Video Math Tutor: Algebra: Formulas From Geometry

ALGEBRA A Self-Tutorial by

Luis Anthony Ast Professional Mathematics Tutor

FORMULAS FROM GEOMETRY

Copyright © 2006

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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.

ANGLES F Two angles are Complimentary Angles if the sum of the measures of their angles is 90°.

α + β = 90° α

β

F Two angles are Supplementary Angles if the sum of the measures of their angles is 180°. α + β = 180° β

α

L

…

In the following formulas, height is also called altitude.

TRIANGLES s1

s2 s3

Perimeter: P = s1 + s2 + s3

2

h = height b = base h

Area:

b

F An Isosceles Triangle has two sides that are of the same length.

F An Equilateral Triangle has all three sides of the same length.

Y The sum of the measures of the interior angles of any triangle is 180°. β

α + β + γ = 180° γ

α

F A Right Triangle has one interior angle equal to 90°. 90°

F The Pythagorean Theorem 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… 3

c

a b

Y Pythagorean Triples. Here are some examples of combinations of integers that make the Pythagorean equation

true:

a

b

c

3 5 7 8 9

4 12 24 15 40

5 13 25 17 41

Y The sides of similar triangles are proportional. b

a

e

d f

c

QUADRILATERALS (Four-sided figures) Y Square:

s

s d

s = side s

d = diagonal

d= Perimeter: P = 4s

s 4

s

Area:

s Y Rectangle:

l

l = length w = width d = diagonal

d

w

w

l

Perimeter: P = 2l + 2w

w

Area:

l Y Parallelogram:

b

s

b= base s = slant height Perimeter: P = 2b + 2s

s b

h = height

h

Area: b

Y Trapezoid:

s2

s1

s3

Perimeter: P = s1 + s2 + s3 + s4

s4 5

b2

b1 = first base b2 = second base h = height

h 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°. β γ α + β + γ + δ = 360° α

δ

CIRCLES r

r = radius

d

d = diameter Diameter: d = 2r

C = circumference Circumference: C = 2πr or C = πd

6

r

π

Area:

3.14159265358979323846264338327950288419716939937510…

π is the number of diameters that can fit on the circumference of a circle. and 3.14 are typical approximations of π.

SOLID FIGURES Y Cube:

s = side of cube d = diagonal of face D = diagonal of cube D

s

d

Surface Area:

Volume: s Y Rectangular Box (or Rectangular Parallelepiped): = length w = width h = height

h

d

d = diagonal of box

w

Surface Area: SA = 2( w + wh + h) 7

h

Volume:

w

Y Prisms (any kind): h = height A = Area of the base h Volume: A The video illustrates other examples of prisms. Y Pyramids (any kind):

h = height A = Area of the base

h

Volume:

A The video illustrates other examples of pyramids. Y Cone (Right Circular Cone): s = slant height h = height r = radius s h Lateral Surface Area (Area of cone not counting the area of base):

r 8

Total Surface Area:

or

Volume: r Y Cylinder (Right Circular Cylinder):

r = radius h = height Lateral Surface Area:

h

Total Surface Area:

r

h

Volume:

r

9

Y Sphere: r = radius r

Surface Area:

r

Volume:

END OF LESSON

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