25 square meters.

Chapter Concept:

Area of Rectangles and Squares

A.

16 m x 4 m

B.

32 m x 2 m

C.

8 m x 8 m

D.

All of the above
Correct Answer: D

Solution:

The area of the first rectangle is 5×10=505 \times 10 = 50 sq m and the second is 2×7=142 \times 7 = 14 sq m. The total area is 50+14=6450 + 14 = 64 sq m. Possible dimensions for a rectangle with this area include 16 m x 4 m, 32 m x 2 m, and 8 m x 8 m, making option D correct.

Chapter Concept:

Perimeter and Area Relationship

A.

The two triangles have different areas.

B.

The two triangles have the same area.

C.

One triangle is twice the area of the other.

D.

The areas depend on the angles of the rectangle.
Correct Answer: B

Solution:

When a rectangle is divided by a diagonal, the two resulting triangles have the same area because they share the same base and height.

Chapter Concept:

Area of Triangles

A.

36 cm

B.

30 cm

C.

42 cm

D.

48 cm
Correct Answer: A

Solution:

A regular hexagon has six equal sides. Therefore, the perimeter is calculated as 6 times the length of one side. Thus, the perimeter is 6×6 cm=36 cm6 \times 6 \text{ cm} = 36 \text{ cm}.

Chapter Concept:

Regular Polygons

A.

10 m

B.

15 m

C.

20 m

D.

25 m
Correct Answer: C

Solution:

The perimeter of a square is 4 times the length of one side. Thus, 4×5 m=20 m4 \times 5 \text{ m} = 20 \text{ m}.

Chapter Concept:

Perimeter of Polygons

A.

36 cm

B.

30 cm

C.

42 cm

D.

48 cm
Correct Answer: A

Solution:

The perimeter of a regular polygon is calculated by multiplying the number of sides by the length of one side. For a hexagon with 6 sides, each 6 cm long, the perimeter is 6×6=366 \times 6 = 36 cm.

Chapter Concept:

Regular Polygons

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Perimeter and Area

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True or False

Correct Answer: True

Solution:

The area of a rectangle is indeed calculated by multiplying its length by its width, as stated in the excerpts.

Chapter Concept :

Area of Rectangles and Squares

Correct Answer: True

Solution:

Grid paper and unit squares are used to estimate the area of shapes by counting the number of squares that fit within the shape.

Chapter Concept :

Area Estimation Techniques

Correct Answer: True

Solution:

The perimeter of a rectangle is given by the formula: 2×(length+breadth)2 \times (\text{length} + \text{breadth}). This formula is derived from adding the lengths of all four sides of the rectangle.

Chapter Concept :

Perimeter of Polygons

Correct Answer: True

Solution:

To estimate or calculate the area of an irregular shape, it can be divided into simpler shapes such as rectangles and triangles, whose areas can be easily calculated.

Chapter Concept :

Area of Irregular Shapes

Correct Answer: False

Solution:

The perimeter of a square is calculated as 4×side length4 \times \text{side length}. For a side length of 3 m, the perimeter is 4×3=124 \times 3 = 12 m, not 9 m.

Chapter Concept :

Practical Applications of Perimeter and Area

Correct Answer: True

Solution:

It is possible for two figures to have the same area but different perimeters, as the arrangement of the sides affects the perimeter.

Chapter Concept :

Perimeter and Area Relationship

Correct Answer: True

Solution:

The perimeter of a square is calculated as 4 times the length of one of its sides.

Chapter Concept :

Comparing Perimeters and Areas

Correct Answer: True

Solution:

The area of a triangle is calculated as half of the base times the height, which is exactly half of the area of a rectangle with the same base and height.

Chapter Concept :

Area of Triangles

Correct Answer: False

Solution:

The perimeter of a square is 4×side length4 \times \text{side length}. For a side length of 1 m, the perimeter is 4×1=44 \times 1 = 4 m.

Chapter Concept :

Practical Applications of Perimeter and Area

Correct Answer: True

Solution:

Irregular shapes can be estimated by dividing them into simpler shapes such as rectangles and triangles, whose areas can be calculated using known formulas.

Chapter Concept :

Area of Irregular Shapes

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