Solved problems for the chapter:

Surface area of 3-D shapes

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Surface area and volume of a sphere

1. Characteristic elements of a sphere

2. Sphere surface area

3. Volume of the sphere

4. Hemisphere

5. Spherical section

6. Forum

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Cone for KS3

1. Cone Properties

1.1. Elements of a cone

1.2. Surface of an upright cone

1.2.1. Vertical Cone Grid

1.2.2. Area of an upright cone

1.3. Volume of an upright cone

1.4. Axial section of an upright cone

2. Types of Cones

2.1. Vertical and oblique cones

2.2. Equilateral cone

3. Forum

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Pyramid for KS4

0.1. Types of pyramids:

0.2. Regular pyramid and irregular pyramids:

0.3. Surface area of a pyramid:

0.4. Volume of a pyramid:

1. Forum

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Surface area and volume of a pyramid

1. Pyramid Properties

1.1. Characteristic elements of the pyramid

1.2. The area of the pyramid

1.2.1. Pyramid Network

1.2.2. Calculating the area of the pyramid

1.3. Volume of an upright pyramid

2. Types of vertical pyramids

2.1. 3, 4, 5, ..., n side of the pyramid

2.2. Equilateral pyramid

2.2.1. Regular base pyramid

2.2.2. Equality of the pyramids

2.3. Special examples of pyramids

2.3.1. Properties of three-sided pyramid

2.3.2. Quadrilateral pyramid

2.3.3. A regular four-sided pyramid

3. Forum

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Probability of compound events

1. Probability of an impossible event

2. Probability of the opposite event

3. The probability of an event that is the mode of another event

4. Probability of event difference

5. Probability of the sum of events

6. The Maximum of Probability of an Event

7. Forum

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Measurement and convert rates for KS4

1. Measuring length (distance)

2. The measurement of area

3. Volume measurement

4. Measurement of (plane) angles

4.1. Measuring angles in angular degrees

4.2. Measuring angles in radians

4.3. Conversion between angular degrees and radians

4.3.1. Unit circle

5. Forum

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Prism for KS4

1. Types of prisms

1.1. Vertical and oblique prisms

1.2. Regular prisms

1.3. Equilateral prism

2. Prism surface area

3. Prism volume

4. Forum

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Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:surface_area_of_3-d_shapes/3667/

Example 1.

The surface area of the sphere measures . Calculate its radius exactly.

Example 2.

A spherical meteoroid has a surface area of . What is the length of the longest straight closed path along its surface?

The diameter of a sphere measures . Calculate the surface area and volume of the sphere in square or cubic centimetres.

The base area and the curved surface area of an equilateral cone measures . What is the radius of the cone?

The length of all the sides of a cuboid is 96 cm. Calculate the length of a cuboid if it is 8 cm wide and 6 cm high .

The surface area of the sphere measures . Calculate its diameter.

Calculate the height of a cuboid, long and wide, if its area is .

The area of the base area of an equilateral cone measures . Calculate the total surface area of the cone.

The base area of a cuboid is . Calculate the total surface area of the cuboid if its lateral surface area is .

10.

How many wafers do we need to make a tall ice cream cone with an opening diameter ?

11.

Calculate the diagonal of the base surface of a cuboid whose base is 15 cm long and 8 cm wide. Use the Pythagorean theorem.

12.

The cone jacket measures . If the radius and the slant height of the cone are in the ratio , calculate the height of the cone.

13.

Calculate the area of a sphere whose area of the main spherical circle measures .

14.

What cone has the area of the circular base and curved surface area in a ratio of ?

15.

Calculate the width of a cuboid, long and high, if its total surface area is

16.

The slant height of a cone is three times its radius and the height of the cone measures . Calculate the diameter of the cone.

17.

The larger sphere has four times the surface area of the smaller sphere. What is the ratio of their radii?

18.

Calculate the width and volume of a cuboid 30 cm long and 10 cm high if its area measures 900 square centimetres.

