Polygons

9000045708

Level: 
B
Given a regular hexagon with the side \(a\), find the radius \(\rho \) of the circle inscribed to this hexagon.
\(\rho = \frac{a} {2\cdot \mathop{\mathrm{tg}}\nolimits 30^{\circ }}\)
\(\rho = 2a\cdot \mathop{\mathrm{tg}}\nolimits 30^{\circ }\)
\(\rho = \frac{2a} {\mathop{\mathrm{tg}}\nolimits 30^{\circ }}\)
\(\rho = 2a\cdot \mathop{\mathrm{tg}}\nolimits 60^{\circ }\)

9000046404

Level: 
B
The parallelogram has sides of the length \(5\, \mathrm{cm}\) and \(4\, \mathrm{cm}\) (see the picture). The area of this parallelogram is \(S = 10\sqrt{2}\, \mathrm{cm}^{2}\). Find the measure of the smaller of the interior angles.
\(45^{\circ }\)
\(30^{\circ }\)
\(60^{\circ }\)

9000046405

Level: 
B
A circle is circumscribed to the regular octagon. The perimeter of the octagon is \(16\, \mathrm{cm}\). Find the radius of the circle and round the result to two decimal places. (The regular octagon is a polygon which has eight sides of equal length. The perimeter of the octagon is the sum of the length of all eight sides.) Circle circumscribed to the regular octagon.
\(2.61\, \mathrm{cm}\)
\(1.08\, \mathrm{cm}\)
\(1.41\, \mathrm{cm}\)

9000046406

Level: 
B
Find the area of the regular octagon of the perimeter \(16\, \mathrm{cm}\). Round the result to two decimal places. (The regular octagon is a polygon which has eight sides of equal length, see the picture. The perimeter of the octagon is the sum of the length of all eight sides.)
\(19.31\, \mathrm{cm}^{2}\)
\(3.31\, \mathrm{cm}^{2}\)
\(20.88\, \mathrm{cm}^{2}\)

9000121802

Level: 
B
Consider a regular polygon with the central angle of \(20^{\circ }\). In the figure the cut of a regular polygon with unspecified number of vertices is shown. The red angle is the central angle of the polygon. Find the number of vertices of this polygon.
\(18\)
\(9\)
\(20\)
\(15\)

9000121803

Level: 
B
Consider a regular polygon with the central angle of \(24^{\circ }\). In the figure the cut of a regular polygon with unspecified number of vertices is shown. The red angle is the central angle of the polygon. Find the number of diagonals in this polygon.
\(90\)
\(15\)
\(72\)
\(45\)