Textbook

1. Word Knowledge

2. Mathematics Knowledge

2.1 Foundational math

2.2 Pre-algebra

2.3 Algebra I

2.4 Geometry basics

2.5 Algebra II

2.6 Advanced geometry

2.6.1 Equations of lines and slope

2.6.2 Right triangles and similar triangles

2.6.3 Distance and circles

2.6.4 Radians and measurement applications

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

2.6.3 Distance and circles

Achievable ASVAB

2. Mathematics Knowledge

2.6. Advanced geometry

Our ASVAB course is currently in development and is a work-in-progress.

Distance and circles

6 min read

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Distance

Now that you’ve learned about the Pythagorean theorem, you can use it to find the distance between two points.

You can either memorize the distance formula or build a right triangle and use the Pythagorean theorem (recommended).

Distance formula:

$(y_{2} - y_{1})^{2} + (x_{2} - x_{1})^{2}$

Since each point is written $(x, y)$ , label the first point $(x_{1}, y_{1})$ and the second point $(x_{2}, y_{2})$ .

Let’s find the distance using both methods.

For example, with points $(3, - 4)$ and $(- 2, 8)$ :

$x_{1} = 3, y_{1} = - 4$

$x_{2} = - 2, y_{2} = 8$

Formula method:

$D i s t an ce = (y_{2} - y_{1})^{2} + (x_{2} - x_{1})^{2}$

$(8 - (- 4))^{2} + (- 2 - 3)^{2}$

$(12)^{2} + (- 5)^{2}$

$144 + 25$

$169$

$13$

Distance $= 13$

Triangle method:

Alternatively, plot the points and imagine a right triangle. The horizontal and vertical changes between the points are the legs of the triangle, and the distance between the points is the hypotenuse. Then use the Pythagorean theorem.

The points (−2, 8) and (3, −4) on a coordinate plane, joined by a segment. A right triangle drawn under it has a vertical leg of 12, a horizontal leg of 5, and the segment, 13, as its hypotenuse. — Distance Example

Circles - Advanced

Definitions

Area: The space inside a circle
Circumference: The distance around a circle

Circles have a few key measurements that show up often on the exam. The two most important are area and circumference. Each has a formula to use.

Area is the space inside the circle:

$A = π r^{2}$

Circumference is the perimeter (distance around the circle):

$C = 2 π r$

Degrees

A full circle goes all the way around and measures $36 0^{\circ}$ .

One way to think about this is to break the circle into parts:

A straight line is $18 0^{\circ}$ (half of a full turn)
A circle is made of two of those halves
So $18 0^{\circ} + 18 0^{\circ} = 36 0^{\circ}$

If a circle is split into sections, all the angle measures must add up to $36 0^{\circ}$ .

You can use this idea to find a missing angle by adding the known angles and subtracting from $36 0^{\circ}$ .

A circle divided by three radii into angles of 130°, 80° and an unknown angle x°. — Solving unknown parts of a circle

Let’s try a couple of examples to practice working with circles.

Example:

What is the value of $x$ in degrees?

Start with the key idea: A full circle adds up to $36 0^{\circ}$ .

This means all the angle pieces in the circle must add up to $36 0^{\circ}$ .

We are given two angles: $13 0^{\circ}$ and $8 0^{\circ}$ , and one unknown angle $x$ .

First, add the known angles:

$130 + 80 = 210$

Now we know:

$210 + x = 360$

To find $x$ , subtract $210$ from both sides:

$360 360 - 210 150 = 210 + x = x = x$

So $x = 15 0^{\circ}$ .

Knowledge check:

A circle is split into angles of $10 0^{\circ}$ , $12 0^{\circ}$ , and $x$ . What is $x$ ?

(spoiler)

$14 0^{\circ}$

Arcs

Definitions

Arcs: A part of the edge of a circle

Segments of the circumference are called arcs, which just means parts of the circle’s edge. We measure arcs in degrees. A full circle is $36 0^{\circ}$ , so the entire edge of the circle also represents $36 0^{\circ}$ . Any arc is just a portion of that total.

