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# Geometry And Measurement - Advanced

In the figure above, a regular polygon has been shaded grey, and partially covered by a piece of paper. If $$s + t = 40 ^ \circ$$, how many sides does the regular polygon have?

A) 6
B) 10
C) 12
D) 15
E) 18

In the $$xy$$ coordinate plane, how many points are at distance of exactly $$5$$ units from the origin?

A) 1
B) 4
C) 8
D) 12
E) Strictly more than 12

A sphere of radius $$r$$ is tangential to all 6 faces of a cube with side length $$s$$. What is the volume of the sphere in terms of $$s$$?

A) $$\pi s^3$$
B) $$\frac{1}{3} \pi s^3$$
C) $$\frac{ 1}{6} \pi s^3$$
D) $$\frac{ 1}{8} \pi s^3$$
E) $$\frac{4}{3} \pi s^3$$

In the above figure, the circle circumscribes square $$ABCD$$. What is the degree measure of arc $$\widehat{BC}$$?

A) $$30 ^ \circ$$
B) $$45 ^ \circ$$
C) $$60 ^ \circ$$
D) $$90 ^ \circ$$
E) $$180 ^ \circ$$

In the above figure, there are 25 congruent right angled isosceles triangles. If the area of $$\triangle DFH$$ is 100, what is the area of $$\triangle AFK$$?

A) 100
B) 400
C) 525
D) 600
E) 625

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