Alternative University

A First Journey Into Math

Part 3: Analytic Geometry


Arc Length

Recall from the Trigonometry lesson that an angle is the arc length on a unit circle perimeter.

The angle for a right angle (perfect corner) is 𝛑╱𝟐 (90 degrees) since it is 1/4th way around a circle and a circle is 2π radians (360°):

Figure 1: The arc length (angle) of a “right angle” (corner that is “just right”) is π ⁄ 2 (90 degrees).

A right angle is denoted with a small square () where the two lines that are subtended join, in this case in the “just right” corner. Other angles may be denoted with an arc at that vertex ().

Note: The point where two lines meet, forming an angle, is called a vertex.

The angle for a half-corner (fold splice) is π ⁄ 4 (45 degrees):

Figure 2: An angle of π ⁄ 4 (45 degrees).

Angles may be denoted θ, and labelled near the vertex joining the two lines being subtended, with the understanding that the angle is actually the arc length distance along the perimeter of a unit circle centered at that vertex and intersecting those lines.

Figure 3: Angle between (subtending) two vectors, denoted with an arc and number of degrees near the subtending vertex. This is the same angle as preceding figure (π ⁄ 4).


A triangle is three lines intersecting at three vertices, more precisely the line segments joining those three vertices:

Figure 4: Example triangle: Line segments ab, bc and ac, joining vertices a, b and c.

Each side (line segment) of a triangle may be labelled as the capital letter of the vertex opposite from the side (the vertex that is not an end point of the side):

Figure 5: Traingle sides opposite each vertex labelled as the capital letter of the vertex label.

Also, each angle may be denoted with the vertex label:

Figure 6: Angles of a triangle denoted as an arc with the vertex label.

In this example, angle a is opposite side A, angle b is opposite side B, and angle c is opposite side C.

Sum of Angles

All three angles of a triangle add up to 180 degrees (π radians). For the angles of this example:

∡a + ∡b + ∡c = 180°

A triangle is a “right triangle” if one of its angles is a right angle (90°). For a right triangle, since one of the angles is 90°, the other two angles must each be less than 90° because they must both add up to 90° for all three angles to equal 180°.

Triangle Inequality

The triangle inequality states that the sum of the lengths of any two sides of a triangle is greater than the length of the remaining side.

The only exception is for a flat triangle in which the two sides equal the third side, referred to as a degenerate triangle (i.e., a triangle flattened into a straight line).

In the practical world, the sum of any two sides of a triangle are longer in length than the remaining side. This is well known intuitively, for example when walking on straight paths, it is always shorter to take a single straight path than two straight paths that end up at the same place. For that reason, diagonal walkways are built to accomodate pedestrians who naturally try to take a diagonal path.

Consider South Park Blocks at Portland State University in Oregon:

Figure 7: Diagonal walkways in South Park Blocks, Portland State University.

In the following map of South Park Blocks, say pedestrians need to walk from point A to Point E:

Figure 8: PSU map.

They could take the blue path, or the red path. Upon reaching where those paths diverge, pedestrians would choose the blue path, intuitively knowing it's shorter.

In this example, where the paths are apart forms a triangle, with the red path taking two sides of the triangle and the blue path taking the diagonal. Intuitive understanding of the triangle inequality causes pedestrians to take the diagonal walkway (single side of a triangle, instead of two sides).


If the triangle is a “right triangle” (a triangle with a right angle as one of the three angles of the triangle), trigonometry can be used to calculate the other two angles as a ratio of the sides of the triangle.

We refer to the long side of a right triangle as the diagonal. It is the longer side of the triangle because it is opposite the larger angle of the triangle.

For a right triangle, the longer side is opposite the right angle, because the other angles are each smaller than a right angle since all three angles must add up to 180° and thus the other two sides must together add up to a right angle.

For an angle other than the right angle, referring to the shorter sides of the right triangle as opposite and adjacent to the angle, the sine of the angle is the opposite side divided by the diagonal, and the cosine is the adjacent side divided by the diagonal:

Figure 9: Trigonometry of a right triangle.


Problem: For a right triangle with diagonal length of 2 meters, and a side length of 1 meter, what is the angle subtending those two sides?

Solution: The adjacent side divided by the diagonal is 1/2. The arccosine of 1/2 is 60°.

To practice with a calculator, press open parentheses ‘(’, then ‘1’, then the divide key (÷ or /), then ‘2’ followed with close paren ‘)’. Then for inverse trigonometry press ‘2nd’ and ‘arccos’.

Figure 9: Arccosine of 1/2 is 60°.

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Analytic Geometry