Transcript Geometry
Pre-College Math
Classifying Solids
Prisms
Prisms
are named for their base shape:
Rectangular
Prism
Triangular
Prism
Hexagonal
Prism
Pentagonal
Octagonal
Prism
Trapezoidal
Square
Prism
Prism
Prism
Prisms
Prisms
are made up of two bases and
rectangular faces connecting the edges
of the bases two each other.
Cylinders
Cylinders
have a circle for a base and
one rectangular face curved around
the circumference of the circle
Pyramids
Pyramids
are named for their bases as well
Rectangular
Pyramid
Triangular
Pyramid
Hexagonal
Pyramid
Pentagonal
Octagonal
Square
Pyramid
Pyramid
Pyramid
Pyramids
Pyramids
are made up of one base and
triangles which converge at a point
(vertex) opposite the base
Spheres and Hemispheres
Spheres
are shaped like a round ball
A hemi-sphere is half a sphere
Cones
Cones
have a circle for a base and a
curved face that converges at a vertex
opposite the base
Homework
Assignment 10-6
Pre-College Math
Volume of Prisms and Cylinders
What is volume?
The
volume of a solid is the number of
cubic units contained in its interior.
Meaning how many cubic units it can
“hold”.
The
volume of an object is measured in
cubic units (i.e. 𝑐𝑚3 , 𝑚3 , 𝑚𝑖 3 , 𝑓𝑡 3 , etc).
Volume of a Prism
The
Volume of a Prism is found by taking
the area of the base shape and multiplying
it by the height if the prism
So
the formula is: 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝐵𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡
This
works because you can think of taking
slices of the base shape and stacking them
on top of each other
Some extras…
Volume of a cube:
The
volume of a cube is the cube of the length of
its side: 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑠 3
Congruence:
If
two polyhedra (solid bounded by polygons) are
congruent, then they have the same volume
Volume Addition:
The
volume of a solid is the sum of the volumes of
all its NON-OVERLAPPING parts.
Helpful Hints
Area
1
2
of a regular polygon:
𝑎𝑝𝑜𝑡ℎ𝑒𝑚 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑖𝑑𝑒𝑠 𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ
Area
of a rectangle: 𝑙𝑒𝑛𝑔𝑡ℎ 𝑤𝑖𝑑𝑡ℎ
Area
of a triangle:
1
(𝑏𝑎𝑠𝑒
2
𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒)(ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒)
Volume of a Cylinder
This
is the same as the volume of a prism
with one little difference, the base is
ALWAYS a circle
Therefore, the formula is:
𝑉𝑜𝑙𝑢𝑚𝑒 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝐶𝑖𝑟𝑐𝑙𝑒 × 𝐻𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝐶𝑦𝑙𝑖𝑛𝑑𝑒𝑟
𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 2 ℎ
Homework
Assignment 10-7
Pre-College Math
Volume of a Pyramid, Cone, and Sphere
Volume of a Pyramid
The
Volume of a Pyramid is given by the
area of the base of the pyramid times the
height of the pyramid all divided by three
Thus, the formula is:
1
𝑉𝑜𝑙𝑢𝑚𝑒 = (𝑏𝑎𝑠𝑒 𝑎𝑟𝑒𝑎)(ℎ𝑒𝑖𝑔ℎ𝑡)
3
Volume of A Cone
The
volume of a cone is given by the
area of the circular base multiplied by
the height od the cone, the whole
quantity dived by three
Thus, the formula is:
1 2
𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 ℎ
3
Volume of a Sphere
The
volume of a sphere is given by
four thirds times pi times the radius of
the sphere cubed (raised to a power of
three).
Thus, the formula is:
4 3
𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟
3
Homework
Assignment 10-8