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8/9/2019 Algebra Formulas From Geometry
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AL G E B R A
A S elf-T utorial
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This document is intended to present typical formulas from geometry.
Students in algebra or calculus (or other higher math classes) will
encounter them when doing word problems. This is NOT meant as a
formal, detailed lesson in geometry, just an informal review. No examples
of the use of the formulas are given, although a few more details will be
provided on the video version of this Lesson.
A NGLES
F Two angles are Com p l im en t a r y Ang l es if the sum of the measures of
their angles is 90°.
α + β = 90°
F Two angles are Supp l em en ta r y Ang l es if the sum of the measures of
their angles is 180°.
α + β = 180°
…In the following formulas, height is also called altitude.
T RIANGLES
s1 s2
Perimeter: P = s1 + s2 + s3
s3 <Add the lengths of all 3 sides>
α β
α β
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h = height b = base
Area:
F An I soscel es Tr i a n g l e has two sides that are of the same length.
F An Equ i l a t er a l T r i a ng l e has all three sides of the same length.
Y The sum of the measures of the interior angles of any triangle is 180°.
α + β + γ = 180°
F A R i g h t T r i a n g l e has one interior angle equal to 90°.
F The Pyth a gor ea n T h eor em states: For any right triangle, the sum of
the squares of the lengths of the legs is equal to the square of the length of
the hypotenuse.
hypotenuse
leg
leg
OR…
h
b
γ α
β
90°
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a
b
Y Pythagorean Triples. Here are some examples of combinations of integers that make the Pythagorean equation true:
a b c
3 4 5
5 12 13
7 24 25
8 15 17
9 40 41
Y The sides of similar triangles are proportional.
a b d
f
c
Q UADRILATERALS (Four-sided figures)
Y Square:
s s = side d = diagonal
s s d =
Perimeter: P = 4 s
s
c
e
d
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s Area:
s
Y Rectangle:l = length w = width d = diagonal
Perimeter: P = 2l + 2w
w Area:
Y Parallelogram:
b b= base s = slant height
s s Perimeter: P = 2b + 2 s
b
h = height
Area:
b
Y Trapezoid: s2
s1 s3 Perimeter: P = s1 + s2 + s3 + s4
<Add the lengths of all 4 sides>
s4
l
l
d
l
h
ww
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b2 b1 = first base b2 = second base
h = height
Area:
b1
In calculus, you may encounter trapezoids “on their sides:”
h1 = first height h2 = second height
b = base
h1 h2 Area:
b
Y The sum of the measures of the interior angles of any quadrilateral is
360°.
α + β + γ + δ = 360°
C IRCLES
r = radius d = diameter C = circumference
Diameter: d = 2r Circumference: C = 2πr
or C = πd
r
h
γ
α
β
δ
d
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Area:
π 3.14159265358979323846264338327950288419716939937510…
π <Pronounced: “pie”> is the number of diameters that can fit on the
circumference of a circle.
and 3.14 are typical approximations of π.
S OLID F IGURES Y Cube:
s = side of cube d = diagonal of face
D = diagonal of cube
s Surface Area:
Volume:
s
Y Rectangular Box (or Rectangular Parallelepiped):
= length w = width h = height
d = diagonal of box
Surface Area: SA = 2( w + wh + h)
r
D
d
dh
w
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Volume:
Y Prisms (any kind):
h = height A = Area of the base
Volume:
The video illustrates other examples of prisms.
Y Pyramids (any kind):
h = height A = Area of the base
Volume:
The video illustrates other examples of pyramids.
Y Cone (Right Circular Cone):
s = slant height h = height
r = radius
Lateral Surface Area (Area of cone
not counting the area of base):
h
w
h
h
s
A
r
h
A
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Total Surface Area:
or
Volume:
Y Cylinder (Right Circular Cylinder):
r = radius h = height
Lateral Surface Area:
Total Surface Area:
Volume:
h
r
h
r
r