The lateral surface area of an upright cone is stretched in a plane. We obtain a circular section equal to half a circle with radius . Calculate the total surface area and volume of this upright cone.

A cuboid and a cube have the same volume of 216 cubic centimetres. Calculate the total surface area of the cuboid and cube if the length of the cuboid is and the width is .

Two spheres have radii in the ratio 2:3. What is the ratio of their volumes?

A ball fits snugly into the cube-shaped cardboard box. The box is wide. How many cubic decimetres of air are in the ball? When calculating, we do not take into account the thickness of the material from which the ball is made.

Calculate the length of the edge of the cube obtained by melting 20 cm high iron cylinder, with the diameter of the base surface 15 cm.

Calculate the volume and surface area of the 30 cm high regular four-sided pyramid with a base of 37 cm square.

We need gold to coat a pendant in the shape of an equilateral cone. Calculate the volume of the pendant.

Calculate the area and the volume of the cuboid in the sketch above and the length of the marked diagonals.

10.

In a cone with a diameter of we can pour liquid. Calculate the slant height of the cone.

11.

The diameter of a sphere measures . Calculate the volume of the sphere in cubic meters.

12.

Calculate the mass of air in a classroom 10 m long, 6 m wide and 4 m high. Note that 20% of the volume of the entire classroom is occupied by furniture. For air density take 1.2 kilograms per cubic meter.

13.

The surface area of a sphere measures . Calculate the volume of the sphere.

14.

The radius of a sphere measures . Calculate the volume of the sphere.

15.

How many litres of water do we need to fill a 25 m long, 18 m wide and 160 cm deep pool? What is the mass of the water in the pool then? Calculate the pressure at the bottom of the pool.

16.

What is the volume of a sphere drawn on a cube with edge ?

17.

The height of a cone is four times its radius and the area of its curved surface measures . Calculate the volume of the cone.

18.

A sphere-shaped scoop of ice cream fits into a cone so that exactly half of the scoop is outside the cone. The cone is rolled from a wafer, and the distance between the tip and the edge is . How much ice cream will we eat?

19.

Calculate the surface area and volume of a square based pyramid with equal edges if the diagonal of the square measures 8 cm.

20.

The parachute has the shape of a semicircle with a radius of . Approximately how much air does it capture?

21.

Three spherical balls have radii , and . If we melt all three into one ball. Calculate its area and volume.

22.

Copper bead has radius . Calculate its mass if ?

23.

The base area of a cylinder of volume is . What is the height of the cylinder?

24.

Calculate the radius of a sphere equal in volume to a cube with edge .

25.

The volume of a cylinder measures and the height . What is the area of the base of the cylinder?

26.

The radius of a soccer ball is 12 cm. At night, in the cold, the volume of the ball decreased by 6%. Calculate the new volume and radius of the ball.

27.

Mom poured 3 spoon of soup into Jerry's plate. The spoon has the shape of a hemisphere with a radius of 3.5 cm. How many decilitres of soup did mom pour for Jerry?

28.

A cylinder with height , has a circle with a diameter of as its base. Calculate the volume of the cylinder in cubic centimetres.

29.

A gold bar in the shape of a square with edges 10 cm, 5 cm and 4 cm is used to make pendants in the shape of a sphere with a diameter of 0.4 cm (the balls are full). At most, how many pendants can they make from one such rod?

30.

In a cube with edge a substitute 27 identical spheres touching each other and the walls of the cube. Calculate the ratio between the volume of the cube and the sum of the volumes of all spheres.

31.

Calculate the area of a sphere with volume .

32.

The body consists of a truncated cone, which has the radii of the base surfaces 10 cm and 5 cm, the height 13 cm, and has a glued hemisphere on the smaller base surface. Calculate the volume and area of the body.

33.

The circumference of the base of an equilateral cylinder is measured . Calculate the total surface area and volume of the cylinder .

34.

Two spheres are melted into the same metal with surfaces area and into one sphere. What is the radius of the new sphere?

35.

