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Lesson 3A: Geometry and Graphing
1. Units of Measure
b. Triangles, angles and lines
In Lesson 3 we will look at geometry and coordinate geometry (graphing).
During the test, you will have a page that contains many useful formulas for making geometry calculations. We will be referring to those formulas here.
UNITS OF MEASURE
The test will ask you to work with measurements that are expressed in units such as feet, kilometers, gallons or pounds.
Distance and length are measured along one dimension, in a straight line from one point to another.
Common units and conversions: Abbreviations:
1 foot = 12 inches inch(es): in. feet:ft.
1 yard = 3 feet yard: yd. yards: yds.
1 meter = 100 centimeters kilometer(s): km meter(s): m
1 kilometer = 100 meters
1 cm = 10 millimeters centimeter(s): cm millimeter(s): mm
Area is a measure of how much space a shape takes up in two dimensions.
A rectangle measures 2 feet by 3 feet. You are asked for the area in square inches.
- The area for a rectangle is length x width:
= 2 x 3
= 6 square feet
- To convert feet to inches, recall that there are 12 inches in a foot so:
1 square foot = (12 x 12) square inches = 144 sq. in.
We convert by multiplying the area (in square feet) times 144 (square inches per square foot):
= (6 x 144) square inches
= 864 sq. in.
Volume is a measure of how much space an object takes up in three dimensions.
A rectangular solid measures 2 feet by 3 feet by 4 feet
It has a volume given by the formula:
“Volume = length x width x height”
= 2 x 3 x 4 = 6 x 4
= 24 cubic feet
Angles are measured in degrees (º). A degree is 1/360th of a full circle.
Some examples of common angles:
We will talk more about angles in our discussion of geometric principles.
The diameter is the distance across the circle.
The radius, which is half the length of the diameter, is the distance from the center to the edge.
The circumference is the distance all the way around the circle.
- In the list of formulas you have:
“Circumference = π x diameter; π is approximately 3.14” or
Circumference = 3.14 x diameter, which can simply be remembered as: C = πd
or, if you have the radius, C = 2πr since d, the diameter, is 2 x r, the radius.
The list of formulas also has this formula for circles:
“Area = π x radius2” or Area = 3.14 x (radius squared)
or A = πr2
You are told that a wheel has a radius of 20 cm. You are asked for the area and circumference of that wheel.
Area = πr2
= 3.14 x (20)2
= 3.14 x (20 x 20)
= 3.14 x 400
=122600 square centimeters
Circumference = 2πr
= 2 x 3.14 x 20
= 6.28 x 20
= 125.6 centimeters
b. Triangles, angles and lines
A triangle has three sides and three vertices (corners).
The angle measures of all three vertices in a triangle always add up to 180º
This can be seen if you cut a triangle apart and bring the vertices together.
They form a straight edge which shows that, together, the three angles span a total angle measure of 180º.
Perpendicular lines intersect to form a 90º corner or right angle: ┼
The right angle symbol always means that an angle is 90º:
Parallel lines are oriented at the same angle: ║
Any angle is formed by two intersecting lines.
A line intersecting another line can bisect that line into two angles. The angles add up to 180º because the
original line was a 180º angle.
We call the two angles supplementary angles:
a + b = 180º
A bisected right angle creates two angles that add up to 90º.
We call those angles complementary angles:
a + b =90º
When a line bisects another line, it can create 4 angles.
Opposite angles have the same degree measure, as indicated by the matching colors
Whenever you have an unknown angle, check to see if it is:
- an opposite angle
- a complementary/supplementary angle
- or, a part of a triangle
Click on the link below to move on to lesson 3B.
Back: Math Lesson 2B | Next: Math Lesson 3B
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