19.

The perimeter of the triangle formed by the projection of a cone is . If the radius and the slant height are in the ratio of , calculate the curved surface area of the cone.

20.

Cube and cuboid have equal volume . At the same time, the edge of the cube is the same length as the width of the cuboid. Calculate the total surface area of the cube and the cuboid if the length of the cuboid is 12.8 cm.

21.

The base area of a cylinder is and its curved surface area is . What is the total surface area of the cylinder?

22.

When playing billiards, players first arrange the balls inside a triangular frame (equilateral triangle). Calculate the radius of each sphere to 2 significant figures if the height of the triangular frame is and the balls are arranged.

23.

The cylindrical roller mesh has a total total surface area of . Its curved surface area measures . What is the area of the circular base?

24.

The surfaces area of a sphere and an equilateral cone are in the ratio . Calculate the ratio of their volumes.

25.

A cylindrical pipe with total surface area has a curved surface area of . What is the value of the base area?

26.

We have a given cone with radius and height . Calculate its volume and total surface area.

27.

Calculate the length of a cuboid, wide and high, if its total surface area is .

28.

The diagonal of a cube measures . Calculate the surface area and volume of the cube.

29.

Calculate the total surface area of a cylinder with base area and height .

30.

We would like to place a decorative concrete ball on top of a concrete column. To prevent it from rolling off, we ground it a little in one place. The resulting flat surface has the area . What is the new height of the ground ball if the original ball was high?

31.

The area of the squares of a cube measures . Calculate the surface area and volume of the cube.

32.

The curved surface area of a cylinder is and the height . Calculate the total surface area of the cylinder.

33.

The sketch of a glass aquarium net without a lid is shown above.

Write down the length, width and height of the aquarium.
How many square meters of glass do we need to make such an aquarium?
Can we pour 10 liters of water into an empty aquarium?

34.

The curved surface area of an equilateral cylinder measures . Calculate the height of the cylinder.

35.

Calculate the surface area and volume of a cube with height .

36.

The curved surface area of a cylinder whose height is is . Calculate the diameter of the cylinder.

37.

Calculate the area of a sphere with volume

38.

The diagonal of the square which form a cube measures . Calculate the surface area and volume of the cube.

39.

If the roller is spread out on a flat surface, it takes up surfaces. What is the height of the cylinder if the area of the base measures ?

40.

Calculate the surface area of a cube whose volume measures .

41.

A cylinder holds water. The height of the cylinder is . Calculate the total surface area of the cylinder.

42.

Calculate the area of the right circular cylinder given it's radius and it's volume, .

43.

A hay bale with a total surface area of fits snugly in a cube-shaped packing box. What is the height the bale?

44.

In a right circular cylinder, the area of the base plane and the curved surface area measured . Calculate the total surface area and volume of the cylinder.

45.

The curved surface area of the cylinder is . What is the area of the base if the radius and height of the cylinder are in the ratio ?

46.

Calculate the surface area of a cube whose side measures 4.87 cm. What will be the length of the side of the cube if its surface area is increased by 43%?

47.

The base area and the casing of a cylinder with a diameter of are in the ratio . Calculate height of the cylinder.

48.

A triangular prism with base area and triangular surface areas of , and is to be cut out from a paper. Calculate the paper needed in .

49.

The sum of all the lengths of the sides of a cuboid is . Calculate the area and volume of this cuboid if the lengths of the sides are in the ratio a: b: c = 6: 2: 5.

50.

The volume of a cuboid measures , and its sides are in the ratio a: b: h = 10: 3: 5. Calculate its surface area.

51.

What is the area of an upright cone that has the same volume as a sphere with area , and the height of the cone is equal to the diameter of the sphere?

52.

Let's have a rectangle with a width b and length a that is units longer than the width.

In the form of a function rule, write the dependence of the side a on the width b.
Express the perimeter of the rectangle P with width b.
If we increase the width by , how much will the perimeter increase?