The sum of the volumes of two spheres is equal to the volume of a cube with edge . What are the volumes of the spheres if their radii are in the ratio ?

36.

In a cube with edge a = 32cm, draw 8 identical spheres in two layers of 4 so that they touch. What is the ratio of the sum of the volumes of the spheres and the volume of the cube? How does the ratio change if we draw 27 identical spheres in three layers of 9 or 64 identical spheres in four layers of 16 in the same cube?

37.

Marjan is building a house for which he needs eight supporting concrete pillars in the shape of a cylinder. What is the mass of a single 2.5 m high pillar with a radius 0.45 m? The density of concrete is .

38.

What is the volume and area of an equilateral cone inscribed with a sphere of radius

39.

Three identical spheres with radius are placed in a cylindrical container having the smallest possible volume and the smallest possible diameter. How much air does the container take up?

40.

The curved surface area of a cylinder measures and the height is five times the diameter of the cylinder. Calculate the volume of the cylinder.

41.

In a rectangular box that is 96 cm long, 64 cm wide and 75.2 cm high, we want to stack as many cans as possible. How many cylindrical cans with a base area and a height h = 15 cm can be stacked in this box?

42.

If we cut an oak log, we get boards. The conveyor belt on the saw must be at least wide for cutting this log. What should be the minimum length of the trailer of the truck that brought the log from the forest? Assume no cutting losses .

43.

Tony bought a cone in the store, the slant height of of the cone is , and the opening at the top of the cone is in diameter. Calculate how much ice cream is poured in the cone if the distance from the top of the cone to the top of the ice cream is .

44.

David has a rectangular garden with sides and long. He wants to extend the shorter side.

Write down the dependence of perimeter and area on the extended side.
How much does the new volume and width differ from the original if the shorter side is extended by ?

45.

What is the radius r of the base of a cone which is high and has a volume ?

46.

Calculate the angle at the top of the axial section of the cone whose curved surface is obtained by turning three-quarters of a circle with radius .

47.

What must be the minimum distance between the chassis of the vehicle and the carriageway at which a road cone of volume standing on of the carriageway can still be driven?

48.

We have two vertical cones that have the same circular base areas. What is the ratio of the volumes of the cones if the curved surface area of the first cone is times its circular base and the curved surface area of the second is times its circular base?

49.

A vertical cone has the area of its circular base as , and its curved surface area is equal to three times the area of the circular base. What is the volume of such a cone?

50.

The volume of a cone measures and the height . What is the radius of the cone?

51.

Calculate the central angle in the curved surface of the unfolded net of a vertical cone with volume and radius of the circular base .

52.

We have three gold cubes with volumes , and . We melt the cubes and get a new, larger cube. What will be the length of the edge of the newly formed cube?

53.

The volume of a cone measures and the base area equals . Calculate the area of the axial section of the cone.

54.

We have three iron cubes with edges , and . We melt the cubes and pour into a mold to make a new larger cube from. What will be the volume of the newly formed cube?

55.

The volume V of a cone measures . If the height h of the cone is , calculate he diameter of the cone.

56.

A pyramid has as its base as a rectangle with an aspect ratio of . The perimeter of the base measures , and the height of the pyramid measures . Calculate its volume.

57.

What is the central angle of an unfolded equilateral cone?

58.

A equilateral cone has the area of the mantle and the radius of the circular base . Calculate its volume.

59.

What is the volume of a cone with the radius of the circular base and curved surface area that is four times the base area?

60.

Calculate the volume and area of an upright cone high whose slant height and circular base are enclosed as .

61.

Calculate the volume of a a high cone having a circular base area and a curved surface area in the ratio .

62.

Calculate the volume and area of the body in the figure.

Figure

63.

At what height should we cut a high metal cone whose diameter of the base surface measures so that both parts will have the same volume?

64.

In the sand pit, they had sand in cone-shaped containers. Before the truck loaded the sand, the sand reached high, and the circumference at the foot measured . How much sand was loaded on the truck if the height of the sand decreased by and the circumference at the base decreased by ?

