Three circles of radius 1 touch each other externally. Find the area of the triangle formed by joining their centers. Find the numerical value if exact solution is
not possible.
Get the complete, step-by-step math solution for: "Three circles of radius 1 touch each other externally. Find the area of the triangle formed by joining their centers. Find the numerical value if exac...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Identify the vertices of the triangle
When three circles of the same radius touch each other externally, the distance between the centers of any two touching circles is equal to the sum of their radii. Since all circles have a radius of 1, the distance between any two centers will be 1+1=2.
Step 2: Determine the type of triangle
Since the distance between any two centers is 2 units, all three sides of the triangle formed by joining the centers are equal. This means the triangle is an equilateral triangle with side length a=2 units.
Step 3: Apply the area formula for an equilateral triangle
The formula for the area of an equilateral triangle with side length a is given by 43a2. We have determined that the side length a is 2 units.
Step 4: Calculate the area
Substitute the side length a=2 into the area formula: Area=43(2)2=43×4=3. The exact area of the triangle is 3 square units.
Step 5: Provide the numerical value
The numerical value of 3 is approximately 1.732.