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Area = bh

P = 2(b+h)

` ````
```Where:
- b is the breadth
- h is the height

A parallelogram is a four-sided shape with opposite sides that are parallel. The altitude of a parallelogram is the perpendicular distance from a base to the opposite side. The base of a parallelogram is any one of its sides.

Area of a Parallelogram

The formula for the area of a parallelogram is given by:

A = b \times h

Where:

- *A* is the area of the parallelogram,

- *b* is the length of the base, and

- *h* is the altitude of the parallelogram.

Perimeter of a Parallelogram

The perimeter of a parallelogram is the sum of the lengths of all its sides. Since opposite sides of a parallelogram are equal in length, we can calculate the perimeter using the following formula:

P = 2 \times (b + s)

Where:

- *P* is the perimeter of the parallelogram,

- *b* is the length of the base, and

- *s* is the length of one of the sides.

The area of a parallelogram can be calculated by multiplying the base by the altitude, while the perimeter can be calculated by summing the lengths of all sides (assuming equal side lengths in a parallelogram). These formulas are crucial in determining the size and dimensions of parallelograms in various mathematical problems.

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Parallelogram

Given a parallelogram with base *b = 4 units* and altitude *h = 3 units*, calculate the area and perimeter of the parallelogram.

1. Calculate the area:

A = 4 \times 3 = 12 square units

Therefore, the area of the parallelogram is *12 square units*.

2. Calculate the perimeter (assume all sides are equal):

P = 2 \times (4 + 4) = 16 units

Therefore, the perimeter of the parallelogram is *16 units*.

Question: Given a parallelogram with sides of length 5 cm and 8 cm, and an angle of 60 degrees between these sides, find the area of the parallelogram.

Solution:

\text{Area of a parallelogram} = \text{base} \times \text{height}

= 8 \times 5 \times \sin(60^\circ)

= 8 \times 5 \times \frac{\sqrt{3}}{2}

= 20\sqrt{3} square cm

Question: The diagonals of a parallelogram are 10 cm and 12 cm in length. If the angle between the diagonals is 45 degrees, calculate the area of the parallelogram.

Solution:

\text{Area of a parallelogram} = \frac{1}{2} \times \text{diagonal}_1 \times \text{diagonal}_2 \times \sin(45^\circ)

= \frac{1}{2} \times 10 \times 12 \times \sin(45^\circ)

= \frac{1}{2} \times 10 \times 12 \times \frac{\sqrt{2}}{2}

= 60\sqrt{2} square cm

Question: If the base of a parallelogram is 6 cm and the height is 4 cm, find the area of the parallelogram.

Solution:

\text{Area of a parallelogram} = \text{base} \times \text{height}

= 6 \times 4

= 24 square cm

Question: The area of a parallelogram is 72 square units and its base is 9 units long. Find the height of the parallelogram.

Solution:

Let the height of the parallelogram be h. From the given information, we have:

72 = 9 \times h

h = \frac{72}{9}

h = 8 units

Question: If the area of a parallelogram is 48 cm squared and its height is 6 cm, calculate the length of its base.

Solution:

Let the length of the base of the parallelogram be b. From the given information, we have:

48 = b \times 6

b = \frac{48}{6}

b = 8 cm

More Worked Examples on Parallelogram
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1. Find the area of a parallelogram with base 8 and height 5.

2. If the base of a parallelogram is 12 and the height is 6, what is its area?

3. Given that the area of a parallelogram is 35 and its height is 7, find the length of its base.

4. Calculate the area of a parallelogram with base 10 and height 4.

5. If the area of a parallelogram is 48 and its height is 8, what is the length of its base?

6. Find the area of a parallelogram with base 6 and height 9.

7. If the base of a parallelogram is 16 and the height is 3, what is its area?

8. Given that the area of a parallelogram is 72 and its height is 6, find the length of its base.

9. Calculate the area of a parallelogram with base 7 and height 10.

10. If the area of a parallelogram is 60 and its height is 5, what is the length of its base?

11. Find the area of a parallelogram with base 9 and height 4.

12. If the base of a parallelogram is 20 and the height is 2, what is its area?

13. Given that the area of a parallelogram is 45 and its height is 9, find the length of its base.

14. Calculate the area of a parallelogram with base 11 and height 8.

15. If the area of a parallelogram is 56 and its height is 7, what is the length of its base?

1. Area = 8 \times 5 = 40

2. Area = 12 \times 6 = 72

3. Base = \frac{35}{7} = 5

4. Area = 10 \times 4 = 40

5. Base = \frac{48}{8} = 6

6. Area = 6 \times 9 = 54

7. Area = 16 \times 3 = 48

8. Base = \frac{72}{6} = 12

9. Area = 7 \times 10 = 70

10. Base = \frac{60}{5} = 12

11. Area = 9 \times 4 = 36

12. Area = 20 \times 2 = 40

13. Base = \frac{45}{9} = 5

14. Area = 11 \times 8 = 88

15. Base = \frac{56}{7} = 8

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