Inscribed equilateral triangle side is a = r√3. Triangle area is (√3/4) x a² = (√3/4) x 3r² = (3√3/4)r². Dividing by circle area πr² yields ratio (3√3) / (4π) ≈ 0.413.
An equilateral triangle is inscribed in a circle of radius r. Show that the ratio of the area of the triangle to the area of the circle is equal to (3√3) / (4π) ≈ 0.413.
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Let an equilateral triangle of side a be inscribed in a circle of radius r.
The circumradius of an equilateral triangle is related to its side by:
r = a / √3, so a = r√3.
Area of the equilateral triangle:
Area(triangle) = (√3 / 4) x a² = (√3 / 4) x (r√3)² = (3√3 / 4) x r².
Area of the circumscribing circle:
Area(circle) = π x r².
Ratio of areas:
Ratio = Area(triangle) / Area(circle) = [(3√3 / 4)r²] / [πr²] = (3√3) / (4π).
Evaluating numerically:
(3 x 1.732) / (4 x 3.1416) = 5.196 / 12.566 ≈ 0.413.
Hence proved.
For more NCERT Solutions of Class 9 Ganita Manjari Chapter 6 Measuring Space: Perimeter and Area Question Answer (2026-27)
https://www.tiwariacademy.com/ncert-solutions/class-9/maths/ganita-manjari-chapter-6/