Textbook

1. Word Knowledge

2. Mathematics Knowledge

2.1 Foundational math

2.2 Pre-algebra

2.3 Algebra I

2.4 Geometry basics

2.4.1 Lines and slopes

2.4.2 Angles and transversals

2.4.3 Basic shapes and area

2.4.4 Circles, polygons, and coordinate geometry

2.5 Algebra II

2.6 Advanced geometry

2.7 Math strategies

3. Paragraph Comprehension

4. Arithmetic Reasoning

5. General Science

6. Mechanical Comprehension

7. Electronics Information

8. Shop Information

9. Auto Information

10. Assembling Objects

Wrapping up

2.4.4 Circles, polygons, and coordinate geometry

Achievable ASVAB

2. Mathematics Knowledge

2.4. Geometry basics

Circles, polygons, and coordinate geometry

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Circles

A circle is a perfectly round shape where every point on the edge is the same distance from the center.

Definitions

Circumference: The distance around the circle (its perimeter)
Radius: The distance from the center to the edge of the circle
Diameter: The distance from one edge of the circle to the other while passing through the center of the circle
Chord: A line segment connecting two points on the circle; a diameter is the longest chord
Arcs: A segment of the circumference

Understanding the parts of a circle will help you solve problems involving distance, area, and measurements around the circle.

Key relationship: The radius goes from the center to the edge, and the diameter goes all the way across the circle, through the center — exactly twice the radius.

$d iam e t er = 2 \times r a d i u s$

A circle with center A, showing its circumference, a radius AC, a diameter BD, and an arc FE. — Parts of a circle

Sidenote

More on circles

We’ll explore the circle more in Advanced Geometry!

Polygons

Definitions

Polygon: A multisided closed shape made up of straight lines, with a minimum of $3$ sides
Regular polygon: A polygon with all equal sides and all equal angles
Irregular polygon: A polygon whose sides or angles aren’t all equal

Polygons can have many different numbers of sides. The simplest polygon is a triangle (3 sides), but polygons can also have 4 sides (quadrilaterals), 5 sides (pentagons), 6 sides (hexagons), 8 sides (octagons), and more.

Sum of angles of a polygon

We know the interior angles in any quadrilateral (such as a rectangle) add up to $36 0^{\circ}$ and the interior angles in a triangle add up to $18 0^{\circ}$ .

For larger polygons, we can break the shape into triangles by drawing lines from a single vertex to all other non-adjacent vertices.

Each triangle contributes $18 0^{\circ}$ , so the total interior angle sum depends on how many triangles are formed.

For a polygon with $n$ sides, you can always form $(n - 2)$ triangles.

A regular pentagon, hexagon and octagon, each divided into triangles by diagonals from one vertex: 3, 4 and 6 triangles, n − 2 for a polygon of n sides. — Common polygons

Interior angle sum of a polygon:

$(n - 2) \times 180$

where $n$ = the number of sides of the polygon.

Let’s look at a problem together.

Example:

What is the value of $∠ y$ in this regular polygon?

$A regular hexagon with the interior angle at its top vertex marked y°.$
Hexagon example
Achievable

This is asking: what is the measure of one interior angle?

To find the total degree value of any polygon, you can either:

Use the formula $(n - 2) \times 180$ , or

Break the polygon into triangles (each triangle has $18 0^{\circ}$ ).

Let’s break this one into triangles to show that we create $4$ triangles.

$A regular hexagon with three diagonals drawn from its top vertex, dividing it into 4 triangles, numbered 1 to 4.$
Hexagon example
Achievable

$180 \times 4 = 72 0^{\circ}$

Because the polygon is regular, all interior angles are equal.

So to find one angle, divide the total by the number of sides:

$\frac{720}{6} = 12 0^{\circ}$

$∠ y = 12 0^{\circ}$

Watch out: $72 0^{\circ}$ is the sum of all six angles, not one angle. And divide by the number of sides ( $6$ ), not the number of triangles ( $4$ ).

Knowledge check: What is the measure of each interior angle in a regular pentagon?

(spoiler)

First, find the total interior angle sum:

$(5 - 2) \times 180 = 3 \times 180 = 54 0^{\circ}$

Because the polygon is regular, divide by the number of sides:

$\frac{540}{5} = 10 8^{\circ}$

Each interior angle is $10 8^{\circ}$ .

