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

Distance and circles

7 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 horizontal leg runs from $x = - 2$ to $x = 3$ , a length of $5$ . The vertical leg runs from $y = - 4$ to $y = 8$ , a length of $12$ .

$5^{2} + 1 2^{2} = 25 + 144 = 169$

$169 = 13$

Distance $= 13$

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

In distance problems, look for common right triangles ( $3$ - $4$ - $5$ , $5$ - $12$ - $13$ , $6$ - $8$ - $10$ ), so the square root comes out to a whole number.

Circles - advanced

Definitions

Area: The space inside a circle.
Circumference: The distance around a circle.
Radius: The distance from the center of a circle to its edge.
Diameter: The distance across a circle through its center, equal to twice the radius ( $d = 2 r$ ).

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, and both formulas use the radius, $r$ .

Area:

$A = π r^{2}$

Circumference:

$C = 2 π r$

If a problem gives you the diameter instead of the radius, divide it by $2$ first. For example, a circle with diameter $10$ has a radius of $5$ , so its area is $A = π (5)^{2} = 25 π$ .

Exam tip: No calculator is allowed on this section. Leave $π$ as a symbol when the answer choices are written with $π$ (like $24 π$ , not $75.36$ ).

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?

All the angles in a circle add up to $36 0^{\circ}$ . Add the two known angles, then subtract from $36 0^{\circ}$ :

$130 + 80 = 210$

$360 - 210 = 150$

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

Arc: A part of the edge of a circle.
Arc measure: The degree measure of an arc, equal to its central angle.
Arc length: The actual distance along an arc, found as a fraction of the circumference.

Segments of the circumference are called arcs, which just means parts of the circle’s edge. An arc’s measure is given in degrees and equals its central angle, while an arc’s length is the actual distance along that piece of the circle - a fraction of the whole circumference.

A circle of any size measures $36 0^{\circ}$ around, so an arc’s degree measure depends only on its central angle, not on the radius.

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

Example:

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

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

$(y_{2} - y_{1})^{2} + (x_{2} - x_{1})^{2}$
Alternative: build right triangle from horizontal/vertical changes between points, apply Pythagorean theorem (leg squared plus leg squared equals hypotenuse squared)
Watch for common Pythagorean triples ( $3$ - $4$ - $5$ , $5$ - $12$ - $13$ , $6$ - $8$ - $10$ ) for whole-number answers

Circle basics: area & circumference

Key terms: radius (center to edge), diameter ( $d = 2 r$ ), area (space inside), circumference (distance around)
Formulas: $A = π r^{2}$ and $C = 2 π r$
If given diameter, divide by 2 to get radius first
No calculator on exam — leave answers in terms of $π$ (e.g., $24 π$ ) rather than converting to decimal

Angles around a circle

Full circle = $36 0^{\circ}$ total (two straight lines of $18 0^{\circ}$ each)
When circle is divided into sections, all angles sum to $36 0^{\circ}$
To find missing angle: add known angles, subtract from $36 0^{\circ}$

Arcs: measure & length

Arc = portion of circle’s edge
Arc measure (degrees) equals its central angle; same for any size circle (depends only on angle, not radius)
Arc length is a fraction of the circumference, found via proportion: central angle over $360$ equals arc length over circumference
Process: find full circumference first ( $C = 2 π r$ ), then solve the proportion using the central angle

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.

The horizontal leg runs from $x = - 2$ to $x = 3$ , a length of $5$ . The vertical leg runs from $y = - 4$ to $y = 8$ , a length of $12$ .

$5^{2} + 1 2^{2} = 25 + 144 = 169$

$169 = 13$

Distance $= 13$

In distance problems, look for common right triangles ( $3$ - $4$ - $5$ , $5$ - $12$ - $13$ , $6$ - $8$ - $10$ ), so the square root comes out to a whole number.

Circles - advanced

Definitions

Area: The space inside a circle.
Circumference: The distance around a circle.
Radius: The distance from the center of a circle to its edge.
Diameter: The distance across a circle through its center, equal to twice the radius ( $d = 2 r$ ).

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, and both formulas use the radius, $r$ .

Area:

$A = π r^{2}$

Circumference:

$C = 2 π r$

If a problem gives you the diameter instead of the radius, divide it by $2$ first. For example, a circle with diameter $10$ has a radius of $5$ , so its area is $A = π (5)^{2} = 25 π$ .

Exam tip: No calculator is allowed on this section. Leave $π$ as a symbol when the answer choices are written with $π$ (like $24 π$ , not $75.36$ ).

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?

All the angles in a circle add up to $36 0^{\circ}$ . Add the two known angles, then subtract from $36 0^{\circ}$ :

$130 + 80 = 210$

$360 - 210 = 150$

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

Arc: A part of the edge of a circle.
Arc measure: The degree measure of an arc, equal to its central angle.
Arc length: The actual distance along an arc, found as a fraction of the circumference.