Sidenote

Important to know!

A central angle has the same degree measure as the arc it points to. For example, if the angle is $6 0^{\circ}$ , the arc is also $6 0^{\circ}$ .

Two circles with the same center, one inside the other. The outer circle and the inner circle are both labeled 360°. — Circle circumference

Example:

In the circle below, what is the measure of arc length $a$ ?

$A circle with radius r = 12. Two radii form a 60° central angle, and the arc between them is labeled a.$
Arc example problem
Achievable

A. $2 π$
B. $4 π$
C. $6 π$
D. $8 π$

An arc is a fraction of the circumference, so start by finding the circumference.

$C C C = 2 π r = 2 π (12) = 24 π$

Now use the fact that the arc is the same fraction of the circumference as its central angle is of $360$ °. Here, arc $a$ corresponds to a $60$ ° central angle, so set up a proportion:

$\frac{360}{60} 6 6 a a a = \frac{24 π}{a} = \frac{24 π}{a} = 24 π = \frac{24 π}{6} = 4 π$

B is the correct answer

Knowledge check: A circle has a radius of $6$ . What is the length of an arc with a central angle of $9 0^{\circ}$ ?

(spoiler)

$3 π$

Distance Between Points

Distance formula: $(y_{2} - y_{1})^{2} + (x_{2} - x_{1})^{2}$ , using labeled points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$
Alternative: build a right triangle from horizontal/vertical changes between points, then apply Pythagorean theorem (legs = coordinate differences, hypotenuse = distance)
Both methods yield the same result (e.g., distance = 13 for points (3,-4) and (-2,8))

Circle Area & Circumference

Area (space inside circle): $A = π r^{2}$
Circumference (distance around circle): $C = 2 π r$

Circle Degrees

Full circle = $36 0^{\circ}$ (two straight lines of $18 0^{\circ}$ each)
All angle sections within a circle must sum to $36 0^{\circ}$
Find missing angle: subtract sum of known angles from $36 0^{\circ}$

Arcs

Arc = a portion of the circle’s edge (circumference), measured in degrees
Central angle measure equals its corresponding arc’s degree measure (e.g., $6 0^{\circ}$ angle → $6 0^{\circ}$ arc)
Arc length is found via proportion: $\frac{360}{central angle} = \frac{circumference}{arc length}$
Steps: calculate circumference first ( $C = 2 π r$ ), then solve proportion for arc length

Next | 2.6.4 Radians and measurement applications

Distance and circles

Distance

Now that you’ve learned about the Pythagorean theorem, you can use it to find the distance between two points.

You can either memorize the distance formula or build a right triangle and use the Pythagorean theorem (recommended).

Distance formula:

$(y_{2} - y_{1})^{2} + (x_{2} - x_{1})^{2}$

Since each point is written $(x, y)$ , label the first point $(x_{1}, y_{1})$ and the second point $(x_{2}, y_{2})$ .

Let’s find the distance using both methods.

For example, with points $(3, - 4)$ and $(- 2, 8)$ :

$x_{1} = 3, y_{1} = - 4$

$x_{2} = - 2, y_{2} = 8$

Formula method:

$D i s t an ce = (y_{2} - y_{1})^{2} + (x_{2} - x_{1})^{2}$

$(8 - (- 4))^{2} + (- 2 - 3)^{2}$

$(12)^{2} + (- 5)^{2}$

$144 + 25$

$169$

$13$

Distance $= 13$

Triangle method:

Alternatively, plot the points and imagine a right triangle. The horizontal and vertical changes between the points are the legs of the triangle, and the distance between the points is the hypotenuse. Then use the Pythagorean theorem.

Circles - Advanced

Definitions

Area: The space inside a circle
Circumference: The distance around a circle

Circles have a few key measurements that show up often on the exam. The two most important are area and circumference. Each has a formula to use.

