The area of the equilateral triangle, in which three coins of radius 1 cm are placed, as shown in the figure, is
- A.(a) sq cm
- B.(b) sq cm
- C.(c) sq cm
- D.(d) sq cm
Show answer and solution
Correct option: A
Solution
Each coin has radius 1 cm. The centers of the three coins form a smaller equilateral triangle. Each center is at distance 1/sin(60°) = 2/√3 from... let me think more carefully.
The triangle has coins of radius 1 placed in the corners. At each corner (60° angle), the incircle of the corner touches both sides. The center of a coin in a 60° corner is at distance r/sin(30°) = 1/(1/2) = 2 cm from the vertex along the angle bisector, and the center is at distance r/tan(30°) = √3 cm from each side.
The side of the equilateral triangle: each side is tangent to two corner coins. The distance from each vertex to the tangent point is r/tan(30°) = 1·√3 = √3 cm. The middle portion of each side (between the two tangent points from adjacent corners) equals the distance between the two centers projected onto the side minus nothing — actually the side length = √3 + √3 + (distance between centers along the side direction).
The two centers near one side are each √3 from the nearest vertex, and the centers are separated by 2r = 2 cm (since both coins touch each other, as shown in figure). So side length = √3 + 2 + √3 = 2 + 2√3.
Area of equilateral triangle with side a = 2 + 2√3:
Area = (√3/4)·a² = (√3/4)·(2 + 2√3)² = (√3/4)·4·(1 + √3)² = √3·(1 +
The triangle has coins of radius 1 placed in the corners. At each corner (60° angle), the incircle of the corner touches both sides. The center of a coin in a 60° corner is at distance r/sin(30°) = 1/(1/2) = 2 cm from the vertex along the angle bisector, and the center is at distance r/tan(30°) = √3 cm from each side.
The side of the equilateral triangle: each side is tangent to two corner coins. The distance from each vertex to the tangent point is r/tan(30°) = 1·√3 = √3 cm. The middle portion of each side (between the two tangent points from adjacent corners) equals the distance between the two centers projected onto the side minus nothing — actually the side length = √3 + √3 + (distance between centers along the side direction).
The two centers near one side are each √3 from the nearest vertex, and the centers are separated by 2r = 2 cm (since both coins touch each other, as shown in figure). So side length = √3 + 2 + √3 = 2 + 2√3.
Area of equilateral triangle with side a = 2 + 2√3:
Area = (√3/4)·a² = (√3/4)·(2 + 2√3)² = (√3/4)·4·(1 + √3)² = √3·(1 +
[AIEEE 2005]