53.

Calculate the surface area and volume of an equilateral triangular prism with side a = 5 cm.

54.

Calculate the ratio of the surface areas and volumes of a regular triangular and a regular hexagonal prism that are 6.7 cm high and their base side measures 3.9 cm.

55.

A group of friends went on a trip by car. After of the regional road, they reached the highway, where they accelerated to a speed of .

Write down the function rule for calculating the total route on the regional road and highway.
Calculate how many kilometers they traveled in total after hours on the highway?
After hours of driving on the highway, they reached the exit and drove on the old road. How long did they drive on the old road if they reached their destination after from the start of their journey?

56.

Hemisphere of a cylinder have a surface area . What is the area and volume of the composite body if its height is ?

57.

The curved surface area and the base area of a cylinder are in a ratio of . What is the area of the curved surface if the radius of the cylinder is equal to ? Give the result obtained in square decimetres.

58.

Calculate the surface area and volume of an equilateral prism with side a = 4 cm.

59.

The area of the base of a square base pyramid measures , and the lateral edge measures . Calculate the total surface area of the pyramid.

60.

Mom bought a vase in the shape of an equilateral triangular prism. She poured 1.3 liters of water and the vase was filled to . Calculate the surface area of this vase.

61.

A cuboid with edges , and is used to form a sphere. What is the area and volume of the delineated sphere?

62.

A circle is inscribed to the base of a equilateral triangular prism. If its radius measures 4 cm, calculate the surface area and volume of the prism when it is 10 cm high.

63.

The shell of a regular triangular base prism with length of its base measures . What is the height of the prism?

64.

Calculate the surface area and volume of a regular octagonal prism with base side a = 4 cm and 10 cm high.

65.

The surface area of the base of a cone with volume measures . What is the height of the cone?

66.

The sketch above shows the axial section of a right-angled circular cylinder. Calculate the area of this cylinder.

67.

A trapezium with sides a = 17 cm, b = 12 cm, c = 8 cm and d = 11 cm is the base of a four-sided prism having a height equal to one third of the base perimeter. Calculate the surface area and volume of the prism.

68.

A trapezium with sides a = 17 mm, b = d = 12 mm and c = 8 mm is the base of a four-sided prism whose rectangular surface area is equal to the base area. Calculate the surface area and volume of the prism.

69.

Given a regular square prism with base edge a = 10 cm and height h = 15 cm. Calculate the total surface area and volume of the hollow cylinder bounded by a prism delineated and inscribed cylinder.

70.

What is the area and volume of a sphere whose radius is r = 6 dm and height h = 8 dm?

71.

The radius of the sphere measures . Calculate the surface area of the sphere in square metres.

72.

The base area of a circular-based cone is and the curved surface area is . What is the total surface area of the cone?

73.

We need a of paper to make a cone. The base of the cone is a circle with area . What will be the area of its curved surface in decimetre square?

74.

The radius of the cone measures . What does the slant height of the cone measures if the area of the circular base and the curved surface area are in the ration ?

75.

The edge of the cube has a length of . Calculate the area and perimeter of the figure .

76.

Place a styrofoam ball with radius on the water surface. The ball sinks for . Calculate the circumference of a small spherical circle.

77.

We have a cube with an edge of . By what percentage do the face and body diagonals increase if the edge of the cube is increased to ?

78.

A sphere and an equilateral cylinder have the same volume. What is the area of the sphere if the radius of the base plane of the cylinder is ?

79.

How much does the area of a cube increases if we increase the length of the edge of by ?

80.

We need cardboard to build a triangular base prism grid. The base is an isosceles triangle with area . How much cardboard do we need to coat this prism?

81.

The volume of an equilateral triangle base prism measures and the base area measures . Calculate the total surface area of the prism. The result should be accurate.

82.

What is the area and volume of an upright cone of a sphere with radius if the height of the cone is equal to the diameter of the base plane?

