Exercise 1

Calculate the surface area and the volume of a regular tetrahedron with an edge of 5 cm.

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Exercise 2

Calculate the diagonal, lateral area, surface area and volume of a cube with an edge of 5 cm.

Exercise 3

Calculate the surface area and the volume of a regular octahedron with an edge of 5 cm.

Exercise 4

Calculate the surface area and volume of a dodecahedron with an edge of 10 cm, knowing that the apothem of one face measures 6.88 cm.

Exercise 5

Calculate the surface area and the volume of a regular icosahedron with an edge of 5 cm.

Exercise 6

Calculate the lateral area, surface area and volume of a prism whose base is a rhombus with diagonals of 12 and 18 cm in length.

Exercise 7

Calculate the lateral area, surface area and volume of a square pyramid whose base edge is 10 cm and its height is 12 cm.

Exercise 8

Calculate the lateral area, surface area and volume of a hexagonal pyramid with a base edge of 16 cm and a side edge of 28 cm.

Exercise 9

Calculate the lateral area, surface area and volume of a truncated square pyramid whose base edges are 24 and 14 cm and whose lateral edge is 13 cm.

Exercise 10

Calculate the lateral area, surface area and volume of a cone whose slant height is 13 cm and base radius is 5 cm.

Exercise 11

Calculate the lateral area, surface area and volume of a cone whose height is 4 cm and base radius is 3 cm.

Exercise 12

Calculate the lateral area, surface area and volume of a truncated cone with radii of 2 and 6 cm and a height of 10 cm.

Exercise 13

Calculate the lateral area, surface area and volume of a truncated cone with radii of 12 and 10 cm and a slant height of 15 cm.

Exercise 14

Calculate the area of the circle resulting from cutting a sphere with a 35 cm radius by a plane whose distance from the centre of the sphere is 21 cm.

Exercise 15

Calculate the area and volume of a sphere inscribed in a cylinder with a height of 2 m.

Exercise 16

Calculate the volume of a hemisphere with a radius of 10 cm.

Exercise 17

Calculate the area and volume of the following spherical cap.

Exercise 18

Calculate the area of the spherical zone and the volume of a spherical segment whose radius circles are 10 and 8 cm and the distance between them is 6 cm.

 

Solution of exercise 1

Calculate the surface area and the volume of a regular tetrahedron with an edge of 5 cm.

{ A }_{ T } = sqrt { 3 } times { 5 }^{ 2 } = 43.30 { cm }^{ 2 }

V = frac { sqrt { 2 } }{ 12 } times { 5 }^{ 3 } = 14.73 { cm }^{ 3 }

 

Solution of exercise 2

Calculate the diagonal, lateral area, surface area and volume of a cube with an edge of 5 cm.

D = { 5 }^{ 2 } + { 5 }^{ 2 } + { 5 }^{ 2 }

D = sqrt { { 5 }^{ 2 } + { 5 }^{ 2 } + { 5 }^{ 2 } } = sqrt { 75 } = 8.66 cm

{ A }_{ L } = 4 times { 5 }^{ 2 } = 100 { cm }^{ 2 }

{ A }_{ T } = 6 times { 5 }^{ 2 } = 150 { cm }^{ 2 }

V = { 5 }^{ 3 } = 125 { cm }^{ 3 }

 

Solution of exercise 3

Calculate the surface area and the volume of a regular octahedron with an edge of 5 cm.

A = 2 sqrt { 3 } times { 5 }^{ 2 } = 86.60 { cm }^{ 2 }

V = frac { sqrt { 2 } }{ 3 } times { 5 }^{ 3 } = 58.92 { cm }^{ 3 }

 

Solution of exercise 4

Calculate the surface area and volume of a dodecahedron with an edge of 10 cm, knowing that the apothem of one face measures 6.88 cm.

A = 30 times 10 times 6.88 = 2,064 { cm }^{ 2 }

V = frac { 1 }{ 4 } (15 + 7 sqrt { 5 } ) times { 10 }^{ 3 } = 7,633.12 { cm }^{ 3 }

 

Solution of exercise 5

Calculate the surface area and the volume of a regular icosahedron with an edge of 5 cm.