65.

Calculate the volume and area of the body in the figure.

66.

The volume of an equilateral triangular base prism is and the height is . What is the length of a side of the base of such a prism?

67.

The length of the side of an equilateral triangular base prism measures and the total surface area measures . Calculate the volume of the prism.

68.

How much soil can be sprinkled in a container of the shape of an upright truncated cone with radii and and the total surface area of ?

69.

How much sand can be sprinkled in a container of the shape of an upright truncated cone with radii and and side ?

70.

The curved surface area of an triangular base equilateral prism measures . Calculate the volume of the prism.

71.

The base of an upright pyramid is a triangle with data , and . Calculate:

side b and the area of the base of the pyramid.
the volume of the pyramid whose height is equal to the radius of a circle inscribed in the triangular base of the pyramid.
the volume of the cone inscribed in the pyramid.

72.

We need to repaint a wood to protect a wooden cube. What is the volume of the cube?

73.

How many litres of water can be poured into a container of the shape of an upright truncated cone with radii and and a coat ?

74.

The curved surface area of an equilateral triangle base prism measures and the height is four times the length of one side of the base. Calculate the height and volume of the prism.

75.

Calculate the central angle in the plane of the vertical cone whose curved surface area is four times the base area.

76.

The length of one side of the base and height of an equilateral triangle base prism are in the ratio 2:3, and the curved surface area measures . Calculate the volume of the prism.

77.

Calculate the center angle into the plane of the unfolded mantle of the cone. The angle at the top of the cone measures and its side measures .

78.

Determine the ratio of the volumes of the cone to the largest regular four-sided pyramid cut from the cone.

79.

Calculate the volume of the cone whose mantle measures and the circumference of the circular base .

80.

Tony made a rattle that has the shape of two joined truncated cones (figure below). Calculate how much rice he put in the rattle if he filled it to .

81.

Calculate the area and volume of a school mushroom 20 cm long, 10 cm wide and 5 cm high.

82.

Calculate the area and volume of a right circular cylinder with the data:

83.

Given triangle ABC:

Calculate the length |AC| and angle BAC to be denoted as .
Calculate the area of the triangle.
Calculate the surface area and volume of a prism whose base is the given triangle and whose height is 1 m.

84.

Calculate the area and volume of an upright cylinder with the data:

85.

Calculate the length of the diagonal of the side ABEF, the volume and surface area of the figure below.

Figure

86.

Calculate the volume of a cube whose surface area measures .

87.

If unit cubes are assembled to form the figure below, calculate its surface area and volume.

Figure

88.

The height of an equilateral three-sided pyramid has the same length as the height of the equilateral triangle at the base surface of this pyramid. Calculate the volume of the pyramid if its surface area is .

89.

Calculate the volume and area of the pyramid whose net is in the figure below, if possible.

90.

A quadrilateral base pyramid has as its base a rectangle with an aspect ratio of 5:3. The area of the base measures and the volume measures . Calculate the height of the pyramid.

91.

Calculate the weight of a wooden kitchen counter with length 3 m, width 8.4 dm and height 8 mm. If the density of the wood .

92.

Calculate the area and volume of a regular four-sided pyramid if its axial section is an equilateral triangle with area and its legs are the heights of the opposite side faces.

93.

Calculate the weight of a wooden slab which is 3 m long, 70 cm wide and 4.3 cm thick. Given wood density .

94.

How many unit cubes do we need to add to the figure below to get a cube with side length 4 cm?

Figure

95.

The side lengths of a cuboid are in the ratio a: b: h = 7: 3: 11. Calculate the surface area and volume of the cuboid if the diagonal of the rectangle with sides b and h is 15 cm long.

96.

Axial section of a wooden pillar in the shape of a cylinder, height 130 cm, has a diagonal length 170 cm. Calculate the area and volume of the column.

97.

The side of the base of a cubic prism measures 23 cm and its surface area is . Calculate the volume of the prism, the diagonal of the base and the surface diagonal.

98.

