Abby W.

asked • 09/01/19

UUUUUUMMMMMM...........

In the figure below, $O$ is the center of the circle, and the radius of the circle is $8$. Angle $\angle BOA$ is right. What is the area of the filled-in (purple) region?


Explain the steps of your solution. Give your answer in exact form (which may involve pi). In other words, don't round off (but do simplify if you can).

[asy] size(4cm); pair o=(0,0); pair a=(-sqrt(2)/2,-sqrt(2)/2); pair b=(sqrt(2)/2,-sqrt(2)/2); path p=Arc(o,1,225,315)--b--a--cycle; fill(p,purple); dot(o); dot(a); dot(b); draw(o--b--a--o); draw(a/8--(a+b)/8--b/8); draw(Circle(o,1)); label("$O$",o,N); label("$A$",a,SW); label("$B$",b,SE); [/asy]

2 Answers By Expert Tutors

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David W. answered • 09/01/19

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Draw a picture of the CIRCLE and the SQUARE:


The area of the complete circle (yellow-blue-purple) is: ∏r2 = ∏(8)2 = 64∏

The yellow area is 3/4 of that: (3/4)64∏ = 48∏


The area of the square (blue-purple-white) is s2 = (8)2 = 64

The area of half of that square (a blue triangle) is: (1/4)64 = 32


The are of the purple region is: (area of complete circle - yellow area - area of blue triangle)

64∏ - 48∏ - 32

16∏ - 32

Simplified: 16(∏-2)


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