1. Algebra 1 Volume of Solid Figures
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2. Warm Up
1) Find the area of a rectangle with width 8 cm and length 7 cm.
2) Calculate the area of a triangle with base 14 cm and height 4 cm.
3) Calculate the area of a trapezoid with bases 14 cm and 15 and height 22 cm.
4) Calculate the perimeter of a circle with radius 42 cm.
5) Calculate the area of a circle with radius 42 cm.
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3. Solid Geometry
Solid geometry is concerned with three-dimensional shapes. Some examples of three-dimensional shapes are cubes, rectangular solids, prisms, cylinders, spheres, cones and pyramids.
The three flat shapes of the triangle, rectangle, and circle may become solids by adding the third dimension  of  depth. The  triangle  becomes  a  cone;  the  rectangle,  a  rectangular  solid;  and  the circle, a cylinder.
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4. Cubes
A cube is a three-dimensional figure with all edges of the same length. s
If s is the length of one of its sides, the
Volume of the cube = s3
Calculate the volume of the cube with length of sides = 7 cm.
Volume of the cube = s3
= (7)3
= 7 x 7 x 7
= 343
Volume of the cube = 343 cm3
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5. Volume of the cuboid = 168 cm3
Rectangular Solids
In a rectangular solid, the length, width and height may be of different lengths. l w h
The volume of the rectangular solid would be the product of the length, width and height. Volume of Rectangular solid = lwh
Calculate the volume of the cuboid with length 7 cm, width 4 cm and height 6 cm.
Volume of the cube = lwh
= 7 x 4 x 6
= 168
Volume of the cuboid = 168 cm3
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6. Now you try!
1) A metal cube of 9 cm is melted and formed into 3 smaller cubes. If the edge of two smaller cubes are 1 cm and 6 cm, find the side of the third smaller cube.
2) Three cuboids of dimensions 2cm x 5cm x 7cm, 4cm x 4cm x 5cm and 2cm x 3cm x 1cm are melted and a cube is formed. Find the side of the cube.
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7. Cylinders
A cylinder is a solid with two congruent circles joined by a curved surface.
In the given figure, the radius of the circular base is r and the height is h. The volume of the cylinder is the area of the base × height.
The volume of the cylinder =Πr2h. r h
Calculate the volume of the cylinder with radius of the base is 8 cm and the height is 10 cm.
The volume of the cylinder =Πr2h
= 3.14 x (8)2 x 14 =
Volume of the cylinder = cm3
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8. Prisms
A prism is a solid that has two congruent parallel bases that are polygons. The polygons form the bases of the prism and the length of the edge joining the two bases is called the height.
The diagrams below show two prisms: s a h a b h
Another with a pentagon-shaped base called a Pentagonal prism.
One with a triangle-shaped base called a Triangular prism
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9. Triangular prism Pentagonal prism
Volume of prism = area of base × height
Triangular prism a b h
Volume of Triangular prism = 1 abh 2
Pentagonal prism s a h
Volume of Pentagonal prism = 5 ash 2
apothem length
height
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10. Volume of the Triangular prism = 200 cm3
Calculate the volume of the Triangular prism with altitude = 5 cm , base = 8 cm and the height is = 10 cm. a b h
Volume of Triangular prism
= 1 abh 2
= 1 x 5 x 8 x 10
= 200
Volume of the Triangular prism = 200 cm3
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11. Volume of the Pentagonal prism = 560 cm3
Calculate the volume of the Pentagonal prism with apothem length = 4 cm , side length = 7 cm and the height is = 8 cm. a h
Volume of Pentagonal prism
= 5 ash 2
= 5 x 4 x 7 x 8
= 560 s
Volume of the Pentagonal prism = 560 cm3
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12. Now you try!
1) Calculate the volume of the cylinder with radius of the base is 10 cm and the height is 13 cm.
2) Calculate the volume of the Triangular prism with altitude = 8 cm , base = 5 cm and the height is = 12 cm.
3) Calculate the volume of the Pentagonal prism with apothem length = 13 cm , side length = 15 cm and the height is = 10 cm.
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13. Spheres
A sphere is a solid with all its points the same distance from the center. r
If r is the length of radius of the sphere ,
Volume of the sphere = 4Πr3 3
Calculate the volume of the sphere with radius = 6 cm.
The volume of the sphere
=4Πr3 = 4 x 3.14 x 6 x 6 x 6 =
Volume of the sphere = cm3
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14. If r is the radius and h is the height,
Cones
A circular cone has a circular base, which is connected by a curved surface to its vertex. A cone is called a right circular cone, if the line from the vertex of the cone to the center of its base is perpendicular to the base. . r h
If r is the radius and h is the height,
Volume of the cone = 1Πr2h 3
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15. Cones
Calculate the volume of the cone with radius 6 cm and the height is = 12 cm. r h
The volume of the cone
=1Π r2h 3
= 1 x 3.14 x 6 x 6 x 12 =
Volume of the cone = cm3
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16. 1) Calculate the volume of the sphere with radius = 13 cm.
