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\text{Given the sides} = a, b,c

\text{Area of triangle} = \sqrt{s (s-a) (s-b) (s-c)}

Given base and altitude( height) of a triangle

A = \frac{1}{2} \times b \times h

\text{Perimeter of triangle} = a + b + c

\text{Semi perimeter of triangle} = \frac{a + b + c}{2}

a, b, c are the lengths of the sides of the triangle

s is the semi perimeter of the triangle (s = (a + b + c)/2)

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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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Triangle

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 Triangle
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Question 1: Find the perimeter of a triangle with sides of lengths 4cm, 7cm, and 9cm.

Question 2: Calculate the area of a triangle with base 10cm and height 6cm.

Question 3: Determine the missing angle in a triangle with angles of 45 degrees and 70 degrees.

Question 4: If the sides of a triangle are 8cm, 15cm, and 17cm, is it a right triangle?

Question 5: What is the area of an equilateral triangle with side length 12cm?

Question 6: If a triangle has angles of 30 degrees, 60 degrees, and x degrees, what is the value of x?

Question 7: Determine the height of a triangle with base 9cm and area 27cm^2.

Question 8: Calculate the area of a triangle with side lengths 6cm, 8cm, and 10cm.

Question 9: Find the perimeter of a right triangle with legs of lengths 5cm and 12cm.

Question 10: Determine the missing side length in a triangle with sides of lengths 3cm, 4cm, and x cm.

Question 11: If two sides of a triangle have lengths 7cm and 9cm, and the included angle is 60 degrees, what is the length of the third side?

Question 12: Find the area of a triangle with sides of lengths 5cm, 12cm, and 13cm.

Question 13: Calculate the perimeter of an isosceles triangle with base 10cm and congruent sides of length 8cm.

Question 14: If the angles of a triangle are in the ratio 3:4:5, what is the measure of the largest angle?

Question 15: Determine the missing angle in a triangle with angles of 60 degrees and 80 degrees.

Answer 1: Perimeter = 4 + 7 + 9 = 20 cm

Answer 2: Area = \frac{1}{2} \times 10cm \times 6cm = 30 cm^2

Answer 3: x = 180 - 45 - 70 = 65 degrees

Answer 4: Yes, it is a right triangle.

Answer 5: Area = \frac{\sqrt{3}}{4} \times 12cm^2 = 36\sqrt{3} cm^2

Answer 6: x = 90 degrees

Answer 7: Height = \frac{2 \times Area}{base} = \frac{2 \times 27}{9} = 6 cm

Answer 8: Area = 24 cm^2

Answer 9: Perimeter = 5 + 12 + 13 = 30 cm

Answer 10: x = 5 cm

Answer 11: The length of the third side is 9cm.

Answer 12: Area = 10 cm^2

Answer 13: Perimeter = 8 + 8 + 10 = 26 cm

Answer 14: The largest angle measures 100 degrees.

Answer 15: Missing \, angle = 180 - 60 - 80 = 40 degrees

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