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Basic Mathematics - Geometry

Just like any other subject in math, Geometry is loaded with a wide variety of subtopics - including the ubiquitous Trigonometry. Just as the previous pages however, we will focus on the basics - again to help stir the memory a bit.

The basic form of Geometry - that is lines, perimeters, areas - is called Euclidean Geometry named after its Greek inventor, Euclid. Another name for Euclidean Geometry is Plane Geometry.

The section will only cover:

Perimeter, Area, and Volume
Angles
Basic Trigonometric Functions

 

Perimeter, Area, and Volume:

A perimeter is the length around the outside of any geometric shape. Simply add the lengths of the sides together.

Area is the measure of space inside any two-dimensional geometric shape.

Volume is the measure of space inside any three-dimensional geometric shape.

Area of a triangle:

Area of a rectangle and parallelogram:

The Circle:

The circumference of a circle is the same as the perimeter of a triangle or rectangle. To find the circumference:

r = radius of a circle

π = pi = 3.14

The area of a circle:

Three dimensional objects:

Area of the rectangle solid:

Angle of the cylinder:

Area of a Sphere:

Volume of a Sphere:

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Angles:

Complimentary Angles:

Two angles are complimentary when the sum of their angles is 90o. In the image on the left, angles A and B are complimentary and angles A and C are also complimentary.

The remaining angle definitions will use the image above.

Supplementary Angles:

Two angles are supplementary is the sum of the angles is 180o. Angles 1 and 2 as well as angles 2 and 4 are supplementary angles.

Opposite (Vertical) Angles:

The intersection of two lines (m1 and m3) form 4 angles. Opposite angles are equal (congruent). Angles 1 and 4 as well as angles 2 and 3 are congruent.

Alternate Angles:

Lines m1 and m2 are parallel. Angles 4 and 5 are alternate interior angles and are congruent. Angles 3 and 6 are also interior angles but are not congruent. Angles 2 and 7 are alternate exterior angles and are congruent. Angles 1 and 8 are also alternate exterior angles but are not congruent.

Triangles:

The three angles of a triangle always total 180o. An equilateral triangle is a triangle with 3 equal sides and all 3 angles are 60o.

An isosceles triangle is a triangle with two equal angles. The two equal angles must each be less than 60o. The image on the left illustrates angles A and B are equal.

The height of a triangle is defined by the base. Once any side of a triangle is chosen, the angle between the base and height can change, but the but measure of the height remains the same.

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Basic Trigonometric Functions:

Trigonometry is a specialty of Geometry that focuses mostly on angles. If we want the sum of all internal angles or if we want a ratio of angles for example, we will turn to Trigonometry.

Two recurrent terms to know are the Sine and Cosine

  • Sine = ratio of the height to the hypotenuse (see image below)
  • Cosine = ratio of the base of the hypotenuse (see image below)

A simple example: A straight line has an angle - its 0 degrees.

Sine = 00
Cosine = 10

To introduce the working formulas for Sine and Cosine, we will use a right triangle.

Pythagorean Theorem:

Sum of Interior Angles:

where n = the number of sides in the shape of the object

Grade and Percent Grade:

 

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