𝐴𝐵𝐶𝐷 is a square inscribed in a
circle of centre 𝑂 and radius 3.2 centimeters. Calculate the area of the shaded
region. Write your answer correct to two
Now, don’t worry too much about the
word “inscribed.” It just means that the vertices —
that’s the corners of the square — sit on the circumference of the circle. Now imagine that we wanted to find
the area of the four segments that sit around the square — that’s the shaded
part. To do this, we’ll need to find the
area of the outer shape, which is the circle, and we’ll need to subtract the area of
the inner shape, which is the square. We’ll find the area of the circle
and take away the area of the square then.
Now, we know that the area of a
circle is given by the formula 𝜋𝑟 squared. The radius of our circle is 3.2
centimeters. So the area is, therefore, 𝜋 times
3.2 squared. That’s 256𝜋 over 25. Now, we could write this in decimal
form. However, it’s much easier to leave
it in terms of 𝜋 to prevent any errors from rounding too early.
Next, we need to find the area of
the square. That’s given by its side length
squared. Now, we don’t actually know the
dimensions of this square. We do, however, know the
measurements of the four triangles that make up this square. Each triangle is made up of two
radii and we said that the radius of the circle was 3.2 centimeters. The area of a triangle is given by
the formula half multiplied by base multiplied by height. Since this is a right-angled
triangle, its base is 3.2 centimeters and its height is also 3.2 centimeters. The area of this triangle then is
given by a half multiplied by 3.2 multiplied by 3.2. That’s 128 over 25.
Now, since four of these triangles
make up the square, the area of the square is four multiplied by 128 over 25. Once again, we’ll leave our answer
as a fraction so we can use the exact value throughout. Now, we said that to find the area
of the four segments, we could subtract the area of the square from the area of the
circle. That’s 256𝜋 out of 25 minus 512
out of 25.
Now, we actually only need to know
the area of three of these segments, that’s three-quarters of the area we just
worked out. Our final line of working is,
therefore, three-quarters of 256𝜋 over 25 minus 512 over 25. That’s 8.7674. Correct to two significant figures,
the area is 8.8 centimeters squared.