83.

The total surface area of an equilateral triangle base prism is five times the area of its base. The curved surface area measures . Calculate the height of the prism.

84.

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

85.

How much area can be paved with base edge cubes cut from granite? Assume no cutting losses.

86.

Jane made a rice bowl in the shape of a right-angled circular cylinder. He filled it with rice (the bowl was full). Wanting to give it to his mother for her birthday, he covered it with decorative paper. Calculate the area of the cylinder (paper used), if the radius and height of the container is in the ratio 3:6?

87.

The radius r of a cone measures and the area of the curved surface measures . Calculate the height of the cone.

88.

We have a given cone with slant height and height . Calculate its total surface area and volume.

89.

The curved surface area CSA of a cone measures and the slant height measures . Calculate the total surface area TSA of the cone.

90.

We have a given cone having the area of the base and the slant height . Calculate its total surface area and volume.

91.

A pyramid has a triangle of size as its base face, and the lateral surface area of its is . What is the to total surface area of this pyramid?

92.

A pyramid has a rectangular base with a base area of size , and lateral surface consists of two pairs of triangles with areas and . What is the total surface area of this pyramid?

93.

What is the maximum number of tennis balls with surface area that can be packed into a cylindrical box? The box has a volume of and is wide enough to fit tennis balls.

94.

The base area of a square base pyramid with total surface area of measures . What is the area of the each face of the pyramid?

95.

The base area of a triangular base pyramid with volume measures . How high is the pyramid?

96.

The grid of a pyramid is bases, of which the lateral faces are occupied by . What is the area of the base of the pyramid?

97.

How much is the radius r of the base area of a cone with total surface area and slant height ?

98.

The volume of a square base pyramid is and the height is . What is the area of the base of the pyramid?

99.

The area of the base of a triangular base pyramid measures and the slant height . Calculate the total surface area of the pyramid.

100.

How much is the slant height l of a cone that is tall and has a circular base area of ?

101.

A pyramid has as its base as a rectangle with sides and . The slant height above the shorter side of the base measures , and above the longer base measures . Calculate the lateral surface area of the pyramid. Leave your result in square centimetres.

102.

The Egyptian pyramid captures of air and creates shadows beneath it. How far above the ground is the highest point of the pyramid?

103.

What is the slant height of a cone with circular base area of and a curved surface area of .

104.

The lateral surface of an equilateral triangular base pyramid measures and the slant height is three times the length of the side of the base. Calculate the total surface area of the pyramid.

105.

The base area of a triangular base pyramid measures . What is the total surface area of a pyramid if the area of the base and the lateral surface area are in the ratio 1:3?

106.

The lateral surface area of a square base pyramid with length of one side of the base as measures . Calculate the height of the pyramid.

107.

In an equilateral triangular base pyramid, there is air. The height of the pyramid measures and its slant height . Calculate the lateral edge of the pyramid.

108.

What are the total surface area and volume of a cone having a slant height and a radius of the circular base ?

109.

The length of one side of the base of a square base pyramid measures and the height . Calculate the total surface area and volume of the pyramid.

110.

An high cone has the circumference c of the base area as . Calculate the total surface area and volume of this cone.

111.

If a quadrilateral base pyramid is unfolded on a flat base, the area of the resulting figure measures . What is the lateral height of the pyramid if the area of the base is ?

112.

A small company has employees 8 men and 4 women. Four of them will attend the seminar.

In how many ways can they choose seminar participants to make gender representation equal?
How many ways can they choose seminar participants if more men than women have to attend the seminar?
What is the probability that all four participants in the randomly selected delegation will be of the same sex.

113.

A vertical cone, whose circular base is of circumference , has a right triangle for its axial section. Find its total surface area and volume.

114.

The shell of a regular three-sided prism measures and its height is . Calculate the total surface area of the prism.

115.

Calculate the total surface area and volume of an equilateral cone having an equilateral triangle with area .

