9000121801
Level:
B
Find the number of diagonals in the regular decagon (a regular polygon with
\(10\)
vertices).
\(35\)
\(7\)
\(10\)
\(70\)
Polygons
9000121805
Level:
B
For a regular pentagon find the interior angle. In the figure a regular pentagon with an interior angle marked in red is shown.
\(108\)
\(144\)
\(112\)
\(72\)
9000121806
Level:
B
The measure of the interior angle in a regular polygon is \(160^{\circ}\). Find the number of vertices of this polygon. In the figure the interior angle (marked in red) of a regular hexagon is shown.
\(18\)
\(16\)
\(32\)
\(9\)
9000121808
Level:
B
The number of diagonals in a polygon is three times bigger than the number of sides
of this polygon. Find the number of vertices of this polygon.
\(9\)
\(12\)
\(6\)
\(15\)
9000124502
Level:
C
A rectangle-shaped land has dimensions
\(3\times 5\, \mathrm{cm}\) on a map with
scale \(1\colon 2\: 000\). The
owner increased the size of his land by buying some land from his neighbor. The new land has
dimensions \(4\times 5\, \mathrm{cm}\)
on the map. Find the actual increase of the perimeter of the land (i.e. find the
increase in the length of the fence required to enclose the whole land). Give your
answer in meters.
\(40\, \mathrm{m}\)
\(20\, \mathrm{m}\)
\(80\, \mathrm{m}\)
\(10\, \mathrm{m}\)
9000045706
Level:
B
Given a regular pentagon with the side
\(a\), find the
radius \(r\) of
the circle circumscribed to this pentagon.
\(r = \frac{a}
{2\cdot \cos 54^{\circ }}\)
\(r = \frac{2a}
{\cos 72^{\circ }}\)
\(r = \frac{2a}
{\cos 54^{\circ }}\)
\(r = \frac{a}
{2\cdot \cos 72^{\circ }}\)
9000045707
Level:
B
Given a regular pentagon with the side
\(a\), find the
radius \(\rho \) of
the circle inscribed to this pentagon.
\(\rho = \frac{a}
{2} \cdot \mathop{\mathrm{tg}}\nolimits 54^{\circ }\)
\(\rho = \frac{2a}
{\mathop{\mathrm{tg}}\nolimits 54^{\circ }}\)
\(\rho = \frac{a}
{2\cdot \mathop{\mathrm{tg}}\nolimits 54^{\circ }}\)
\(\rho = 2a\cdot \mathop{\mathrm{tg}}\nolimits 54^{\circ }\)
9000045708
Level:
B
Given a regular hexagon with the side
\(a\), find the
radius \(\rho \) of
the circle inscribed to this hexagon.
\(\rho = \frac{a}
{2\cdot \mathop{\mathrm{tg}}\nolimits 30^{\circ }}\)
\(\rho = 2a\cdot \mathop{\mathrm{tg}}\nolimits 30^{\circ }\)
\(\rho = \frac{2a}
{\mathop{\mathrm{tg}}\nolimits 30^{\circ }}\)
\(\rho = 2a\cdot \mathop{\mathrm{tg}}\nolimits 60^{\circ }\)
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.
\(2.61\, \mathrm{cm}\)
\(1.08\, \mathrm{cm}\)
\(1.41\, \mathrm{cm}\)
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.
\(45^{\circ }\)
\(30^{\circ }\)
\(60^{\circ }\)