Area is the space inside the circle:

$A = π r^{2}$

Circumference is the perimeter (distance around the circle):

$C = 2 π r$

Degrees

A full circle goes all the way around and measures $36 0^{\circ}$ .

One way to think about this is to break the circle into parts:

A straight line is $18 0^{\circ}$ (half of a full turn)
A circle is made of two of those halves
So $18 0^{\circ} + 18 0^{\circ} = 36 0^{\circ}$

If a circle is split into sections, all the angle measures must add up to $36 0^{\circ}$ .

You can use this idea to find a missing angle by adding the known angles and subtracting from $36 0^{\circ}$ .

Let’s try a couple of examples to practice working with circles.

Example:

What is the value of $x$ in degrees?

Start with the key idea: A full circle adds up to $36 0^{\circ}$ .

This means all the angle pieces in the circle must add up to $36 0^{\circ}$ .

We are given two angles: $13 0^{\circ}$ and $8 0^{\circ}$ , and one unknown angle $x$ .

First, add the known angles:

$130 + 80 = 210$

Now we know:

$210 + x = 360$

To find $x$ , subtract $210$ from both sides:

$360 360 - 210 150 = 210 + x = x = x$

So $x = 15 0^{\circ}$ .

Knowledge check:

A circle is split into angles of $10 0^{\circ}$ , $12 0^{\circ}$ , and $x$ . What is $x$ ?

(spoiler)

$14 0^{\circ}$

Arcs

Definitions

Arcs: A part of the edge of a circle

Sidenote

Important to know!

A central angle has the same degree measure as the arc it points to. For example, if the angle is $6 0^{\circ}$ , the arc is also $6 0^{\circ}$ .

Example:

In the circle below, what is the measure of arc length $a$ ?

A. $2 π$
B. $4 π$
C. $6 π$
D. $8 π$

An arc is a fraction of the circumference, so start by finding the circumference.

$C C C = 2 π r = 2 π (12) = 24 π$

Now use the fact that the arc is the same fraction of the circumference as its central angle is of $360$ °. Here, arc $a$ corresponds to a $60$ ° central angle, so set up a proportion:

$\frac{360}{60} 6 6 a a a = \frac{24 π}{a} = \frac{24 π}{a} = 24 π = \frac{24 π}{6} = 4 π$

B is the correct answer

Knowledge check: A circle has a radius of $6$ . What is the length of an arc with a central angle of $9 0^{\circ}$ ?

(spoiler)

$3 π$

Achievable ASVAB

2. Mathematics Knowledge

2.6. Advanced geometry

Our ASVAB course is currently in development and is a work-in-progress.

Distance and circles

6 min read

Font

Discuss

Feedback

Distance

Now that you’ve learned about the Pythagorean theorem, you can use it to find the distance between two points.

You can either memorize the distance formula or build a right triangle and use the Pythagorean theorem (recommended).

Distance formula:

$(y_{2} - y_{1})^{2} + (x_{2} - x_{1})^{2}$

Since each point is written $(x, y)$ , label the first point $(x_{1}, y_{1})$ and the second point $(x_{2}, y_{2})$ .

Let’s find the distance using both methods.

For example, with points $(3, - 4)$ and $(- 2, 8)$ :

$x_{1} = 3, y_{1} = - 4$

$x_{2} = - 2, y_{2} = 8$

Formula method:

$D i s t an ce = (y_{2} - y_{1})^{2} + (x_{2} - x_{1})^{2}$

$(8 - (- 4))^{2} + (- 2 - 3)^{2}$

$(12)^{2} + (- 5)^{2}$

$144 + 25$

$169$

$13$

Distance $= 13$

Triangle method:

Alternatively, plot the points and imagine a right triangle. The horizontal and vertical changes between the points are the legs of the triangle, and the distance between the points is the hypotenuse. Then use the Pythagorean theorem.