The axial section area 2.1 m of a high concrete column having the shape of a right circular cylinder measures . What is its area and volume?

99.

Calculate the total surface area and volume of the cylinder formed by a square with side 17 cm rotate about a symmetrical axis by .

100.

Calculate the total surface area and volume of the cylinder formed if a rectangle with sides 13 cm and 10 cm, rotate around the longer side for .

101.

Calculate the volume and surface area of a four-sided pyramid that has a square with side 8 cm as its base. The side surface is enclosed with the base surface at an angle of .

102.

Calculate the area and volume of a four-sided pyramid with a side edge s = 40.2 cm. The base is a rectangle with sides a = 43 cm and b = 20 cm.

103.

A 12 cm high triangular prism has its base triangle sides measure a = 17 cm, b = 10 cm and c = 12 cm. Calculate the surface area and volume of the prism.

104.

A regular three-sided pyramid with a height of length 11 cm has a volume of . Calculate the length of the base edge of the pyramid.

105.

On the table stands a right circular cylindrical vessel with an inner diameter 30 mm. Calculate by how much the water level rises if we immerse a cube with an edge in this container 14 mm?

106.

Calculate the length of the base edge of a regular four-sided pyramid with height 11 cm and volume .

107.

Calculate the base edge of a regular six-sided based pyramid with height 11 cm and a volume of .

108.

Fill in a right circular cylinder with the largest possible volume from an equilateral three-sided prism. Calculate the ratio of the volumes and areas of both bodies.

109.

A rectangle with sides 11 cm and 4.5 cm is the base of a pyramid with a side edge 23 cm. Find the surface area and volume of the pyramid.

110.

Calculate the length of the body diagonal of the cube obtained by melting a metal regular four-sided pyramid with a base edge 4 cm and lateral area of .

111.

The coordinates of a rectangle are given by the points A (1,1), B (7,1), C (7,3) and D (1,3 ).

Draw the figure in a coordinate system and calculate the perimeter of the rectangle ABCD.
Point T lies on side AB, so that ratio , and point S bisects side BC. Draw the points T and S in the given coordinate system and calculate the length of the line TS.
Rectangle ABCD represents the body surface area of the prism with an equilateral triangle base. If the height of the prism is h = 2, calculate the volume of the prism.

112.

A four-sided pyramid has a square with lenght 15 cm as its base surface. Calculate the surface area and volume of the pyramid if the angle between the base and side edge is .

113.

The regular four-sided pyramid has a base edge of length 9 dm, and its side edge measures 13.6 dm. Calculate the area of the lateral area and the volume of the pyramid.

114.

A three-sided pyramid with volume has a triangle with sides 22 cm, 48 cm and 66 cm as its base. Calculate its height.

115.

A cuboid has the same surface area as a cube with a body diagonal . The lengths of side of the cuboid are in the ratio a: b: c = 2: 1: 3. Calculate the volume of the cuboid.

116.

Calculate the surface area and volume of the figure below:

Figure

117.

A trapezium with sides a = 21 cm, b = 19 cm, c = 10 cm and d = 12 cm, represents the base of a four-sided prism. If the rectangular surface area of the prism is twice its base area, calculate the surface area and volume of the prism.

118.

A triangular prism is formed in a cube prism of side a = 11 cm and height h = 12 cm. What percentage does the volume of the three-sided prism represent of the four-sided prism?

119.

The base side of an oblique four-sided prism is a square which measures 5 cm. The angle between the base and side surfaces . Calculate the surface area and volume of this prism if its slant side measures 15 cm.

120.

The pyramid net in the figure below is to be cut out. Calculate the length of the edge of the paper if the volume of the pyramid is

121.

Calculate the angle between the base and side edges of a regular three-sided pyramid of volume if its side edges are perpendicular to each other.

122.

The lateral area of a regular four-sided pyramid measures . Calculate the volume of this pyramid if the area of the base area is equal to one third of the lateral area.

123.

What is the volume of a four-sided pyramid with a lateral area of , if its base is a square with side 6 cm?