A circle of any size measures $36 0^{\circ}$ around, so an arc’s degree measure depends only on its central angle, not on the radius.

Example:

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

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

Distance and circles

7 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.

The horizontal leg runs from $x = - 2$ to $x = 3$ , a length of $5$ . The vertical leg runs from $y = - 4$ to $y = 8$ , a length of $12$ .

$5^{2} + 1 2^{2} = 25 + 144 = 169$

$169 = 13$

Distance $= 13$

In distance problems, look for common right triangles ( $3$ - $4$ - $5$ , $5$ - $12$ - $13$ , $6$ - $8$ - $10$ ), so the square root comes out to a whole number.

Circles - advanced

Definitions

Area: The space inside a circle.
Circumference: The distance around a circle.
Radius: The distance from the center of a circle to its edge.
Diameter: The distance across a circle through its center, equal to twice the radius ( $d = 2 r$ ).

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, and both formulas use the radius, $r$ .

Area:

$A = π r^{2}$

Circumference:

$C = 2 π r$

If a problem gives you the diameter instead of the radius, divide it by $2$ first. For example, a circle with diameter $10$ has a radius of $5$ , so its area is $A = π (5)^{2} = 25 π$ .

Exam tip: No calculator is allowed on this section. Leave $π$ as a symbol when the answer choices are written with $π$ (like $24 π$ , not $75.36$ ).

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?

All the angles in a circle add up to $36 0^{\circ}$ . Add the two known angles, then subtract from $36 0^{\circ}$ :

$130 + 80 = 210$

$360 - 210 = 150$

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

Arc: A part of the edge of a circle.
Arc measure: The degree measure of an arc, equal to its central angle.
Arc length: The actual distance along an arc, found as a fraction of the circumference.

A circle of any size measures $36 0^{\circ}$ around, so an arc’s degree measure depends only on its central angle, not on the radius.

Example:

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

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

$(y_{2} - y_{1})^{2} + (x_{2} - x_{1})^{2}$
Alternative: build right triangle from horizontal/vertical changes between points, apply Pythagorean theorem (leg squared plus leg squared equals hypotenuse squared)
Watch for common Pythagorean triples ( $3$ - $4$ - $5$ , $5$ - $12$ - $13$ , $6$ - $8$ - $10$ ) for whole-number answers

Circle basics: area & circumference

Key terms: radius (center to edge), diameter ( $d = 2 r$ ), area (space inside), circumference (distance around)
Formulas: $A = π r^{2}$ and $C = 2 π r$
If given diameter, divide by 2 to get radius first
No calculator on exam — leave answers in terms of $π$ (e.g., $24 π$ ) rather than converting to decimal

Angles around a circle

Full circle = $36 0^{\circ}$ total (two straight lines of $18 0^{\circ}$ each)
When circle is divided into sections, all angles sum to $36 0^{\circ}$
To find missing angle: add known angles, subtract from $36 0^{\circ}$

Arcs: measure & length

Arc = portion of circle’s edge
Arc measure (degrees) equals its central angle; same for any size circle (depends only on angle, not radius)
Arc length is a fraction of the circumference, found via proportion: central angle over $360$ equals arc length over circumference
Process: find full circumference first ( $C = 2 π r$ ), then solve the proportion using the central angle

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.

The horizontal leg runs from $x = - 2$ to $x = 3$ , a length of $5$ . The vertical leg runs from $y = - 4$ to $y = 8$ , a length of $12$ .

$5^{2} + 1 2^{2} = 25 + 144 = 169$

$169 = 13$

Distance $= 13$

In distance problems, look for common right triangles ( $3$ - $4$ - $5$ , $5$ - $12$ - $13$ , $6$ - $8$ - $10$ ), so the square root comes out to a whole number.

Circles - advanced

Definitions

Area: The space inside a circle.
Circumference: The distance around a circle.
Radius: The distance from the center of a circle to its edge.
Diameter: The distance across a circle through its center, equal to twice the radius ( $d = 2 r$ ).

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, and both formulas use the radius, $r$ .

Area:

$A = π r^{2}$

Circumference:

$C = 2 π r$

If a problem gives you the diameter instead of the radius, divide it by $2$ first. For example, a circle with diameter $10$ has a radius of $5$ , so its area is $A = π (5)^{2} = 25 π$ .

Exam tip: No calculator is allowed on this section. Leave $π$ as a symbol when the answer choices are written with $π$ (like $24 π$ , not $75.36$ ).

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?

All the angles in a circle add up to $36 0^{\circ}$ . Add the two known angles, then subtract from $36 0^{\circ}$ :

$130 + 80 = 210$

$360 - 210 = 150$

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

Arc: A part of the edge of a circle.
Arc measure: The degree measure of an arc, equal to its central angle.
Arc length: The actual distance along an arc, found as a fraction of the circumference.

A circle of any size measures $36 0^{\circ}$ around, so an arc’s degree measure depends only on its central angle, not on the radius.

Example:

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

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)

$3 π$