Chapter III
Final Comments
Mathematical Voyage
The Dimensions Program
P=π
⎛
⎜2
⎜⎝
(
)
2⎞
a2 + b2 − (a − b)
2.2
⎟
⎟⎠
Geometry Formulas
Logos
Pedagogical License
Section A
Geometry Formulas
Part 1
Perimeter and Circumference
Perimeter of a Polygon
P = a + b + c +…
Circumference of a Circle
C = 2πr
Perimeter / Circumference of an Ellipse
P = π 2(a 2 + b 2 )
Part 2
Area
Area of a Triangle
A=
1
bh
2
Area of an Equilateral Triangle
A=
s2 3
4
Quadrilaterals
Area of a Rectangle
A = lw
Area of a Parallelogram
A = bh
Area of a Square
A=s
Area of a Trapezoid
A=
2
1
(B + b)h
2
Area of a Circle
A = πr
Area of an Ellipse
A = π ab
2
Part 3
Volume
Volume of a Rectangular Solid
V = lwh
Volume of a Cube
V=s
Volume of an Equilateral Triangle-
V=
1
Ah
3
Volume of a Square-Based Pyramid
V=
1 2
s h or
3
Volume of a Regular Octagon
V = 4.84s 2 h
Volume of a Right Circular Cylinder
V = πr 2 h
Volume of a Cone
V=
1 2
πr h
3
Volume of a Sphere
V=
4 3
πr
3
Volume of an Ellipsoid
V=
4π
abc
3
Volume of a Cone Frustum
or a Truncated Cone
where R = radius of bottom
r = radius of top
h = height
V = 1.0472h(R 2 + Rr + r 2 )
3
Based Pyramid
V=
1
Ah
3
or
V = 0.2618h(D 2 + Dd + d 2 )
where D = diameter of bottom
d = diameter of top
h = height
Part 4
Surface Area
Surface Area of a Rectangular Solid
SA = 2hw + 2lw + 2lh
Surface Area of a Cube
SA = 6s
Surface Area of a Square-Based Pyramid
SA = s + 2bh
Surface Area of a Right Circular Cylinder
SA = 2πr 2 + 2πrh
Surface Area of a Cone
SA = πr
Surface Area of a Sphere
SA = 4πr 2
Surface Area of an Ellipse
SA =
Area of a Cone Frustum
or a Truncated Cone
where R = radius of bottom
r = radius of top
h = height
A = π (R + r) (R − r) + h 2
2
2
(r
2
+ h2 )
4π
b a2 + b2
2
or
A = 1.5708(D + d) (R − r) + h 2
where D = diameter of bottom
d = diameter of top
h = height
Section B
Logos
Mathematical Voyage
The logo represents the four components of the Mathematical Voyage strategic pedagogy. The
longest cone represents the Mathematical Voyage “Program”, and the cone crossing the
program cone represents the science, technology, and engineering “Projects”.
The cone crossing the project cone represents the
“Laboratories” within each project and the smallest
cone represents the “Question Reservoir” within
each laboratory.
The Dimensions Program
This logo is composed of an Ellipse with its Perimeter Formula in the center. The Ellipse
Perimeter Formula can be represented by
P = π 2 (a
2
+b
2
) , but a closer approximation is the
formula1 in the center of the “The Dimensions Program” Logo Ellipse located below.
P=π
1
⎛
⎜2
⎜⎝
(
a2
+ b2
)
2⎞
(a
−
b)
−
⎟
2.2 ⎟⎠
Machinery’s Handbook, The Industrial Press, The Machinery Publishing Company, LTD., London, 1937 – Erik Oberg and F. D. Jones,
Section C
Pedagogical License
You are a Strategic Pedagogical Designer Who Possesses
the Knowledge and Ability to Think Creatively, and
Explore in Innovative and Unique Ways
Use Your Pedagogical License to Explore
Mathematics Voyage – The Dimensions Program
as You Would Like with No Limitations!
⎛
(
)
P = π ⎜ 2 a2 + b2 −
⎝
(a − b)2 ⎞
2.2
⎟⎠
John, Pat, and Dennis