Three circles of unit radii touch each other externally. They are inscribed in triangle. What will the area after leaving the area of those circles.
You can start here. Once you know the side length of the triangle, you can get its area.
https://owlcation.com/stem/Inscribed-Circles-and-Equilateral-Triangles-7-Hard-Geometry-Problems-and-Solutions
To find the area left after excluding the area of the three circles from the triangle, we can first calculate the area of the triangle and then subtract the areas of the circles.
Let's break down the solution step by step:
Step 1: Calculate the area of the triangle.
We know that the circles are externally tangent to each other. Therefore, we can draw three radii from the center of each circle to the points of tangency, forming an equilateral triangle. Since the radii of the circles are of unit length, the sides of the equilateral triangle will also be of unit length.
The area of an equilateral triangle can be calculated using the formula: Area = (square root of 3 / 4) * (side)^2
In this case, the side length is 1, so the area of the equilateral triangle is:
Area of the triangle = (sqrt(3) / 4) * (1)^2 = sqrt(3) / 4
Step 2: Calculate the area of one circle.
The area of a circle can be calculated using the formula: Area = pi * (radius)^2
In this case, the radius of each circle is 1, so the area of one circle is:
Area of one circle = pi * (1)^2 = pi
Step 3: Calculate the total area of three circles.
Since there are three circles, we need to multiply the area of one circle by 3 to get the total area of three circles:
Total area of three circles = 3 * pi
Step 4: Calculate the area left after excluding the circles.
To find the area left after excluding the circles, we subtract the total area of three circles from the area of the triangle:
Area left = Area of the triangle - Total area of three circles
Area left = sqrt(3) / 4 - 3 * pi
So, the area left after excluding the area of those circles from the triangle is sqrt(3) / 4 - 3 * pi.