Question

If △ A B C has side lengths a, b, f and f is the radhas of its circumeircle, then Area △ABC = abc/4r

Answer 1

The formula for the area of △ABC, given its side lengths a, b, and c, and the radius of its circumcircle r, is expressed as Area
`△ABC = \((abc)/(4r)\)`.

Step-by-step explanation:

The formula
`\(Area \, \triangle ABC = (abc)/(4r)\)` relates the area of a triangle to its side lengths (a, b, c) and the radius of its circumcircle (r). The derivation of this formula involves the use of the circumradius, which is the radius of the circle circumscribing the triangle.

The formula is derived from the general formula for the area of a triangle \(Area = \frac{1}{2} \times \text{base} \times \text{height}\). In the case of △ABC, the base and height are represented by the side lengths a and b, respectively.

The circumradius (r) comes into play as it is related to the height of the triangle. The use of the circumradius is crucial for triangles that may not have a perpendicular height from one side to the opposite vertex.

This formula is particularly useful in geometry and trigonometry applications where the circumradius and side lengths of a triangle are known, enabling the calculation of its area. Understanding and utilizing such formulas contribute to a deeper comprehension of geometric relationships and facilitate solving problems related to triangles and circles.