If the circumradius R of triangle ABC is 5 cm, what is the maximum area of the t

If the circumradius R of triangle ABC is 5 cm, what is the maximum area of the triangle?

If the circumradius R of triangle ABC is 5 cm, what is the maximum area of the triangle?

Step 1: Understand that the circumradius R of a triangle is the radius of the circle that passes through all three vertices of the triangle.
Step 2: Note that the circumradius R is given as 5 cm.
Step 3: Recall the formula for the area of a triangle in terms of its circumradius R: Area = (abc)/(4R), where a, b, and c are the lengths of the sides of the triangle.
Step 4: Recognize that to achieve the maximum area for a triangle with a given circumradius, the triangle must be equilateral.
Step 5: For an equilateral triangle, the area can also be calculated using the formula: Area = (3√3/4) * (R^2).
Step 6: Substitute R = 5 cm into the area formula: Area = (3√3/4) * (5^2).
Step 7: Calculate 5^2, which is 25.
Step 8: Now substitute this value back into the area formula: Area = (3√3/4) * 25.
Step 9: Simplify the expression: Area = (75√3)/4 cm².
Step 10: Conclude that the maximum area of triangle ABC with circumradius 5 cm is (75√3)/4 cm².

Circumradius and Area of Triangle – Understanding the relationship between the circumradius of a triangle and its area, particularly for maximizing the area with respect to the triangle's shape.
Equilateral Triangle Properties – Recognizing that the maximum area for a given circumradius occurs when the triangle is equilateral.
Area Formula for Triangles – Applying the formula for the area of a triangle in terms of its sides and circumradius.