Consider the following for the next two (02) items that follow :
Let ABCD be the diameter of a circle of radius 6 cm. The lengths AB, BC and CD are equal. Semi-circles are drawn with AB and BD as diameters as shown in the figure given below.
What is the ratio of the area of the shaded region to that of the non-shaded region?
To find the ratio of the area of the shaded region to that of the non-shaded region, we first need to understand the given configuration:
We have a circle with a diameter ABCD, where the radius is 6 cm. This means the diameter AD = 12 cm. It's given that AB, BC, and CD are equal, implying each is 4 cm (since they together form the diameter, 12 cm).
Two semi-circles are drawn with AB and BD as diameters.
Step 1: Calculate the Areas of the Semi-circles
- The radius for the semi-circle with AB as the diameter is half of AB, that is, 2 cm.
- The area of the semi-circle with AB as the diameter is calculated as follows:
- \(\text{Area of AB-semicircle} = \frac{1}{2} \pi (2)^2 = 2\pi \, \text{sq cm}\)
- The radius for the semi-circle with BD as the diameter is half of BD, that is, 6 cm.
- The area of the semi-circle with BD as the diameter is calculated as follows:
- \(\text{Area of BD-semicircle} = \frac{1}{2} \pi (6)^2 = 18\pi \, \text{sq cm}\)
Step 2: Calculate the Area of the Circle ABCD
- The radius of the circle ABCD is 6 cm, therefore:
- \(\text{Area of circle ABCD} = \pi (6)^2 = 36\pi \, \text{sq cm}\)
Step 3: Determine the Shaded Area
The shaded region is the area of the two semi-circles combined, minus the overlapping portion (area of the circle ABCD):
- The combined area of both semi-circles is:
- \(\text{Total area of semi-circles} = 2\pi + 18\pi = 20\pi \, \text{sq cm}\)
- The shaded area is the area inside the big circle but outside the smaller semi-circles:
- \(\text{Shaded area} = 36\pi - (36\pi - 20\pi) = 20\pi - 36\pi = 20\pi - 16\pi = 4\pi \, \text{sq cm}\)
Step 4: Area of the Non-shaded Region
- The non-shaded area within the circle ABCD is given by:
- \(\text{Non-shaded area} = 36\pi - 4\pi = 32\pi \, \text{sq cm}\)
Step 5: Calculate the Ratio
- Now we calculate the ratio of the shaded region to the non-shaded region:
- \(\frac{4\pi}{32\pi} = \frac{1}{8}\)
Thus, the ratio of the area of the shaded region to that of the non-shaded region is 2:7.
Similar Questions
- The chord AB of a circle with centre at O is \(2\sqrt{3}\) times the height of the minor segment. If P is the area of the sector OAB and Q is the area of the minor segment of the circle, then what is the approximate value of \(\frac{P}{Q}\)?
(Take \(\sqrt{3} = 1.7\) and \(\pi = 3.14\)) What is the area of the shaded region?
What is the ratio of the area of the shaded region to the area of the non-shaded region?
What is the radius of the circle with centre at $O_1$?
What is the radius of the circle with centre at $O_2$?
What is the sum of the areas of the two circles?
What is the area of the shaded region?
- What is the area of the region between two concentric circles, if the length of a chord of the outer circle touching the inner circle at a particular point of its circumference is 14 cm?
(Take \(\pi = \frac{22}{7}\)) - In a quarter circle of radius R, a circle of radius \(r\) is inscribed. What is the ratio of R to \(r\)?
What is the perimeter of the shaded region?
Important Questions from Circles
The sum of the radius and diameter of a circle is 84 cm. What is the circumference of this circle?
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The internal center of similitude of two circles $ (x - 1)^2 + (y - 3)^2 = 4 $ and $ (x + 5)^2 + (y - 9)^2 = 16 $ is