A = 5 times sqrt { 3 } times { 5 }^{ 2 } = 216.51 { cm }^{ 3 }

V = frac { 5 }{ 12 } (3 + sqrt { 5 } ) times { 5 }^{ 3 } = 272.71 { cm }^{ 3 }

 

Solution of exercise 6

Calculate the lateral area, surface area and volume of a prism whose base is a rhombus with diagonals of 12 and 18 cm in length.

{ l }^{ 2 } = { 9 }^{ 2 } + { 6 }^{ 2 }

l = sqrt { { 9 }^{ 2 } + { 6 }^{ 2 } } = 10.82 cm

{ A }_{ L } = 4 times (24 times 10.82) = 1,038.72 { cm }^{ 2 }

{ A }_{ T } = 1,038.72 + (2 times frac { 18 times 12 }{ 2 }) = 1,254.72 { cm }^{ 2 }

V = frac { 18 times 12 }{ 2 } times 24 = 2592 { cm }^{ 3 }

Solution of exercise 7

Calculate the lateral area, surface area and volume of a square pyramid whose base edge is 10 cm and its height is 12 cm.

{ Ap }^{ 2 } = { 12 }^{ 2 } + { 5 }^{ 2 }

Ap = sqrt { { 12 }^{ 2 } + { 5 }^{ 2 } } = 13 cm

{ P }_{ B } = 4 times 10 = 40 cm

{ A }_{ L } = frac { 40 times 13 }{ 2 } = 260 { cm }^{ 2 }

{ A }_{ T } = 260 + { 10 }^{ 2 } = 360 { cm }^{ 2 }

V = frac { 100 times 12 }{ 3 } = 400 { cm }^{ 3 }

 

Solution of exercise 8

Calculate the lateral area, surface area and volume of a hexagonal pyramid with a base edge of 16 cm and a side edge of 28 cm.

{ 28 }^{ 2 } = { Ap }^{ 2 } + { 8 }^{ 2 }

Ap = sqrt { { 28 }^{ 2 } - { 8 }^{ 2 } } = 26.83 cm

{ A }_{ L } = frac { 6 times 16 times 26.83 }{ 2 } = 1,287.84 { cm }^{ 2 }

 

Solution of exercise 9

Calculate the lateral area, surface area and volume of a truncated square pyramid whose base edges are 24 and 14 cm and whose lateral edge is 13 cm.

Ap = sqrt { { 13 }^{ 2 } - { 5 }^{ 2 } } = 12 cm

{ 12 }^{ 2 } = { h }^{ 2 } + { 5 }^{ 2 }

h = sqrt { { 12 }^{ 2 } - { 5 }^{ 2 } } = 10.91 cm

P = 24 times 4 = 96 cm

{ P }' = 14 times 4 = 56 cm

{ A }_{ L } = frac { 96 + 59 }{ 2 } times 12 = 912 { cm }^{ 2 }

A = { 24 }^{ 2 } = 576 { cm }^{ 2 }

{ A }' = { 14 }^{ 2 } = 196 { cm }^{ 2 }

{ A }_{ T } = 912 + 576 + 196 = 1684 { cm }^{ 2 }

V = frac { 10.91 }{ 3 } times (576 + 196 + sqrt { 576 times 196 }) = 4029.43 { cm }^{ 3 }

 

Solution of exercise 10

Calculate the lateral area, surface area and volume of a cone whose slant height is 13 cm and base radius is 5 cm.

 

 

{ A }_{ l } = pi times 13 times 5 = 204.20 { cm }^{ 2 }

{ A }_{ T } = pi times 13 times 5 + pi times { 5 }^{ 2 } = 282.74 { cm }^{ 2 }

{ 13 }^{ 2 } = { h }^{ 2 } + { 5 }^{ 2 }

h = sqrt { { 13 }^{ 2 } - { 5 }^{ 2 } } = 12 cm

V = frac { pi times { 5 }^{ 2 } times 12 }{ 3 } = 314.159 { cm }^{ 3 }

Solution of exercise 11

Calculate the lateral area, surface area and volume of a cone whose height is 4 cm and base radius is 3 cm.

