- 97. The sum of the angles will always be 98*180°. Each reflex angle can be 181°, and the last 3 angles share the last 80°.
- 34. We can have every third angle be nearly 360°, and the rest by 90°. However, it is actually possible to get a shape like a triangle with curved sides that has an extra right angle. Also, we can make another right angle with the two leftover sides. 2⌊50/3⌋+1+1=16*2+2=34.
- π√(2)/4cm2. The area of an ellipse is πab, where a is the major radius, and b, the minor radius. The minor radius is 0.5 cm. The length from a foci to a point on the minor axis is √(2)/2, and that always equals the major radius. π * √(2)cm/2 * 0.5cm equals π√(2)/4cm2, the area.
- 2340°. We know that the sum of the exterior angles of a polygon is 180°. Looking at the triangles on the edge of a star, the sum of their angles is 180° times the number of points. But 180°*2 = 360° of these aren’t in the points. Our answer is thus 15*180 = 2700° minus 360° equals our answer, 2340°.
- 15. The only way to do this is by trial and error. 15 is in the Pythagorean triples 8-15-17, 9-12-15, 15-20-25, 15-36-39, and 15-112-113.
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Challenges cover a variety of topics