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

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

Options:

  1. 12.5 cm²
  2. 15 cm²
  3. 20 cm²
  4. 25 cm²

Correct Answer: 20 cm²

Solution: 

The maximum area of a triangle with circumradius R is given by the formula Area = (abc)/(4R). For maximum area, the triangle should be equilateral, thus Area = (3√3/4) * (R^2) = (3√3/4) * (5^2) = 25√3/4 cm².

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

Practice Questions

Q1
If the circumradius R of triangle ABC is 5 cm, what is the maximum area of the triangle?
  1. 12.5 cm²
  2. 15 cm²
  3. 20 cm²
  4. 25 cm²

Questions & Step-by-Step Solutions

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.
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