116.

A building is 10 meters long and 5 meters wide. The roof of the building has the shape of an equilateral triangle base prism. How many square meters of bricks will the owner need to replace the roof?

117.

A sheet of paper has the shape of a rectangle with a length of 30 cm and a width of 20 cm.

Roll the sheet into a cylinder so that the shorter side is the height of the cylinder. Calculate the area of the resulting cylinder.
Let the sheet be transformed into a regular 4-sided prism with a height equal to the length of the shorter side. Calculate the surface area of the prism.
By what percentage is the area of the cylinder larger than the area of the prism?

118.

We have a cone with an axial cross-sectional area and a circular base diameter . Calculate its total surface area and volume.

119.

Calculate the total surface area and volume of the body obtained if the trapezium ABCD with sides , , and , rotate around the side a for full angle.

120.

Calculate the area of an equilateral cone with volume .

121.

Calculate the slant height, height and volume of an upright cone with total surface area and radius of circular base .

122.

An Iron crate with length , width and height is filled with sand which is then shaken into high upright cone. Calculate the diameter of the circular base of the resulting cone.

123.

The surface area of a cube and a cuboid is equal. If the side lengths of the cuboid are a = 12.5 cm, b = 4.5 cm and h = 3.5 cm, calculate the length of the cube and its volume.

124.

One of the sides of a squared base pyramid measures , and its height . We also have a cone with the radius of the circular base as and the height . Calculate:

The total surface area of the pyramid and the cone.
what angle is enclosed by the slant height and the base surface of the pyramid.
by what percentage it is necessary to increase the height of the pyramid so that the volumes of both bodies are equal.

125.

The volume of the truncated cone measures and the radius and . Calculate its total surface area.

126.

The volume of a equilateral triangular based prism is and the height is . Calculate the curved surface area of the prism.

127.

Kate made a metal base in the shape of a cylinder and glued a wooden rocket to it (see picture). Calculate the total surface area of the resulting body and calculate how much wood is used to make the rocket.

Sketch

128.

Calculate the surface area and volume of the figure below:

Figure

129.

Calculate the surface area and volume of a cube whose length is .

130.

A right-sided pyramid has an equilateral triangle with side 15 cm as the base surface. How much does the side edge of this pyramid measure if the lateral area is 10 times the area of the base? Also, calculate the angle between the base and side edges.

131.

Calculate the area of the shaded triangle drawn in a cube with side 7 cm as shown in the figure below:

Figure

132.

Calculate the length of the hypotenuse in the right triangle ABC, the volume and surface area of the figure below:

Figure

133.

Calculate the area of a right circular cylinder, given the area of its curve surface, and the volume, .

134.

The edge of a regular four-sided pyramid measures 30 dm and its height is 20 dm. Calculate the area of the lateral surface and the volume of the pyramid.

135.

What is the surface area of a regular six-sided pyramid with a base edge a = 12 cm and height 30 cm?

136.

What is the surface area of a regular six-sided pyramid with a base edge a = 10 cm and a side edge s = 15 cm?

137.

What is the surface area of a regular six-sided pyramid with height h = 5 cm and side edge s = 15 cm?

138.

Calculate the surface area of a triangular prism with base side lengths a = 12.7 cm, b = 14.5 cm and c = 21 cm when its volume is .

139.

A right-angled triangular prism has two of its base lengths as 4 cm and 7 cm. If the third side is the hypotenuse and the prism is 10 cm high, calculate its surface area and volume.

140.

The pyramid has a square with a side 25 cm as its base, and its side edge measures 20 cm. Calculate the slant height, the angle between the base and side surfaces, and the surface area of the pyramid.

141.

Calculate the height of a right circular cylinder given it's curve surface area, and its total surface area, .

142.

How high is a regular hexagonal pyramid, whose base edge measures 5 cm and the lateral surface area is .

143.

Calculate the area of a cylindrical box whose volume measures and the area of the curve surface is equal to the area of the base surface.

