. In the diagram, the five circles have the same radii and touch each other as shown. The square joins the centres of the four outer circles. What is the ratio of the area of the shaded parts of all five circles to the area of the unshaded parts of all five circles? With full solution
To find the ratio of the shaded parts to the unshaded parts of all five circles, we need to determine the areas of both.
Let's start by finding the area of one circle. Since all the circles have the same radius, we only need to find the area of one and multiply it by five to represent all five circles.
The formula to find the area of a circle is A = πr^2, where A is the area and r is the radius.
Let's assume that the radius of each circle is represented by 'r'. Therefore, the area of one circle is A1 = πr^2.
To find the area of the shaded region, we need to subtract the area of the square from the area of the circle. The square has sides equal to the diameter of the outer circle (which is 2r in this case). The formula to find the area of a square is A = s^2, where A is the area and s is the side length.
Therefore, the area of the square is A_square = (2r)^2 = 4r^2.
Now, the shaded area of one circle is A_shaded = A1 - A_square.
The unshaded area of one circle is simply the remaining area, which is A_unshaded = A1 - A_shaded.
The ratio we are looking for is the ratio of the total shaded areas (5 times A_shaded) to the total unshaded areas (5 times A_unshaded).
Therefore, the ratio of the shaded parts to unshaded parts of all five circles is:
(A_shaded * 5) / (A_unshaded * 5)
Simplifying the expression, we get:
A_shaded / A_unshaded
Now, let's substitute the values we found for A_shaded and A_unshaded:
A_shaded = A1 - A_square
A_unshaded = A1 - A_shaded
Substituting these values, we get:
(A1 - A_square) / (A1 - (A1 - A_square))
Simplifying this expression further, we get:
A_square / A1
Substituting the values for A_square and A1:
(4r^2) / (πr^2)
The 'r^2' terms cancel out, and we are left with:
4 / π
So, the ratio of the shaded parts of all five circles to the unshaded parts of all five circles is 4/π.