Note: If are the sides of the triangle and s is the semi perimeter, then its area if given by where . This is **Heron’s Formula**.

Question 1: Find the area of a triangle whose sides are respectively and .

Answer:

Here and

and

Therefore area of triangle

Question 2: Find the area of a triangle whose sides are and .

Answer:

Here and

and

Therefore area of triangle

Question 3: Find the area of a triangle two sides of which are and and the perimeter is .

Answer:

Perimeter

Therefore third side

Here and

and

Therefore area of triangle

Question 4: In a and . Find the area of and hence its altitude on .

Answer:

Here and

and

Therefore area of triangle

Altitude

Question 5: The perimeter of a triangular field is and its sides are in the ratio . Find the area of the triangle.

Answer:

Ratio of sides

Perimeter

Therefore

Hence the sides are and

Here and

and

Therefore area of triangle

Question 6: The perimeter of a right triangle is . If its sides are in the ratio . Find the area of the triangle.

Answer:

Ratio of sides

Perimeter

Therefore

Hence the sides are and

Here and

and

Therefore area of triangle

Question 7: The perimeter of a triangular field is . If two of its sides are and , find the length of perpendicular on the side of length from the opposite vertex.

Answer:

Perimeter

Therefore third side

Here and

and

Therefore area of triangle

Altitude

Question 8: A triangle has sides and long. Find its area. Also, find the smallest of its altitudes.

Answer:

Here and

and

Therefore area of triangle

Smallest Altitude

Question 9: The lengths of the sides of a triangle are in the ratio and its perimeter is . Find the area of the triangle and the height corresponding to the longest side.

Answer:

Ratio of sides

Perimeter

Therefore

Hence the sides are and

Here and

and

Therefore area of triangle

Smallest Altitude

Question 10: The perimeter of an isosceles triangle is and its base is times each of the equal sides. Find length of each side of the triangle, area of the triangle and the height of the triangle.

Answer:

Perimeter

Therefore

Hence the sides are and

Here and

and

Therefore area of triangle

Altitude

Question 11: Find the area of a quadrilateral is which and

Answer:

Area of Area of Area of

For

Here and

and

Therefore area of triangle

For

Here and

and

Therefore area of triangle

Therefore Area of

Question 12: The sides of a quadrangular field, taken in order are are respectively. The angle contained by the last two sides is a right angle. Find its area.

Answer:

Area of Area of Area of

For

Here and

and

Therefore area of triangle

For

Here and

and

Therefore area of triangle

Therefore Area of

Question 13: The sides of a quadrilateral, taken in order are and meters respectively, and the angle contained by the first two sides is a right angle. Find its area.

Answer:

Area of Area of Area of

For

Here and

and

Therefore area of triangle

For

Here and

and

Therefore area of triangle

Therefore Area of

Question 14: A park, in the shape of a quadrilateral , has and . How much area does it occupy?

Answer:

Area of Area of Area of

For

Here and

and

Therefore area of triangle

For

Here and

and

Therefore area of triangle

Therefore Area of

Question 15: Two parallel side of a trapezium are and and other sides are and . Find the area of the trapezium.

Answer:

Area of Area of Area of

For

Here and

and

Therefore area of triangle

Altitude

Area of

Therefore Area of

Question 16: Find the area of a rhombus whose perimeter is and one of whose diagonal is .

Answer:

Perimeter

Therefore Side

Area of Area of Area of

For

Here and

and

Therefore area of triangle

For

Here and

and

Therefore area of triangle

Therefore Area of

Question 17: A rhombus sheet, whose perimeter is and whose one diagonal is long, is painted on both sides at the rate of . Find the cost of painting.

Answer:

Perimeter

Therefore Side

Area of Area of Area of

For

Here and

and

Therefore area of triangle

For

Here and

and

Therefore area of triangle

Therefore Area of

Therefore cost of painting

Question 18: Find the area of a quadrilateral in which and forms an equilateral triangle whose each side is equal to .

Answer:

Area of Area of Area of

For

Here and

and

Therefore area of triangle

For

Here and

and

Therefore area of triangle

Therefore Area of

Question 19: Find the area of a quadrilateral in which and diagonal .

Answer:

Area of Area of Area of

For

Here and

and

Therefore area of triangle

For

Here and

and

Therefore area of triangle

Therefore Area of

Question 20: Find the perimeter and area of the quadrilateral in which and .

Answer:

Perimeter

Area of Area of Area of

For

Here and

and

Therefore area of triangle

For

Here and

and

Therefore area of triangle

Therefore Area of

Question 21: The adjacent sides of a parallelogram measures and , and the diagonal measures . Find the area of the parallelogram.

Answer:

Area of Area of Area of

For

Here and

and

Therefore area of triangle

For

Here and

and

Therefore area of triangle

Therefore Area of