Calculate the volumes of prisms, cylinders, pyramids, and cones

1. Volume
The student is able to (I can):
• Calculate the volumes of prisms, cylinders, pyramids, and
cones

2. facefacefaceface – the flat polygonal surface on a three-dimensional
figure.
edgeedgeedgeedge –––– the segment that is the intersection of two faces.
vertexvertexvertexvertex – the point that is the intersection of three or more
edges.
face
edge
vertex•

3. polyhedronpolyhedronpolyhedronpolyhedron – a three-dimensional figure composed of
polygons. (plural polyhedrapolyhedrapolyhedrapolyhedra)
prismprismprismprism – two parallel congruent polygon bases connected by
faces that are parallelograms.
cylindercylindercylindercylinder – two parallel congruent circular bases and a curved
surface that connects the bases.

4. pyramidpyramidpyramidpyramid – a polygonal base with triangular faces that meet at
a common vertex.
coneconeconecone – a circular base and a curved surface that connects the
base to a vertex.

5. rightrightrightright prismprismprismprism – a prism whose faces are all rectangles.
obliqueobliqueobliqueoblique prismprismprismprism – a prism whose faces are not rectangles.
altitudealtitudealtitudealtitude – a perpendicular segment joining the planes of the
bases (the height).

6. VolumeVolumeVolumeVolume
Let’s consider a deck of cards. If a deck is stacked neatly, it
resembles a right rectangular prism. The volume of the
prism is
V = Bh,
where B is the area of one card, and h is the height of the
deck.
If we shift the deck so that it becomes an
oblique prism, does it have the same
number of cards?

7. For any prism, whether right or oblique, the volume is
V = Bh
where h is the altitude, not the length of the lateral edge.

8. Likewise, for cylinders, it doesn’t matter whether the cylinder
is right or oblique, the volume is
V = Bh = πr2h

9. Examples
Find the volume of each figure:
1.
2.
10 ft.
8 ft.
3 m
19 m

10. Examples
Find the volume of each figure:
1.
2.
10 ft.
8 ft.
3 m
19 m
( )2 2
3 9 m= π = πB
3
(9 )(19) 171 mV = π = π
( )( )2
5 10
172.05
180
4tan
5
= =
 
 
 
B
V = (172)(8) = 1376 ft3

11. The volume of a pyramid with base area B and height h is
1
3
V Bh=
The volume of a cone is
21 1
3 3
V Bh r h= = π

12. Examples
Find the volume of the following:
1.
2.
10 mm
10 mm
13 mm
5 mm
12 mm
(Pyth. triple)
2 yd
3 yd
2 yd

13. Examples
Find the volume of the following:
1.
2.
2
22 3
3 yd
4
B = =
31
( 3)(3) 3 yd
3
V = =
10 mm
10 mm
13 mm
5 mm
12 mm
(Pyth. triple)
21
(10 )(12)
3
V =
3
400 mm=
2 yd
3 yd
2 yd

14. Examples
3.
4.
7 ft.
21 ft.
25 mi
20 mi

15. Examples
3.
4.
7 ft.
21 ft. 2
3
1
(7 )(21)
3
343 ft
V = π
= π
25 mi
20 mi
2
3
1
(10 )(25)
3
2500
mi
3
V = π
= π