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Geometry Warmups

Mathematics is filled with shapes that are kaleidoscopic in variety. Wielded since ancient times, the power of geometry helps us examine and measure these shapes.

Geometry Warmups: Level 3 Challenges


In the figure above, we have a square and a circle inside a larger square.

Find radius of the circle. Give your answer to 3 decimal places.

Find the area (in \(\text{cm}^2\)) of the shaded green square in the blue isosceles right triangle, whose legs have length 6 cm.

\[ \tan6^{\circ} \times \tan42^{\circ} \times \tan66^{\circ} \times \tan78^{\circ}=\, ?\]

The 5 circles have the same size. If the side of the large square is 1, what is the radius of each circle?

Three balls are placed inside a cone such that each ball is in contact with the edge of the cone and the next ball. If the radii of the balls are 20 cm, 12 cm, and \(r\) cm from top to bottom, what is the value of \(r\)?


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