Circles

1103077203

Level: 
B
The tip of a minute hand is at distance of \( 15\,\mathrm{mm} \) from the clock centre. Calculate the length of the path the tip travels in \( 42 \) minutes. Round the result to two decimal places.
\( 65.97\,\mathrm{mm} \)
\( 94.20\,\mathrm{mm} \)
\( 35.27\,\mathrm{mm} \)
\( 72.12\,\mathrm{mm} \)

1103077202

Level: 
C
Let \( ABCDEF \) be a regular hexagon. Six circles of equal radii are drawn touching each other with their centres at the hexagon vertices (see the picture). Calculate the area of the coloured region inside the hexagon if you know that the perimeter of the hexagon \( ABCDEF \) is \( 36\,\mathrm{cm} \). Round the result to two decimal places.
\( 36.98\,\mathrm{cm}^2 \)
\( 93.53\,\mathrm{cm}^2 \)
\( 65.26\,\mathrm{cm}^2 \)
\( 25.37\,\mathrm{cm}^2 \)

1103077201

Level: 
B
The flower bed has the shape of a circle sector of radius \( 3\,\mathrm{m} \) with central angle \( 75^{\circ} \). Calculate the area of this flower bed. Round the result to two decimal places.
\( 5.89\,\mathrm{m}^2 \)
\( 1.96\,\mathrm{m}^2 \)
\( 11.78\,\mathrm{m}^2 \)
\( 9.34\,\mathrm{m}^2 \)

1103256901

Level: 
C
The farmer tied two goats on the meadow. The distance of the stakes \( K_1 \), \( K_2 \) to which the goats are tied is \( 5\,\mathrm{m} \) and the ropes have lengths of \( 3\,\mathrm{m} \) and \( 4\,\mathrm{m} \). What is the area of the grassland which is common for both goats? Round the result to two decimal places.
\( 6.64\,\mathrm{m}^2 \)
\( 0.57\,\mathrm{m}^2 \)
\( 0.35\,\mathrm{m}^2 \)
\( 1.52\,\mathrm{m}^2 \)

1103021612

Level: 
C
Consider two circles: the circle \( k \) with centre \( S_1 \) and radius \( 3\,\mathrm{cm} \), and the circle \( n \) with centre \( S_2 \) and radius \( 8\,\mathrm{cm} \). The distance between \( S_1 \) and \( S_2 \) is \( 22\,\mathrm{cm} \). Common internal tangents of the circles intersect at point \( A \). Calculate the distance of the point \( A \) from the centre \( S_1 \). (See the picture.)
\( 6\,\mathrm{cm} \)
\( 16\,\mathrm{cm} \)
\( 11\,\mathrm{cm} \)
\( 5\,\mathrm{cm} \)

1103021602

Level: 
C
The side of an equilateral triangle is \( 6\,\mathrm{cm} \) long. Find the area of the annulus between the incircle and circumcircle of the given triangle. (See the picture.)
\( 9\pi\,\mathrm{cm}^2 \)
\( 6\pi\,\mathrm{cm}^2 \)
\( 12\pi\,\mathrm{cm}^2 \)
\( 8\pi\,\mathrm{cm}^2 \)

1103021511

Level: 
A
An acute triangle \( ABC \) is inscribed in the circle of radius \( r=4\,\mathrm{cm} \). Determine the measure of the angle \( ACB \), if the length of side \( c \) is \( 6\,\mathrm{cm} \). Round the result to two decimal places. (See the picture.)
\( 48.59^{\circ} \)
\( 97.18^{\circ} \)
\( 24.30^{\circ} \)
\( 41.41^{\circ} \)

1103021510

Level: 
A
A regular nonagon \( ABCDEFGHI \) is inscribed in a circle. Calculate the measures of all interior angles of the quadrilateral \( ABEH \). (See the picture.)
\( \alpha=120^{\circ};\ \beta=100^{\circ};\ \gamma=60^{\circ};\ \delta=80^{\circ} \)
\( \alpha=100^{\circ};\ \beta=120^{\circ};\ \gamma=60^{\circ};\ \delta=80^{\circ} \)
\( \alpha=100^{\circ};\ \beta=100^{\circ};\ \gamma=80^{\circ};\ \delta=60^{\circ} \)
\( \alpha=110^{\circ};\ \beta=130^{\circ};\ \gamma=70^{\circ};\ \delta=50^{\circ} \)

1103021509

Level: 
A
A regular dodecagon \( ABCDEFGHIJKL \) is inscribed in a circle. Find the measures of all interior angles of the quadrilateral \( ABHJ \). (See the picture.)
\( \alpha=120^{\circ};\ \beta=75^{\circ};\ \gamma=60^{\circ};\ \delta=105^{\circ} \)
\( \alpha=105^{\circ};\ \beta=60^{\circ};\ \gamma=75^{\circ};\ \delta=120^{\circ} \)
\( \alpha=120^{\circ};\ \beta=30^{\circ};\ \gamma=60^{\circ};\ \delta=105^{\circ} \)
\( \alpha=105^{\circ};\ \beta=75^{\circ};\ \gamma=75^{\circ};\ \delta=105^{\circ} \)

1003021508

Level: 
A
A triangle is inscribed in a circle. Its vertices divide the circle into three arcs whose lengths are in the ratio \( 2:4:9 \). Determine the measures of the interior angles of the triangle.
\( 24^{\circ};\ 48^{\circ};\ 108^{\circ} \)
\( 30^{\circ};\ 40^{\circ};\ 110^{\circ} \)
\( 48^{\circ};\ 15^{\circ};\ 117^{\circ} \)
\( 15^{\circ};\ 60^{\circ};\ 105^{\circ} \)