Circles - Advanced

Definitions

Area: The space inside a circle
Circumference: The distance around a circle

Circles have a few key measurements that show up often on the exam. The two most important are area and circumference. Each has a formula to use.

Area is the space inside the circle:

$A = π r^{2}$

Circumference is the perimeter (distance around the circle):

$C = 2 π r$

Degrees

A full circle goes all the way around and measures $36 0^{\circ}$ .

One way to think about this is to break the circle into parts:

A straight line is $18 0^{\circ}$ (half of a full turn)
A circle is made of two of those halves
So $18 0^{\circ} + 18 0^{\circ} = 36 0^{\circ}$

If a circle is split into sections, all the angle measures must add up to $36 0^{\circ}$ .

You can use this idea to find a missing angle by adding the known angles and subtracting from $36 0^{\circ}$ .

Let’s try a couple of examples to practice working with circles.

Example:

What is the value of $x$ in degrees?

Start with the key idea: A full circle adds up to $36 0^{\circ}$ .

This means all the angle pieces in the circle must add up to $36 0^{\circ}$ .

We are given two angles: $13 0^{\circ}$ and $8 0^{\circ}$ , and one unknown angle $x$ .

First, add the known angles:

$130 + 80 = 210$

Now we know:

$210 + x = 360$

To find $x$ , subtract $210$ from both sides:

$360 360 - 210 150 = 210 + x = x = x$

So $x = 15 0^{\circ}$ .

Knowledge check:

A circle is split into angles of $10 0^{\circ}$ , $12 0^{\circ}$ , and $x$ . What is $x$ ?

(spoiler)

$14 0^{\circ}$

Arcs

Definitions

Arcs: A part of the edge of a circle

Sidenote

Important to know!

A central angle has the same degree measure as the arc it points to. For example, if the angle is $6 0^{\circ}$ , the arc is also $6 0^{\circ}$ .

Example:

In the circle below, what is the measure of arc length $a$ ?

$A circle with radius r = 12. Two radii form a 60° central angle, and the arc between them is labeled a.$
Arc example problem
Achievable

A. $2 π$
B. $4 π$
C. $6 π$
D. $8 π$

An arc is a fraction of the circumference, so start by finding the circumference.

$C C C = 2 π r = 2 π (12) = 24 π$

Now use the fact that the arc is the same fraction of the circumference as its central angle is of $360$ °. Here, arc $a$ corresponds to a $60$ ° central angle, so set up a proportion:

$\frac{360}{60} 6 6 a a a = \frac{24 π}{a} = \frac{24 π}{a} = 24 π = \frac{24 π}{6} = 4 π$

B is the correct answer

Knowledge check: A circle has a radius of $6$ . What is the length of an arc with a central angle of $9 0^{\circ}$ ?

(spoiler)

$3 π$

Distance Between Points

Distance formula: $(y_{2} - y_{1})^{2} + (x_{2} - x_{1})^{2}$ , using labeled points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$
Alternative: build a right triangle from horizontal/vertical changes between points, then apply Pythagorean theorem (legs = coordinate differences, hypotenuse = distance)
Both methods yield the same result (e.g., distance = 13 for points (3,-4) and (-2,8))

Circle Area & Circumference

Area (space inside circle): $A = π r^{2}$
Circumference (distance around circle): $C = 2 π r$

Circle Degrees

Full circle = $36 0^{\circ}$ (two straight lines of $18 0^{\circ}$ each)
All angle sections within a circle must sum to $36 0^{\circ}$
Find missing angle: subtract sum of known angles from $36 0^{\circ}$

Arcs

Arc = a portion of the circle’s edge (circumference), measured in degrees
Central angle measure equals its corresponding arc’s degree measure (e.g., $6 0^{\circ}$ angle → $6 0^{\circ}$ arc)
Arc length is found via proportion: $\frac{360}{central angle} = \frac{circumference}{arc length}$
Steps: calculate circumference first ( $C = 2 π r$ ), then solve proportion for arc length

Next | 2.6.4 Radians and measurement applications

Distance and circles

Distance

Now that you’ve learned about the Pythagorean theorem, you can use it to find the distance between two points.

