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√34π ≈ 0.413.
Answer: Joining O to the vertices makes 3 triangles with two sides r and a 120° angle; each has height r2 and base √3r, area √34r2. Triangle = 3√34r2; ratio = 3√34π ≈ 0.413.
Step-by-step solution
Idea: Split the triangle from the centre into three equal triangles; use half of an equilateral triangle (a 30°-60°-90° triangle) to get their height and base.
- Join O to the vertices A, B, C. The three triangles OAB, OBC, OCA are congruent (all sides r, r, side of triangle), so each angle at O is 360° ÷ 3 = 120°.1 mark
- Drop OM ⊥ BC. It bisects ∠BOC, so ∠BOM = 60°: triangle OMB is half of an equilateral triangle of side r. So OM = r2 and BM = √(r2 − r24) = √32r, BC = √3r. Area(△OBC) = 12 × √3r × r2 = √34r2.1 mark
- Area(△ABC) = 3 × √34r2 = 3√34r2. Ratio = 3√34r2 ÷ πr2 = 3√34π ≈ 5.19612.566 ≈ 0.413.1 mark
Area of triangle : area of circle = (3√3/4)r² : πr² = 3√3/(4π) ≈ 0.413.
Check: Side √3r in √34 × side2 gives √34 × 3r2 = 3√34r2 ✓.
Answer to write in the exam
∠BOC = 360°3 = 120°; OM ⊥ BC ⇒ ∠BOM = 60°
OM = r2, BM = √32r ⇒ BC = √3r
ar(△OBC) = 12 × √3r × r2 = √34r2
ar(△ABC) = 3 × √34r2 = 3√34r2
∴ Ratio = 3√34π ≈ 0.413
Common mistakes that cost marks
- Taking the side of the triangle equal to r. That is true for a hexagon, not a triangle; the side is √3r.
- Taking the distance from O to a side as r; it is r2.
- Writing the ratio upside down (circle : triangle).
How this can come in the exam
MCQ (1 mark)
The side of an equilateral triangle inscribed in a circle of radius 6 cm is
- 6 cm
- 6√3 cm
- 3√3 cm
- 12 cm
Show answer
(B) 6√3 cm
Side = √3r = 6√3 cm.
Try one yourself
An equilateral triangle is inscribed in a circle of radius 4 cm. Find its area.
Show answer
3√34 × 16 = 12√3 ≈ 20.78 cm2.
More questions like this
- A square is inscribed in a circle of radius r. Show that the ratio of the area of the square to the area of the circle is equal to 2π ≈ 0.637.
- A hexagon is inscribed in a circle of radius r. Show that the ratio of the area of the hexagon to the area of the circle is equal to 3√32π ≈ 0.827. Can you see why the answer is exactly twice the answer for the inscribed equilateral triangle?
- Identities in algebra can sometimes be shown as area relationships. For example: The figure shown corresponds to the identity (a + b)2 = a2 + 2ab + b2. Do you see how? Draw figures corresponding to the identities (a + b)(a − b) = a2 − b2 and (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca.
- An isosceles triangle has perimeter 40 cm; the equal sides are 15 cm each. Find the area of the triangle.
- An isosceles triangle has base 10 cm, and its area is 60 cm2. What are the lengths of the equal sides?