Now you try!
1) Calculate the volume of the sphere with radius = 13 cm.
2) Calculate the volume of the cone with radius 5 cm and the height is = 9 cm.
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17. Pyramid
A pyramid is a solid with a polygon base and connected by triangular faces to its vertex. A pyramid is a regular pyramid if its base is a regular polygon and the triangular faces are all congruent isosceles triangles.
Vertex b h s
slant height
Altitude
Face
Base
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18. Regular triangular pyramid Regular square pyramidh a b
Volume of the Regular triangular pyramid = 1abh 6
a =apothem length
Regular square pyramid b h s
Volume of the Regular square pyramid = 1b2h 3
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19. Regular triangular pyramid
Calculate the volume of the regular triangular pyramid with base 10 cm, apothem length 8 and the height 12 cm.
Volume of the Regular triangular pyramid
= 1abh 6
= 1 x 10 x 8 x 12
= 160 h a b
Volume of the Regular triangular pyramid = 160 cm3
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20. Regular square pyramid
Calculate the volume of the regular square pyramid with base 11 cm and the height 12 cm. b h s
Volume of the Regular square pyramid
= 1b2h 3
= 1 x (11)2 x 12
= 484
Volume of the Regular square pyramid = 484 cm3
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21. Now you try!
1) Calculate the volume of the regular triangular pyramid with base 5 cm, apothem length 4 and the height 7 cm.
2) Calculate the volume of the regular square pyramid with base 7 cm and the height 9 cm.
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22. Assessment
1) The carpentry class has agreed to help the physical education teacher by building a box to store the athletic equipment on the edge of the field. The plans called for the box to be 3 feet high, 4 feet long, and 4 feet wide. (Call this Box A.) Two students proposed that the box would hold more equipment if it were taller and not as wide. They want the box to be 3 feet wide, 4 feet high, and 4 feet long. (Call this Box B.)
What is the volume of Box A?
Which box would hold the most,
Box A or Box B? 4 3
Box A
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23. 2) Suppose the radius of a cone is doubled
2) Suppose the radius of a cone is doubled. How could you change the height so that the volume will remain the same?
3) Calculate the volume of the cylinder with radius of the base is 12 cm and the height is 21 cm.
4) Calculate the volume of the Triangular prism with altitude = 7.6 cm , base = 11 cm and the height is = 15 cm.
5) Calculate the volume of the Pentagonal prism with apothem length = 15.4 cm , side length = 19 cm and the height is = 21 cm.
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24. 6) Calculate the volume of the sphere with radius = 34 cm.
7) Calculate the volume of the cone with radius = 12 cm and the height is = 17 cm.
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25. 8) Calculate the volume of the regular triangular pyramid with base 15 cm, apothem length 8 and the height 13 cm.
9) Calculate the volume of the regular square pyramid with base 13 cm and the height 15 cm.
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26. 16 13 9
10) The given cylinder has radius = 13 cm and the height = 16 cm. A cone of radius = 9 cm and height = 13 cm is cut from the cylinder. What is the volume of the remaining piece.
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27. Let’s review Cylinders
A cylinder is a solid with two congruent circles joined by a curved surface.
In the given figure, the radius of the circular base is r and the height is h. The volume of the cylinder is the area of the base × height.
The volume of the cylinder =Πr2h. r h
Calculate the volume of the cylinder with radius of the base is 8 cm and the height is 10 cm.
The volume of the cylinder =Πr2h
= 3.14 x (8)2 x 14 =
Volume of the cylinder = cm3
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28. review Triangular prism Pentagonal prismb h
Volume of Triangular prism = 1 abh 2
Pentagonal prism s a h
Volume of Pentagonal prism = 5 ash 2
apothem length
height
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29. review
Spheres
A sphere is a solid with all its points the same distance from the center. r
If r is the length of radius of the sphere ,
Volume of the sphere = 4Πr3 3
Calculate the volume of the sphere with radius = 6 cm.
The volume of the sphere
=4Πr3 = 4 x 3.14 x 6 x 6 x 6 =
Volume of the sphere = cm3
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30. If r is the radius and h is the height,
review
Cones
A circular cone has a circular base, which is connected by a curved surface to its vertex. A cone is called a right circular cone, if the line from the vertex of the cone to the center of its base is perpendicular to the base. . r h
If r is the radius and h is the height,
Volume of the cone = 1Πr2h 3
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31. Regular triangular pyramid Regular square pyramid
review
Regular triangular pyramid h a b
Volume of the Regular triangular pyramid = 1abh 6
a =apothem length
Regular square pyramid b h s
Volume of the Regular square pyramid = 1b2h 3
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32. You did a great job today!
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