124.

The figure below shows three-sided prism ABCDEF. , angle ABC and . If the height of the prism is 56 cm,

Figure 1

Calculate the length of the lines BD.
Calculate the volume of the prism.
Calculate the triangular surface area of the prism.

125.

The figure is made of two identical cubes with side 15 cm, as shown below. If the distance between the points and is 14 cm.

Calculate the volume of the figure.
Calculate surface area of the figure.
Calculate the distance between the points and .

Figure 1

126.

What is the ratio of the volume of a regular four-sided pyramid to the volume of a regular four-sided prism having the same base edge and surface area, if the base edge of the prism is three times its height?

127.

Matthias is making a model of a regular six-sided prism, which he will make from a 20 cm high cylinder with a volume of . Calculate the length of the base edge of the largest possible formed regular hexagonal prism .

128.

A right four-sided pyramid has a base edge of length 2.5 dm, and its side edge is 3 times longer. Calculate:

the angle between the base and the side surfaces,
the angle between the base and side edges,
the angle between the side edge and the base surface,
volume of the pyramid.

129.

Two similar right circular cylinder are of heights 16 cm each. If the the first one with inner diameter of 4 cm is filled with a liquid and the liquid is then poured into the second cylinder of inner diameter 6 cm, find the depth of the liquid in the second container.

130.

A rod with a diameter of and a height of is trimmed so that its diameter is reduced by . By how much cm should we reduce the height h of the original rod to have the same volume as the reduced diameter?

131.

The base of a pyramid is a regular hexagon and is three times smaller than the lateral surface area of the pyramid. Calculate the total surface area and volume of the pyramid if the length of the base edge is 10 cm.

132.

Calculate the volume of a regular six-sided pyramid forming an isosceles triangle with a cross sectional base of 10 cm and arm 16 cm.

133.

John got a drawn and outlined cylinder 5 cm high, three-sided prisms in his math class for homework. The edges of the base plane, represented by a right-angled triangle, measured a = 6 cm, b = 2.6 cm and c = 4.2 cm. He also had to determine the ratio of the areas and volumes of the drawn cylinders.

134.

A wooden regular eight-sided pyramid, 30 cm high with a base edge 10 cm, was transformed into the largest possible regular four-sided pyramid. Calculate what percentage of the volume of the octagonal pyramid the volume of the newly formed pyramid is.

135.

The radius of the circumscribed circle to the base plane of a regular octagonal pyramid measures 12 cm. Calculate the total surface area and volume of the pyramid if it extends 48 cm high.

136.

What is the total surface area and the volume of a regular octagonal pyramid with a base edge of 8 cm and a side edge that is twice as long as the base?

137.

Four-equilateral-sided pyramids with edges of 2.5 cm were added to each surface of a cube of edge a = 5 cm. Calculate the total surface area and volume of the resulting body and what percentage of the total volume is the volume of the added pyramids?

138.

From a wooden regular six-sided prism, a pyramid was carved out with the same base surface as the prism and with a height equal to one third of the base edge of the prism. Calculate the volume and surface area of the resulting wooden body if the base edge of the prism is 15 cm and its height is twice as large.

139.

Calculate the volume and area of a regular hexagonal pyramid with the base edge . The area of the base surface is equal to a quarter of the lateral area.

140.

Calculate the the volume of a three-sided frustum of a pyramid with a height of 12 cm and the base edges and . The base planes are equilateral triangles.

141.

The total surface area of a right -angled circular cylinder measures . Calculate the volume of the cylinder if the diameter is 3 cm smaller than the height of the cylinder. Also determine the axial cross-sectional area of the cylinder.

142.

We round 30 m copper wires on a coil. What is the diameter of this wire if we used 10 kg copper for it, the density of which is .

143.

A tree trunk of length 6m and diameter of 6.5dm is cut down from the forest. A large possible beam (four -sided prism) will be made from it. Calculate the percentage of the beam and what percentage is waste.

144.

