C

1103235603

Level: 
C
The base of a right pyramid is a regular hexagon of side \( 4\,\mathrm{m} \) and its slant surfaces are inclined to the horizontal at an angle of \( 30^{\circ} \) (see the picture). Find the volume.
\( 16\sqrt3\,\mathrm{m}^3 \)
\( 72\sqrt3\,\mathrm{m}^3 \)
\( 48\sqrt3\,\mathrm{m}^3 \)
\( 24\sqrt3\,\mathrm{m}^3 \)

1103235602

Level: 
C
Find the surface area of a right pyramid whose base is a regular hexagon each side of which is \( 6\,\mathrm{cm} \) and height \( 9\,\mathrm{cm} \) (see the picture).
\( 162\sqrt3\,\mathrm{cm}^2 \)
\( 15\sqrt3\,\mathrm{cm}^2 \)
\( 9\left(\sqrt3+6\sqrt{13}\right)\,\mathrm{cm}^2 \)
\( 117\sqrt3\,\mathrm{cm}^2 \)

1103235601

Level: 
C
Find the volume of a right pyramid whose base is a regular hexagon each side of which is \( 6\,\mathrm{cm} \) and height \( 8\,\mathrm{cm} \) (see the picture).
\( 144\sqrt3\,\mathrm{cm}^3 \)
\( 72\sqrt3\,\mathrm{cm}^3 \)
\( 48\sqrt3\,\mathrm{cm}^3 \)
\( 24\sqrt3\,\mathrm{cm}^3 \)

1103077109

Level: 
C
Two quarter circles are inscribed in the square with sides of \( 2\,\mathrm{dm} \). The centres of the quarter circles are at the opposite vertices of the square (see the picture). Calculate the area of the region between the quarter circles. Round the result to two decimal places.
\( 2.28\,\mathrm{dm}^2 \)
\( 3.14\,\mathrm{dm}^2 \)
\( 21.12\,\mathrm{dm}^2 \)
\( 1.72\,\mathrm{dm}^2 \)

1103077108

Level: 
C
The figure shows an equilateral triangle whose side is \( 10\,\mathrm{cm} \) long. The circular sector inside the triangle has the centre at one of the vertices of the triangle, and the arc touches the opposite side. Calculate the area of the sector. Round the result to one decimal place.
\( 39.3\,\mathrm{cm}^2 \)
\( 37.5\,\mathrm{cm}^2 \)
\( 14.4\,\mathrm{cm}^2 \)
\( 3.75\,\mathrm{cm}^2 \)

1103077107

Level: 
C
The figure shows an equilateral triangle whose side is \( 10\,\mathrm{cm} \) long. The circular sector inside the triangle has the centre at one of the vertices of the triangle, and the arc touches the opposite side. Find the ratio of the circumference of the sector to the perimeter of the triangle. Round the result to one decimal place.
\( 0.9 \)
\( 0.5 \)
\( 0.8 \)
\( 1.5 \)

1103077106

Level: 
C
Let an equilateral triangle have a side of length of \( 10\,\mathrm{cm} \). Suppose there is a circular sector inside the triangle that has the centre at one of the vertices of the triangle, and the arc touches the opposite side (see the picture). Calculate the length of the arc of the sector. Round the result to two decimal places.
\( 9.07\,\mathrm{cm} \)
\( 8.62\,\mathrm{cm} \)
\( 8.93\,\mathrm{cm} \)
\( 9.05\,\mathrm{cm} \)

1103077105

Level: 
C
In a triangle \( ABC \), \( a=7\,\mathrm{cm} \), \( b=8\,\mathrm{cm} \), \( c=11\,\mathrm{cm} \). What is the radius of a circle circumscribed about this triangle? Round the result to two decimal places.
\( 5.51\,\mathrm{cm} \)
\( 6.11\,\mathrm{cm} \)
\( 4.92\,\mathrm{cm} \)
\( 6.52\,\mathrm{cm} \)

1103077104

Level: 
C
Three equal circles, each of radius \( 6\,\mathrm{cm} \), touch each other as shown in the figure. Find the area of the region bounded by the circles. Round the result to one decimal place.
\( 5.8\,\mathrm{cm}^2 \)
\( 62.3\,\mathrm{cm}^2 \)
\( 6.2\,\mathrm{cm}^2 \)
\( 8.4\,\mathrm{cm}^2 \)

1103077103

Level: 
C
The length of the shortest diagonal in a regular polygon is \( 8\,\mathrm{cm} \). The measure of the angle between this diagonal and the side of the polygon is \( 20^{\circ} \). Calculate the radius of a circle circumscribed about this polygon. Round the result to two decimal places.
\( 6.22\,\mathrm{cm} \)
\( 5.22\,\mathrm{cm} \)
\( 4.26\,\mathrm{cm} \)
\( 11.69\,\mathrm{cm} \)