144.

Calculate the area of the polyhedron in the figure below.

145.

What is the ratio of the surface area of a cuboid to the surface area of a cube if the cuboid dimensions are 300 cm, 2.5 dm and 0.4 m and the volume of the cube and the cuboid are equal?

146.

Calculate the area of the polyhedron in the figure below.

147.

What is the length of the side edge of a four-sided pyramid with surface area of if the base surface is a square with side 13 cm?

148.

The radius of a cylinder is 7 cm and it's volume is . calculate the number of times the area of the base surface is smaller than the right-angled circular cylindrical.

149.

A triangle with sides 15 cm, 22 cm, and 24 cm is the base of a pyramid of height 13 cm. Calculate the side edge s.

150.

Calculate the surface area of a three-sided pyramid with the lateral area of . The side edges are perpendicular to each other, and the base is an equilateral triangle.

151.

Calculate the base area and area of the diagonal section of a cube with volume .

152.

Calculate the surface area of a rectangular-based pyramid with a volume of if the base measures 10 cm by 4 cm.

153.

Calculate the area of the figure below:

154.

The height of a pyramid is 8 cm, the length of the side edge is 15 cm. Draw a sketch of the pyramid and calculate the value of the angle marked between the base surface and the side edge.

155.

In the rectangles appearing in the following tasks, we denote and .

The circumference of the rectangle is 880 cm. Calculate the sides a and b so that the area of the rectangle is the largest.
The rectangle (a = 8 cm, b = 3 cm) is rotated by around an axis outside the rectangle that is parallel to the side and away from it by d = 2 cm. Calculate the volume and area of the spindle. The result should be accurate.
Point M lies on side BC so that | BM | : | MC | = 1: 3. The intersection of the line DM with the diagonal AC is denoted by P. In what proportion does the point P diagonal AC divide? Write down the ratio | AP | : | PC |.

156.

A candy box shape is a regular six-sided prism. The base side length of the prism is 6 cm and 5 cm high as shown in the figure below:

Figure 1

Calculate the area of the base of the prism and the size of the marked angle in the figure.
Calculate the surface area of the prism.
If the total volume of candies in the box is approximately , calculate the proportion of the volume occupied by the candies in the box.

157.

Alexander bought a cylindrical bottle with a radius of 1.5 cm and a height of 15 cm.

He poured 30ml of perfume into the bottle. How high above the bottom is the level of the perfume?
The bottle was in a square shape. Calculate the possible area of the smallest packaging.
After a month, Alexander consumed 15% of the perfume. How high was the perfume level after one month?

158.

The radius of a vertical cone measures and the slant height of the cone is . Sketch the cone and mark the axial section. Calculate the area of the axial section.

159.

What is the height of a regular three-sided pyramid with the same volume as a cube with the surface area of , If the base edge of the pyramid is equal to three-quarters of the side diagonal of the cube.

160.

The base of a prism is a rhombus with diagonals and as shown in the figure below. If the diagonal of the side surface measures , calculate the surface area of the prism.

Figure

161.

What is the total surface area of a truncated cone with radii and and slant height ?

162.

What is the area of a truncated cone with radii and and height ?

163.

Parallelogram ABCD with sides and rotate by around a longer diagonal f with length . How much does the total surface area and volume of the resulting body measure?

164.

An isosceles trapezium ABCD with sides is given:

Prove the height of the trapezium measures . Calculate the volume of an upright prism with height , which has the trapezium ABCD as its base.
Calculate the area of the trapezium obtained by the central extension of the trapezium . The stretch has its centre at the point A and the factor .
The trapezium ABCD rotates around its bisector for . Calculate the area and volume of the resulting revolution.

165.

Manny bought a new flower pot in the shape of a truncated cone. The height of the flower pot is , and the radii of the circular base of the pot are in a ratio of . How much is the circumference of the smaller circular base if we can put of soil in the pot?