{ s }^{ 2 } = { 4 }^{ 2 } + { 3 }^{ 2 }

s = sqrt { { 4 }^{ 2 } + { 3 }^{ 2 } } = 5 cm

{ A }_{ l } = pi times 3 times 5 = 47.12 { cm }^{ 2 }

{ A }_{ T } = pi times 3 times 5 + pi times { 3 }^{ 2 } = 75.39 { cm }^{ 2 }

V = frac { pi times { 3 }^{ 2 } times 4 }{ 3 } = 37.70 { cm }^{ 3 }

Solution of exercise 12

Calculate the lateral area, surface area and volume of a truncated cone with radii of 2 and 6 cm and a height of 10 cm.

{ s }^{ 2 } = { 10 }^{ 2 } + { (6 - 2) }^{ 2 }

s = sqrt { { 10 }^{ 2 } + { (6 - 2) }^{ 2 } } = 9.165 cm

{ A }_{ L } = pi times (6 + 2) times 9.165 = 230.43 { cm }^{ 2 }

{ A }_{ T } = 230.43 + (pi times { 6 }^{ 2 }) + (pi times { 2 }^{ 2 }) = 356.14 { cm }^{ 2 }

V = frac { 1 }{ 3 } pi times 10 ({ 6 }^{ 2 } + { 2 }^{ 2 } + sqrt { { 6 }^{ 2 } times { 2 }^{ 2 } }) = 544.54 { cm }^{ 3 }

 

Solution of exercise 13

Calculate the lateral area, surface area and volume of a truncated cone with radii of 12 and 10 cm and a slant height of 15 cm.

{ A }_{ L } = pi times (12 + 10) times 15 = 1,036.73 { cm }^{ 2 }

{ A }_{ T } = 1036.72 + pi times { 12 }^{ 2 } + pi times { 10 }^{ 2 } = 1,803.27 { cm }^{ 2 }

{ 15 }^{ 2 } = { h }^{ 2 } + { (12 - 10) }^{ 2 }

h = sqrt { { 15 }^{ 2 } - { 2 }^{ 2 } } = 14.866 cm

V = frac { 1 }{ 3 } pi times 14.866 ({ 12 }^{ 2 } + { 10 }^{ 2 } + sqrt { { 12 }^{ 2 } times { 10 }^{ 2 } }) = 5,666.65 { cm }^{ 3 }

Solution of exercise 14

Calculate the area of the circle resulting from cutting a sphere with a 35 cm radius by a plane whose distance from the centre of the sphere is 21 cm.

 

{ 35 }^{ 2 } = { 21 }^{ 2 } + { r }^{ 2 }

A = pi times { 28 }^{ 2 } = 2,461.76 { cm }^{ 2 }

 

Solution of exercise 15

Calculate the area and volume of a sphere inscribed in a cylinder with a height of 2 m.

A = 4 pi { 1 }^{ 2 } = 12.57 { m }^{ 2 }

V = frac { 4 }{ 3 } pi { 1 }^{ 3 } = 4.19 { m }^{ 3 }

 

Solution of exercise 16

Calculate the volume of a hemisphere with a radius of 10 cm.

V = frac { 2 }{ 3 } pi times { 10 }^{ 3 } = 209.44 { cm }^{ 3 }

 

Solution of exercise 17

Calculate the area and volume of the following spherical cap.

A = 2 pi times 7 times 5 = 219.91 { cm }^{ 2 }

V = frac { 1 }{ 3 } pi times { 5 }^{ 2 } times (3 times 7 - 5 ) = 418.88 { cm }^{ 3 }

 

Solution of exercise 18

Calculate the area of the spherical zone and the volume of a spherical segment whose radius circles are 10 and 8 cm and the distance between them is 6 cm.

A = 2 pi times 10 times 6 = 376.99 { cm }^{ 2 }

V = frac { 1 }{ 6 } pi times 6 ({ 6 }^{ 2 } + 3 times { 10 }^{ 2 } + 3 times { 8 }^{ 2 } ) = 1658.76 { cm }^{ 3 }

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.