perimeter and area class 6

 Easy Level

Perimeter

1.     Find the perimeter of a square with side 6 cm.

2.     Find the perimeter of a rectangle with length 10 cm and breadth 5 cm.

3.     Find the perimeter of an equilateral triangle with side 7 cm.

4.     Find the perimeter of a regular pentagon with side 3 cm.

5.     A square has a perimeter of 20 cm. Find the length of one side.

Area

6.     Find the area of a square with side 8 cm.

7.     Find the area of a rectangle with length 12 cm and breadth 4 cm.

8.     Find the area of a triangle with base 9 cm and height 6 cm.

9.     A rectangle has an area of 48 cm². If its length is 8 cm, find its breadth.

10.                        Find the area of a square with perimeter 36 cm.

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Moderate Level

Perimeter

1.     The perimeter of a rhombus is 40 cm. Find the length of each side.

2.     Find the perimeter of a regular hexagon with side 5 cm.

3.     A rectangular field is 50 m long and 30 m wide. Find the cost of fencing the field at ₹15 per meter.

4.     A wire is bent into a square of side 10 cm. If the wire is rebent into a rectangle of length 12 cm, find the breadth of the rectangle.

5.     The perimeter of a triangle is 36 cm. If two sides are 12 cm and 14 cm, find the third side.

Area

6.     Find the area of a triangle with base 10 cm and height 8 cm.

7.     A rectangular garden is 15 m long and 10 m wide. A path of uniform width 1 m runs around the garden. Find the area of the path.

8.     The area of a square is 144 cm². Find its perimeter.

9.     A room is 12 m long and 8 m wide. A square carpet of side 6 m is laid on the floor. What fraction of the room's area is not carpeted?

10.                        Find the area of a rectangle whose perimeter is 40 cm and length is 12 cm.

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Hard Level

Perimeter

1.     A piece of string is 96 cm long. What will be the length of each side if the string is used to form:

o    An equilateral triangle

o    A regular hexagon

o    A square

2.     The perimeter of a rectangular plot is 240 m. If the length is twice the breadth, find the area of the plot.

3.     A wire is bent into a circle of radius 14 cm. If the same wire is rebent into a square, find the side of the square.

Area

4.     A rectangular field is 80 m long and 60 m wide. Two paths, each 3 m wide, run through the middle of the field, one parallel to the length and the other parallel to the breadth. Find:

o    The area of the paths.

o    The area of the remaining portion of the field.

5.     The length and breadth of a rectangular park are in the ratio 5:3. If the perimeter of the park is 160 m, find its area.

6.     A square and a rectangle have the same perimeter. If the side of the square is 15 cm and the length of the rectangle is 18 cm, find the breadth of the rectangle.

7.     A room is 15 m long and 10 m wide. A square carpet of side 8 m is laid on the floor. What fraction of the room's area is not carpeted?

Hard Level

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Perimeter

1. A piece of string is 96 cm long. Find the length of each side if the string is used to form:

a) An equilateral triangle

• Concept: An equilateral triangle has all three sides equal. The perimeter of the triangle is equal to the length of the string.

• Formula: $\text{Perimeter}=3\times \text{side}$

• Calculation: $96\text{ cm}=3\times \text{side}$ $\text{side}=\frac{96\text{ cm}}{3}=32\text{ cm}$

• Answer: The length of each side of the equilateral triangle is 32 cm.

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b) A regular hexagon

• Concept: A regular hexagon has six equal sides. The perimeter of the hexagon is equal to the length of the string.

• Formula: $\text{Perimeter}=6\times \text{side}$

• Calculation: $96\text{ cm}=6\times \text{side}$ $\text{side}=\frac{96\text{ cm}}{6}=16\text{ cm}$

• Answer: The length of each side of the regular hexagon is 16 cm.

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c) A square

• Concept: A square has four equal sides. The perimeter of the square is equal to the length of the string.

• Formula: $\text{Perimeter}=4\times \text{side}$

• Calculation: $96\text{ cm}=4\times \text{side}$ $\text{side}=\frac{96\text{ cm}}{4}=24\text{ cm}$

• Answer: The length of each side of the square is 24 cm.

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2. The perimeter of a rectangular plot is 240 m. If the length is twice the breadth, find the area of the plot.

• Concept: Let the breadth be $x$ meters. Then, the length is $2x$ meters. The perimeter of a rectangle is given by $2\times (\text{length}+\text{breadth})$.

• Formula: $\text{Perimeter}=2\times (\text{length}+\text{breadth})$

• Calculation: $240\text{ m}=2\times (2x+x)$ $240\text{ m}=2\times 3x$ $240\text{ m}=6x$ $x=\frac{240\text{ m}}{6}=40\text{ m}$

  So, the breadth is $40\text{ m}$ and the length is $2\times 40\text{ m}=80\text{ m}$.

