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Heron’s Formula
Now, let us consider a scalene triangle where the lengths of its sides are known but the height is not known.
To find its area we require the height of corresponding to a base. But height is not known. Heron (10 A.D. – 75 A.D.), an encyclopedic writer in Applied Mathematics gave a formula for finding the area of a triangle when we know the lengths of all three sides. It is called "Heron's Formula" after Hero of Alexandria. Heron's formula relates to the side lengths, perimeter and area of a triangle.
Heron’s Formula –
Let a, b, c denotes the lengths of the sides of a triangle ABC.
Interesting Fact: This formula is applicable to all types of triangles whether it is right angled or equilateral or isosceles.
Example 1: Find the area of a triangle whose sides are 12cm, 14cm and 16cm.
Example 2: Find the area of a triangle, two sides of which are 13cm and 14cm and the perimeter is 42cm.
Example 3: An isosceles triangle has perimeter 50cm and each of the equal sides is 15cm. Find the area of the triangle.
Example 4: The perimeter of a triangular plot is 360 m and its sides are in the ratio 10:12:14. Find the area of the triangle.
Example 5: A triangle has sides 9cm, 12cm and 15cm. Find the length of perpendicular from the opposite vertex to the side whose length is 15 cm.
Example 6: A triangular park has sides 100m, 100m and 160m. A gardener has to put a fence all around it and also plant grass inside. Find the cost of fencing it with barbed wire at the rate of rupees 10 per meter leaving a space 2m wide for a gate on one side and how much area does he need to plant?
Example 7: There is a triangular sign board in an area with a message “Danger Deep Excavations”. If the three sides of the board are 120cm, 80cm, and 50cm, find the area of the board.