In the diagram, and are diameters of a circle with radius 4. If and are perpendicular, what is the area of the shaded region?
Calling the center of the circle "C".
The area of the whole circle is: A = pi·42 = 16pi.
The area of the sector PCR is one-fourth the whole circle ---> ¼·16pi = 4pi.
Similarly, the area of the sector SCQ is 4 pi.
To find the area of the shaded triangular area in sector PCS:
this is a right triangle whose side are each 4: A = ½·4·4 = 8.
Similarly, the area of the shaded trianglular area in sector RCQ is 8.
We need to add these four areas together: 4pi + 4pi + 8 + 8.
Call the center of the circle C. Since the diameters PQ and PR are perpendicular, we know that sectors PCR and SCQ are quarters of the circle. The area of the circle is 16 pi, so 1/4 of that is 4 pi and since there are two quarter sectors, that means that the area of them both is 4 pi * 2= 8pi. The right triangle PCS and RCQ have legs of 4. The area of the triangls are (4 * 4)/2= 8, since there are 2 triangles, the total area is 16. Combine this area with the sectors to get 8 pi + 16 as your answer.