  $\text{Area}=\text{length}\times \text{breadth}=80\text{ m}\times 40\text{ m}=3,200{\text{ m}}^2$

• Answer: The area of the plot is 3,200 m².

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3. A wire is bent into a circle of radius 14 cm. If the same wire is rebent into a square, find the side of the square.

• Concept: The circumference of the circle is equal to the perimeter of the square. First, find the circumference of the circle, then use it to find the side of the square.

• Formulas: $\text{Circumference of circle}=2\pi r$ $\text{Perimeter of square}=4\times \text{side}$

• Calculation: $\text{Circumference}=2\times \frac{22}{7}\times 14\text{ cm}=88\text{ cm}$

  The perimeter of the square is equal to the circumference of the circle: $88\text{ cm}=4\times \text{side}$ $\text{side}=\frac{88\text{ cm}}{4}=22\text{ cm}$

• Answer: The side of the square is 22 cm.

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Area

4. A rectangular field is 80 m long and 60 m wide. Two paths, each 3 m wide, run through the middle of the field, one parallel to the length and the other parallel to the breadth. Find:

a) The area of the paths.

• Concept: The total area of the paths is the sum of the areas of the two paths minus the overlapping area (since the intersection is counted twice).

• Calculation: $\text{Area of path parallel to length}=80\text{ m}\times 3\text{ m}=240{\text{ m}}^2$ $\text{Area of path parallel to breadth}=60\text{ m}\times 3\text{ m}=180{\text{ m}}^2$ $\text{Overlapping area}=3\text{ m}\times 3\text{ m}=9{\text{ m}}^2$

  $\text{Total area of paths}=240{\text{ m}}^2+180{\text{ m}}^2-9{\text{ m}}^2=411{\text{ m}}^2$

• Answer: The area of the paths is 411 m².

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b) The area of the remaining portion of the field.

• Concept: Subtract the area of the paths from the total area of the field.

• Calculation: $\text{Total area of field}=80\text{ m}\times 60\text{ m}=4,800{\text{ m}}^2$ $\text{Remaining area}=4,800{\text{ m}}^2-411{\text{ m}}^2=4,389{\text{ m}}^2$

• Answer: The area of the remaining portion of the field is 4,389 m².

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5. The length and breadth of a rectangular park are in the ratio 5:3. If the perimeter of the park is 160 m, find its area.

• Concept: Let the length be $5x$ meters and the breadth be $3x$ meters. Use the perimeter formula to find $x$, then calculate the area.

• Formula: $\text{Perimeter}=2\times (\text{length}+\text{breadth})$

• Calculation: $160\text{ m}=2\times (5x+3x)$ $160\text{ m}=2\times 8x$ $160\text{ m}=16x$ $x=\frac{160\text{ m}}{16}=10\text{ m}$

  So, the length is $5\times 10\text{ m}=50\text{ m}$ and the breadth is $3\times 10\text{ m}=30\text{ m}$.

  $\text{Area}=\text{length}\times \text{breadth}=50\text{ m}\times 30\text{ m}=1,500{\text{ m}}^2$

• Answer: The area of the park is 1,500 m².

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6. A square and a rectangle have the same perimeter. If the side of the square is 15 cm and the length of the rectangle is 18 cm, find the breadth of the rectangle.

• Concept: First, find the perimeter of the square, then use it to find the breadth of the rectangle.

• Formulas: $\text{Perimeter of square}=4\times \text{side}$ $\text{Perimeter of rectangle}=2\times (\text{length}+\text{breadth})$

• Calculation: $\text{Perimeter of square}=4\times 15\text{ cm}=60\text{ cm}$

  The perimeter of the rectangle is equal to the perimeter of the square: $60\text{ cm}=2\times (18\text{ cm}+\text{breadth})$ $30\text{ cm}=18\text{ cm}+\text{breadth}$ $\text{breadth}=30\text{ cm}-18\text{ cm}=12\text{ cm}$

• Answer: The breadth of the rectangle is 12 cm.

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7. A room is 15 m long and 10 m wide. A square carpet of side 8 m is laid on the floor. What fraction of the room's area is not carpeted?

• Concept: Find the area of the room and the carpet, then subtract the carpet area from the room area to find the uncarpeted area. Finally, express the uncarpeted area as a fraction of the room's area.

• Calculation: $\text{Area of room}=15\text{ m}\times 10\text{ m}=150{\text{ m}}^2$ $\text{Area of carpet}=8\text{ m}\times 8\text{ m}=64{\text{ m}}^2$ $\text{Uncarpeted area}=150{\text{ m}}^2-64{\text{ m}}^2=86{\text{ m}}^2$ $\text{Fraction}=\frac{86{\text{ m}}^2}{150{\text{ m}}^2}=\frac{43}{75}$

• Answer: The fraction of the room's area that is not carpeted is 43/75.