You can either memorize the distance formula or build a right triangle and use the Pythagorean theorem (recommended).

Distance formula:

$(y_{2} - y_{1})^{2} + (x_{2} - x_{1})^{2}$

Since each point is written $(x, y)$ , label the first point $(x_{1}, y_{1})$ and the second point $(x_{2}, y_{2})$ .

Let’s find the distance using both methods.

For example, with points $(3, - 4)$ and $(- 2, 8)$ :

$x_{1} = 3, y_{1} = - 4$

$x_{2} = - 2, y_{2} = 8$

Formula method:

$D i s t an ce = (y_{2} - y_{1})^{2} + (x_{2} - x_{1})^{2}$

$(8 - (- 4))^{2} + (- 2 - 3)^{2}$

$(12)^{2} + (- 5)^{2}$

$144 + 25$

$169$

$13$

Distance $= 13$

Triangle method:

Alternatively, plot the points and imagine a right triangle. The horizontal and vertical changes between the points are the legs of the triangle, and the distance between the points is the hypotenuse. Then use the Pythagorean theorem.

Circles - Advanced

Definitions

Area: The space inside a circle
Circumference: The distance around a circle

Circles have a few key measurements that show up often on the exam. The two most important are area and circumference. Each has a formula to use.

Area is the space inside the circle:

$A = π r^{2}$

Circumference is the perimeter (distance around the circle):

$C = 2 π r$

Degrees

A full circle goes all the way around and measures $36 0^{\circ}$ .

One way to think about this is to break the circle into parts:

A straight line is $18 0^{\circ}$ (half of a full turn)
A circle is made of two of those halves
So $18 0^{\circ} + 18 0^{\circ} = 36 0^{\circ}$

If a circle is split into sections, all the angle measures must add up to $36 0^{\circ}$ .

You can use this idea to find a missing angle by adding the known angles and subtracting from $36 0^{\circ}$ .

Let’s try a couple of examples to practice working with circles.

Example:

What is the value of $x$ in degrees?

Start with the key idea: A full circle adds up to $36 0^{\circ}$ .

This means all the angle pieces in the circle must add up to $36 0^{\circ}$ .

We are given two angles: $13 0^{\circ}$ and $8 0^{\circ}$ , and one unknown angle $x$ .

First, add the known angles:

$130 + 80 = 210$

Now we know:

$210 + x = 360$

To find $x$ , subtract $210$ from both sides:

$360 360 - 210 150 = 210 + x = x = x$

So $x = 15 0^{\circ}$ .

Knowledge check:

A circle is split into angles of $10 0^{\circ}$ , $12 0^{\circ}$ , and $x$ . What is $x$ ?

(spoiler)

$14 0^{\circ}$

Arcs

Definitions

Arcs: A part of the edge of a circle

Sidenote

Important to know!

A central angle has the same degree measure as the arc it points to. For example, if the angle is $6 0^{\circ}$ , the arc is also $6 0^{\circ}$ .

Example:

In the circle below, what is the measure of arc length $a$ ?

A. $2 π$
B. $4 π$
C. $6 π$
D. $8 π$

An arc is a fraction of the circumference, so start by finding the circumference.

$C C C = 2 π r = 2 π (12) = 24 π$

Now use the fact that the arc is the same fraction of the circumference as its central angle is of $360$ °. Here, arc $a$ corresponds to a $60$ ° central angle, so set up a proportion:

$\frac{360}{60} 6 6 a a a = \frac{24 π}{a} = \frac{24 π}{a} = 24 π = \frac{24 π}{6} = 4 π$

B is the correct answer

Knowledge check: A circle has a radius of $6$ . What is the length of an arc with a central angle of $9 0^{\circ}$ ?

(spoiler)

$3 π$