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G.GMD.3 STUDENT NOTES WS #4
1
THE PYRAMID
The third dimension has a new vocabulary that we need to familiarize ourselves with before we can begin
looking at formulas or calculations.
A Solid – A three dimensional closed spatial figure.
A Polyhedron – a geometric solid with polygons as faces.
A Face of a Polyhedron – One of the polygons that form the
polyhedron. Sometimes these get called sides but the better
term is face.
An Edge – The intersection of two faces of a polyhedron.
A Vertex – The intersection of two or more edges.
To establish the pyramid volume formula we can approach it
informally a few different ways. To derive the formula rigorously
it requires more mathematics than we have at this level and so
here we will simply discuss the informal arguments for this
formula.
VOLUMEPYRAMID =
1
Bh
3
PYRAMID VOLUME – POURING
The classic demonstration is to fill a pyramid with sand or water and then pour that sand or water into a prism
that has the same base and height. It is truly a mathematical thing of beauty that exactly three of these fill the
prism. From this we informally learn that the volume of a pyramid is 1/3 the volume of prism with the same
base and height.
PYRAMID CALCULATION
The formula for the volume of a pyramid follows directly from the volume of a
prism. The pyramid will always be 1/3 the volume of the prism with the same
base and height.
Example #1 – Square Pyramid
1
Bh
3
1
= ( Base Area)( height )
3
1
1
= (8)(8) (10) = 213 cm3
3
3
Example #2 – Rectangular Pyramid
1
Bh
3
1
= ( Base Area)( height )
3
1
= (4)(6) (8) = 64 cm3
3
VPYRAMID
Example #3 – Triangular Pyramid
1
Bh
3
1
= ( Base Area)( height )
3
1  1 

=    (3)(5)  (8) = 20 cm3
3 2 

VolumePYRAMID =
VolumePYRAMID =
VolumePYRAMID =
VolumePYRAMID
VolumePYRAMID
VolumePYRAMID
VolumePYRAMID
VolumePYRAMID
1
= Bh
3
VolumePYRAMID
G.GMD.3 STUDENT NOTES WS #4
2
Sometimes with pyramids and cones the height of the solid is not given and we need to calculate it using the
Pythagorean Theorem. In this example we have been given the slant height (l) – the height of the triangular
face instead of the height of the pyramid. We are also missing the distance x but it is the apothem of the
square and so it is half of the side, 3 cm.
(3)2 + h2 = 52
h = 4 cm
Now we can use the volume formula
1
Bh
3
1
= ( Base Area)( height )
3
1
= (6)(6) (4) = 48 cm3
3
VolumePYRAMID =
VolumePYRAMID
VolumePYRAMID
Another given instead of the height might be to have the length of the
lateral edge. To do this type of question we can use the right triangle on
the face of the pyramid to find the slant height, l and then use the slant
height value to determine h.
(5)2 + l2 = 132
l = 12 cm
(5)2 + h2 = 122
h = 10.91 cm
1
Bh
3
1
= ( Base Area)( height )
3
1
= (10)(10) (10.91) = 363.67 cm3
3
VolumePYRAMID =
VolumePYRAMID
VolumePYRAMID
G.GMD.3 WORKSHEET #4
NAME: ____________________________ PERIOD ________
1
1. Match the following terms to the diagram.
Given the square pyramid.
_________ 1. Slant Height
_________ 2. Apex
_________ 3. Height
_________ 4. Lateral Edge
_________ 5. Face
_________ 6. Vertex
2. Jeff missed class and Dillon is explaining the notes. “The slant height and the height
of the pyramid basically mean the same thing.” Is this summary of height correct?
Explain.
3. Properly name the pyramid.
a)
b)
c)
d)
Name:
Name:
Name:
Name:
_____________________ _____________________ _____________________
_____________________
4. Two pyramids with the same base are side by side. One is a right pyramid and the other is an oblique
pyramid. If the oblique pyramid has been tilted to an angle of 80°°, what is volume relationship between the
two pyramids?
G.GMD.3 WORKSHEET #4
2
5. Determine the volume of the pyramid.
a) Square Pyramid
b) Rectangular Pyramid
c) Regular Hexagonal Pyramid
Volume = ________________
Volume = _____________ (2 dec.)
Volume = _________________ (E)
d) Square Pyramid
e) Equilateral Triangular Pyramid
f) Square Pyramid
Volume = ______________ (2 dec.) Volume = _________________ (E)
Volume = _________________ (E)
G.GMD.3 WORKSHEET #4
3
6. Determine the volume of the pyramid.
a) Square Pyramid
b)
c)
Volume = ________________
Volume = _____________
Volume = _______________
d)
e)
f)
Volume = _________________ (E)
Volume = _________________
Volume = _________________ (E)
G.GMD.s woRKsHEEr
#4
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1. Slant Height
2. Apex
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2. lett missed class and Dillon is explaining the notes. 'The slant height and the
height of the pyramid basically mean the same thing." ls this summary of height
correct? Explain.
3. Properly name the pyramid.
a)
b)
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4. Two pyramids with the same base are side by side. One is a right pyramid and the other is an oblique
pyramid. lf the oblique pyramid has been tilted to an angle of 80o, what is volume retationship between the
two pyramids?
2
G.GMD.sWORKSHEET #4
5. Determine the volume of the pyramid'
c) Regular Hexagonal PYramid
b) Rectangular PYramid
a)Square Pyramid
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G.GMD.3 WORKSHEET #4
a
6. Determine the volume of the pyramid.
a)Square Pyramid
b)
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fi: Li
h=7
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d)
Volume =
Volume
e)
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Volume =
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