Area of the shaded region of a circle

The parallelogram ABCD has a larger altitude of 4 cm and a shorter altitude of 3 cm. What is the area of the shaded region?

The figure doesn't show to which side each of the altitudes are related, however, it´s perceived that the shorter altiude must be the one with side b as the base simply by looking at it.

As for the problem, I've managed to find the relation between the sides: 4a = 3b; but I can't see how that can help. My idea was to find the area of the parallelogram and somehow use it to discover the radius of the circle, so that I could, subsequently, calculate the area of the circle.

Is it possible to solve the problem this way? If not, how?

1 Answer

The circle fits snuggly between one pair of parallels, but is two large for the other pair. It follows that the diameter of the circle is the larger of the two distances between parallels (aka. altidudes). In other words, the diameter of the circle is $4\,\text{cm}$ and the height of the shaded region is the difference $4\,\text{cm}-3\,\text{cm}=1\,\text{cm}$. Can you take it from here?

• Yes. I was able to solve it using the formulas for chord lenght and area of a circular segment. – Henrique Silva Apr 17 '16 at 14:48
• Typo: the last line should read 4 cm - 3 cm (you have 2 instead of 3). – mathguy Apr 17 '16 at 15:03
• ....Snuggly....:) for sure +1 – imranfat Apr 17 '16 at 17:47