Question

Two smaller circles touch a large circle internally and pass through the center o of the larger circle. if the diameter of the bigger circle is 28 cm, then what is the area of the bigger circle which is not enclosed by the two smaller circles (in cm2)?

Answer 1

To find the area of the bigger circle not enclosed by the two smaller circles, we can use the Pythagorean theorem to find the radius of the smaller circles and calculate their combined area. The area not enclosed by the smaller circles is the difference between the area of the bigger circle and the combined area of the two smaller circles.

To find the area of the bigger circle that is not enclosed by the two smaller circles, we first need to find the radius of the bigger circle. Since the diameter is given as 28 cm, the radius is half of that, which is 14 cm.

The two smaller circles touch the larger circle internally and pass through its center. This means that the radii of the smaller circles are perpendicular to the radius of the larger circle. We can draw radii from the center of the larger circle to the points where the smaller circles touch it. This will form a right triangle, where the hypotenuse is the radius of the larger circle, and the two shorter sides are the radii of the smaller circles.

Using the Pythagorean theorem, we can find the length of the shorter sides of the triangle, which are the radii of the smaller circles. Let's denote this length as r:

r2 + r2 = 142

Simplifying the equation, we get:

2r2 = 196

r2 = 98

r = √98

Now that we have the radius of the smaller circles, we can find their areas. The area of a circle is given by the formula πr2, where π is a constant approximately equal to 3.14159.

The area of each smaller circle is therefore π(√98)2. To find the combined area of the two smaller circles, we multiply this by 2.

The area of the bigger circle that is not enclosed by the two smaller circles is simply the difference between the area of the bigger circle and the combined area of the two smaller circles.

Let's calculate:

Area of the bigger circle = π(14)2

Area of the two smaller circles = 2π(√98)2

Area not enclosed by the two smaller circles = Area of the bigger circle - Area of the two smaller circles

Answer: Area not enclosed by the two smaller circles = π(14)2 - 2π(√98)2

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