A round (Cylindrical) birthday cake with a diameter of 24 cm and height 11 cm is cut into different pieces. Calculate the volume and area of a piece of cake with a centred angle

145.

What is the volume of a three-sided frustum of a pyramid with a side edge of 25 cm and the base edges and . The top and base surfaces are equilateral triangles.

146.

A regular triangular frustum of a pyramid that is 30 cm high has a base edge with lengths and . Calculate its surface area and volume.

147.

Mom poured 1.5 l juice into a jug of right-angled circular cylinder shape. How high does the juice reach in a 2.75 l jug that is 35 cm high?

148.

Find the area and volume of the frustum of a pyramid, which has a square with sides , as its base and its height measures 7 cm.

149.

The cylindrical wine container is filled to a height of 63 cm. How many cups of wine can be poured if the radius of the cup is 5 cm and its height 16 cm? Pour the cup up to 1 cm under the edge. The barrel of wine is 1.3 m high and has a diameter of 90 cm.

150.

Calculate the surface area and volume of frustum of a pyramid having a square with sides and and a lateral area of .

151.

A flower pot in the shape of a frustum of a four-sided pyramid with a rectangular base area was given. The smaller base area has sides 5 dm and 3 dm long, while the larger base has both sides that are 30% longer than the lower smaller area. Both surfaces are connected by an edge of length 6 dm. Calculate how many liters of soil go into the trough.

152.

Jessica makes a wooden model of a right-angled circular cylinder at school, the diagonal of the axial section is inclined towards the base surface by , and the area of the axial section is . Calculate how many liters of paint it will use per cylinder if we use approximately on .

153.

How much is the volume of an isosceles oblique cylinder whose axial cross section has a plane . The vertical angle in the axial section is

154.

Calculate the volume and area of the largest regular four-sided prism inscribed in an equilateral cylinder with a diameter of 14 cm.

155.

The base face of an upright pyramid is a rectangle with sides and , and the height of the pyramid is 8. Sketch and mark as , the angle between the side margin and the base face. Calculate the volume of the pyramid and the magnitude of the angle to a tenth of a degree.

156.

The length of the base edge of the regular 4 - side of the pyramid is 4.2 m. The side face of the pyramid is inclined towards the base face by an angle . Draw a sketch of the pyramid, mark the angle and calculate the volume of the pyramid.

157.

James has 5 cylindrical oak wood. The length of an individual wood is 3.5 m and the radius is 0.25 m. How many euros does he get for wood if the price of oak wood is 86 euros per cubic metre?

158.

The base of an upright pyramid ABCDV is a rectangle with sides | AB | = a = 60 cm and | BC | = b = 80 cm. The side edge of the pyramid measures 1.3 m.

Sketch the pyramid and a sketch of its net.
Calculate the volume of the pyramid.
Calculate the area of the side plane BCV.

159.

We have two empty cylindrical tanks standing on their base planes.

The first tank has the shape of an upright cylinder with a radius of . It is filled to two-thirds full with apple juice. Calculate the height of the tank. Round the result to the nearest tenth of a decimetre.
The second tank has the shape of an equilateral cylinder (the axial cross-section is square). Pour of apple juice into it and fill it to the top. Calculate the radius of the tank. Round the result to the nearest tenth of a decimetre..

160.

A regular four-sided pyramid has an edge of the base plane of length 2. The adjacent side edges of the pyramid are enclosed at an angle of .

Express the surface area of the pyramid in terms of .
Calculate the value of , where the volume of the pyramid is .
Show that the volume of the pyramid is .
Let . Show that at this angle the opposite side edges of the pyramid are perpendicular.

161.

The base of an upright four-sided pyramid is a square with a side of length 2 dm. The height of the pyramid is 6 dm.

Sketch and calculate the volume of the pyramid.
Calculate the area of the pyramid.
Calculate the length of the side edge of the pyramid.

162.

The sketch shows the axial section of the vertical (circular) cone. Calculate the height of the cone and the angle at the top of the axial section of the cone.

163.