Coordinate plane

Definitions

Coordinate plane: A flat plane defined by two perpendicular axes — the horizontal $x$ -axis and the vertical $y$ -axis
Origin: The point $(0, 0)$ where the $x$ -axis and $y$ -axis intersect

A coordinate plane with a horizontal x-axis and a vertical y-axis, each numbered from −10 to 10, crossing at the origin, 0. — Coordinate plane

Plotting points

To plot a point on the coordinate plane:

Move horizontally along the $x$ -axis first, then
Move vertically along the $y$ -axis

Write the point as $(x -value, y -value)$ .

For example,

The point (2, 3) plotted on a coordinate plane, 2 units to the right of the origin and 3 units up. — Coordinate point

Start at the origin. This point is $2$ units in the $x$ direction and $3$ units in the $y$ direction, so it’s written $(2, 3)$ .

Understanding direction

$x$ : how far right (positive) or left (negative) to move
$y$ : how far up (positive) or down (negative) to move

Quadrants

The coordinate plane is divided into four sections called quadrants.

Quadrants are numbered counterclockwise, starting with Quadrant I in the upper right.

Each quadrant has a pattern of positive and negative values:

Quadrant I: $(+, +)$
Quadrant II: $(-, +)$
Quadrant III: $(-, -)$
Quadrant IV: $(+, -)$

Midpoint

Definitions

Midpoint: The point exactly halfway between two points, found by averaging their $x$ -values and $y$ -values

$(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$

Example:

What is the midpoint of $(- 3, 8)$ and $(5, - 2)$ ?

This is asking: what point is exactly halfway between the two given points?

Substitute the values into the formula:

$= = = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2}) (\frac{- 3 + 5}{2}, \frac{8 + ( - 2 )}{2}) (\frac{2}{2}, \frac{6}{2}) (1, 3)$

The midpoint is $(1, 3)$ .

Knowledge check: What is the midpoint of $(2, 4)$ and $(6, 8)$ ?

(spoiler)

The midpoint is $(4, 6)$ .

Circles

Key parts: circumference (perimeter), radius (center to edge), diameter (edge to edge through center), chord (segment connecting two points; diameter is longest chord), arc (segment of circumference)
Core relationship: $d iam e t er = 2 \times r a d i u s$

Polygons

Polygon: closed straight-sided shape, minimum 3 sides; regular = equal sides/angles, irregular = not equal
Interior angle sum formula: $(n - 2) \times 180$ , where $n$ = number of sides
For regular polygons, divide total angle sum by $n$ to get each interior angle (e.g., pentagon: $54 0^{\circ} \div 5 = 10 8^{\circ}$ per angle)
Watch out: don’t confuse total sum, number of triangles, and number of sides when solving

Coordinate plane

Defined by horizontal $x$ -axis and vertical $y$ -axis; origin = $(0, 0)$
Points written as $(x, y)$ : move horizontally first, then vertically; positive/negative indicate direction (right/up vs. left/down)
Four quadrants (counterclockwise from upper right):
- Quadrant I: $(+, +)$
- Quadrant II: $(-, +)$
- Quadrant III: $(-, -)$
- Quadrant IV: $(+, -)$

Midpoint

Midpoint = average of $x$ -values and average of $y$ -values: $(\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})$
Represents point exactly halfway between two given points

Next | 2.5.1 FOIL and multiplying polynomials

Circles, polygons, and coordinate geometry

Circles

A circle is a perfectly round shape where every point on the edge is the same distance from the center.

Definitions

Circumference: The distance around the circle (its perimeter)
Radius: The distance from the center to the edge of the circle
Diameter: The distance from one edge of the circle to the other while passing through the center of the circle
Chord: A line segment connecting two points on the circle; a diameter is the longest chord
Arcs: A segment of the circumference

Understanding the parts of a circle will help you solve problems involving distance, area, and measurements around the circle.

Key relationship: The radius goes from the center to the edge, and the diameter goes all the way across the circle, through the center — exactly twice the radius.

$d iam e t er = 2 \times r a d i u s$

Sidenote

More on circles

We’ll explore the circle more in Advanced Geometry!