The formula for the area of an equilateral triangle with side length a is given by (√3)/(4) a². We have determined that the side length a of our triangle is 2 units.

Substitute the side length a=2 into the area formula: Area = (√3)/(4) (2)² = (√3)/(4) × 4 = √3. Thus, the area of the triangle is √3 square units.

Three circles of radius 1 touch each other externally. Find the area of the triangle formed by joining their centers.

Answer: The area of the triangle formed by joining their centers is $\sqrt{3}$ square units.

Step-by-step solution

Step 1: Identify the vertices of the triangle

When three circles touch each other externally, the distance between the centers of any two circles is equal to the sum of their radii. Since all three circles have the same radius, the triangle formed by joining their centers will have equal side lengths.

Step 2: Determine the side length of the triangle

Each circle has a radius of 1 unit. Since the circles touch externally, the distance between the centers of any two circles is $1 + 1 = 2$ units. Therefore, the triangle formed by joining their centers is an equilateral triangle with side length 2 units.

Step 3: Apply the formula for the area of an equilateral triangle

The formula for the area of an equilateral triangle with side length $a$ is given by $\frac{\sqrt{3}}{4} a^2$ . We have determined that the side length $a$ of our triangle is 2 units.

Step 4: Calculate the area

Substitute the side length $a=2$ into the area formula: $\text{Area} = \frac{\sqrt{3}}{4} (2)^2 = \frac{\sqrt{3}}{4} \times 4 = \sqrt{3}$ . Thus, the area of the triangle is $\sqrt{3}$ square units.

More from this chapter

Solve your own maths question free →