Points are given in the plane

How far is the point C away from the line through the points A and B? The result should be accurate.
Calculate the equation of the line of the beam of the center of gravity per side AC of the triangle ABC and the size of the angle at the angle A.
Calculate the vertex D so that the quadrilateral ABCD is a parallelogram.
Calculate the volume of the vortex created by rotating the parallelogram ABCD around the side AB.

164.

In a regular four-sided pyramid, the lateral height is longer than the height h of the pyramid. Calculate the exact length of the base edge of the pyramid if the volume measures .

165.

An upright circular cone with radius of base area and height .

Calculate the total surface area of the cone.
Draw a regular four-sided pyramid to a given cone (the base surface of the pyramid is inscribed to the base surface of the cone). Calculate the volume of this pyramid.
We draw a sphere for a given cone. Calculate its radius.
The given cone is cut by two mutually perpendicular planes passing through the axis of the cone so as to divide it into four equal parts. Calculate the area of one part. The result should be accurate. (The axis of the cone is a line passing through its apex and the center of the base plane.)

166.

A right triangle with sides and rotate around the side a for .

Calculate the angle at the top of the axial section of the resulting cone.
Calculate the total surface area of the cone.
Calculate the volume of the cone and express it in cubic decimetres.

167.

Mike bought in the store long pencil in the shape of a regular six-sided prism with a basic edge . At home, he sharpened a pencil on one side into the shape of a cone with a slant height of . Calculate what percentage of the pencil is sharpened?

168.

Sonia poured her father a drink into a glass that has the shape of an inverted upright cone with a height of . If she filled it to the brim, there would be of liquid in it. Calculate the height of the drink in the glass if the father drinks half to the rim of the poured glass.

169.

Calculate the volume and area of the body in the figure.

Figure 1

170.

The area of the axial section of a cone whose side is equal to three times the radius of the base plane measures . Calculate the total surface area and volume of this cone and the inscribed angle in the plane of the unfolded lateral surface.

171.

The regular triangular based pyramid is inscribed in the base of a cone. The length of the base of the pyramid is , the length of its slant height l is . Calculate the volume and total surface area of the resulting cone.

172.

We make a cone from the three-third and half of a circle. Calculate the ratio of their volumes.

173.

Calculate the inscribed angle in the lateral surface of the cone whose radius and height are in the ratio .

174.

The area of the base of a cone is . Calculate the radius of the parallel cross section bisecting the height of the cone h. Calculate what part of the volume is the volume of the upper part of the cone.

175.

The ratio between the base area and the lateral surface area of the cone is . What is the volume and total surface area of this cone if its height is equal to ?

176.

Zach made two cones out of clay. Both have the same base area with a radius . The first cone is . What is the change in the total surface area of the cones if the volume of the second cone smaller than the volume of the first cone?

177.

Alexia makes a hat in the shape of an equilateral cone. How tall will the hat be if the lateral surface area measures . She will fill the hat with cotton wool. How many of wool does she need?

178.

Greg makes a metal candlestick in the shape of a truncated cone with diameters of base surfaces as and and a slant height of . How much of metal does he need?

179.

Mac made of fabric for birthday hats in the shape of an oblique cone. The longest side of the hat measures , the shortest and the circumference of the base is . To make the hats look good, he will fill them with a special filler. How much filler does it need for one of such hat?

180.

From the cylinder and cone we assemble the body in the figure. The angle at the top of the axial section of the cone, it measures .

Calculate body height.
Calculate the radius of the base of the cylinder.
Calculate total surface area and volume of the body.

181.

Mira sets up a polygon with cones. It has two types of cones that have the same base area with radius . The first cone has the lateral surface area exactly equal to 6 times the area of its base, and the second cone has the lateral surface area equal to 4 times the area of its base. Determine the ratio of their volumes.

182.

The base face of an upright pyramid with volume is a rectangle with sides of lengths a and b. The length of the height of this pyramid is denoted by v. The numbers a, b and v in this order form an increasing arithmetic sequence with a common difference . Calculate the lengths of the base edges and the height of the pyramid.