In the given figure, 𝑂 is the
centre of a circle with diameter 𝐴𝐵 equal to 13 centimeters and chord 𝐴𝐶 equal
to 12 centimeters. Find the total area of the shaded
regions. Take 𝜋 equal to 3.14.
As the line 𝐴𝐵 is the diameter of
the circle, we know that angle 𝐴𝐶𝐵 is equal to 90 degrees as the angle in any
semicircle is equal to 90 degrees. The area of the shaded regions can
be calculated by subtracting the area of the right-angled triangle from the area of
the semicircle. As the area of any circle is equal
to 𝜋𝑟 squared, the area of a semicircle will be half of this area. The area of the triangle can be
calculated by multiplying the base by the height and then dividing by two.
We were told in the question that
the diameter of the circle 𝐴𝐵 was equal to 13 centimeters. The radius of a circle is half of
the diameter. Therefore, we need to divide 13 by
two. 13 divided by two is equal to
6.5. Therefore, the radius of the circle
is 6.5 centimeters.
This means that we can calculate
the area of the semicircle by multiplying a half by 𝜋 by 6.5 squared. The right-angled triangle has a
hypotenuse of 13 centimeters from 𝐴 to 𝐵 and a height of 12 centimeters from 𝐴 to
𝐶. We can, therefore, use Pythagoras’s
theorem to calculate the missing length of the triangle.
Pythagoras’s theorem states that 𝑎
squared plus 𝑏 squared is equal to 𝑐 squared, where 𝑐 is the hypotenuse. Substituting in our values gives us
𝑥 squared plus 12 squared is equal to 13 squared. 12 squared is equal to 144 and 13
squared is equal to 169. Subtracting 144 from both sides of
this equation gives us 𝑥 squared is equal to 25. And finally, square rooting both
sides of this equation gives us a value for 𝑥 equal to five. This means that the length of 𝐵𝐶
is five centimeters.
The base of the right-angled
triangle is five and its height is 12. We can, therefore, calculate its
area by multiplying five by 12 and dividing the answer by two. We were told to use a value for 𝜋
of 3.14. A half multiplied by 3.14
multiplied by 6.5 squared is equal to 66.3325. Five multiplied by 12 is equal to
60 and 60 divided by two is equal to 30. Therefore, the area of the triangle
is 30 centimeters squared.
Subtracting these two values gives
us an answer of 36.3325. Rounding this to a suitable degree
of accuracy gives us a shaded area of 36.33 centimeters squared to two decimal