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

2.6.4 Radians and measurement applications

Achievable ASVAB

2. Mathematics Knowledge

2.6. Advanced geometry

Radians and measurement applications

9 min read

Font

Discuss

Feedback

Radians

Definitions

Radian: The angle at the center of a circle that cuts off an arc as long as the radius.

Another way to measure angles in a circle is in radians. Radians measure a circle as fractions of $2 π$ .

Here’s why $2 π$ shows up: if a circle has radius $1$ , then its circumference is

$2 π r = 2 π (1) = 2 π$

A full turn sweeps the whole circumference, which is $2 π$ radii long, so $36 0^{\circ} = 2 π$ radians.

That means:

Half a circle ( $18 0^{\circ}$ ) is $π$
One fourth of a circle ( $9 0^{\circ}$ ) is $\frac{π}{2}$

This works for any circle, no matter the size.

These relationships let us convert between degrees and radians. Since $18 0^{\circ} = π$ , we can use this to switch between the two units.

Converting degrees to radians

How to convert degrees to radians

Multiply by

$\frac{π}{180}$

This works because $18 0^{\circ} = π$ , so $\frac{π}{180}$ converts degrees into radians.

Example 1:

What is $12 0^{\circ}$ in radians?

$120 \cdot \frac{π}{180} = \frac{120 π}{180}$

Simplify the fraction by dividing top and bottom by $60$ :

$\frac{120 π}{180} = \frac{2 π}{3}$

So, $12 0^{\circ} = \frac{2 π}{3}$ radians.

Example 2:

What is $\frac{5 π}{4}$ radians in degrees?

To convert radians → degrees, multiply by $\frac{180}{π}$ . Cancel the $π$ , then divide $180$ by $4$ before multiplying:

$\frac{5 π}{4} \cdot \frac{180}{π} = 5 \cdot \frac{180}{4} = 5 \cdot 45 = 225$

So, $\frac{5 π}{4}$ radians = $22 5^{\circ}$ .

Volume

Definitions

Volume: The amount of space inside a $3$ -dimensional shape

The volume of an object tells you how much space it takes up. You can think of it like how much something can hold (like water in a box or container).

For the ASVAB, it helps to think of volume as: Area of a flat face × height

Rectangular prisms

To find the volume of a box:

$v o l u m e = l e n g t h \times w i d t h \times h e i g h t$

A box (rectangular prism) has a flat, rectangular base.

Find the base area (length × width), then stack it up through the height.

The base is the bottom face of the shape. The height is how tall the shape is from bottom to top.

A rectangular prism, a box, with its length, width and height labeled. Its hidden back edges are dashed. — Volume of a rectangular prism

Cylinders

To find the volume of a cylinder:

$V o l u m e = π \times r^{2} \times h e i g h t$

A cylinder follows the same idea as a box, but instead of a rectangular base, it has a circular base.

So instead of length × width, we use the area of a circle for the base, then multiply by the height.

In the example cylinder below, $π r^{2}$ is the area of the circular base, and $h$ is the height of the cylinder.

Example:

Find the volume of a cylinder with radius $3$ and height $5$ .

$A cylinder with the radius of its circular top and its height labeled.$
Volume of a cylinder
Achievable

Use the formula:

$V = π r^{2} h$

Start by identifying the values:

$r = 3$

$h = 5$

First, square the radius. Then multiply by the height. Finally, multiply by $π$ .

$V = π r^{2} h = π (3)^{2} (5) = π (9) (5) = 45 π$

The volume is $45 π$ .

Exam tip: No calculator is allowed, and answer choices usually keep $π$ in them, so leave your answer as a multiple of $π$ (like $45 π$ ) unless the choices are decimals. Canceling common factors first - like $\frac{120}{180}$ or $\frac{180}{4}$ - also saves time.

Sidenote

Squaring the radius

Do not forget to square the radius. The formula uses $r^{2}$ , so make sure you square the radius before multiplying.

Surface area

Definitions

Surface area: The outside area of a $3$ -dimensional shape

Surface area is the total area covering only the outside of an object. Think of it as the area you can touch.

Different shapes have different formulas, but the idea is always the same: add up the area of every face.

A cube has 6 equal square faces
A rectangular prism has 6 rectangular faces
Other shapes follow the same idea: find each face, then add them together

Sidenote

Drawing the shape

It is strongly recommended to draw all geometry problems, especially surface area problems! These are easy to get correct, but also easy to miss by forgetting a side. Draw the shape and label each face.

Example:

What is the surface area of a rectangular prism with length $4$ cm, width $3$ cm, and height $2$ cm?

A. $26$ cm²
B. $38$ cm²
C. $46$ cm²
D. $52$ cm²

Find the area of each pair of opposite faces, then add them together.

$Front and back Left and right Top and bottom = 2 \times (4 \times 2) = 16 = 2 \times (3 \times 2) = 12 = 2 \times (4 \times 3) = 24$

Add the three pairs:

$16 + 12 + 24 = 52$

The surface area is $52$ cm², so the answer is D.

Density

Definitions

Density: The ratio of an object’s mass to its volume (mass ÷ volume)

Density describes how much mass is packed into a given amount of space.

You can think of it like this:

More mass in the same space = higher density
Less mass in the same space = lower density

The formula for density is:

$D e n s i t y = \frac{ma ss}{v o l u m e}$

Sidenote

Matching units

Make sure your units match. Volume is usually in $c m^{3}$ and mass is in grams, so density will be in $\frac{g}{c m ^{3}}$ .

Example:

A rectangular prism with a width of $4$ centimeters, a length of $3$ centimeters, and a height of $5$ centimeters has a mass of $180$ grams. What is its density?

A. $3 \frac{g}{c m ^{3}}$
B. $6 \frac{g}{c m ^{3}}$
C. $12 \frac{g}{c m ^{3}}$
D. $18 \frac{g}{c m ^{3}}$

Start by finding the volume of the rectangular prism.

$V = 4 \times 3 \times 5 = 60 cm^{3}$

Now divide mass by volume.

$D e n s i t y = \frac{180}{60} = 3 \frac{g}{c m ^{3}}$

The density is $3 \frac{g}{c m ^{3}}$ .

Area of irregular shapes

To find the area of an irregular shape, break it into familiar shapes (like rectangles, squares, or triangles). Find the area of each smaller piece, then add them together.

This works because the total area is just the sum of all the smaller parts.

Sidenote

Splitting the shape

Look for straight lines where you can split the shape into rectangles. Label each piece so you don’t miss any areas.

Example:

What is the area of the figure?

$An irregular shape made of right angles. Going clockwise from the top left: 12 across the top, down 2, in 2, down 3, 6 across the bottom, up 2, out 4, and up 3 to the start.$
Irregular shape example
Achievable

A. $32 cm^{2}$
B. $40 cm^{2}$
C. $46 cm^{2}$
D. $54 cm^{2}$

Split the figure into smaller rectangles.

$The irregular shape split into three rectangles: a 3 by 4 rectangle of area 12 on the left, a 6 by 5 rectangle of area 30 in the middle, and a 2 by 2 rectangle of area 4 on the right.$
Irregular shape answer explanation
Achievable

Find the area of each piece:

Rectangle 1: $3 \times 4 = 12$

Rectangle 2: $2 \times 2 = 4$

Large rectangle: $6 \times 5 = 30$

Add the areas:

$12 + 4 + 30 = 46$

The total area is $46 cm^{2}$ , so the answer is C.

Knowledge check: A solid metal cylinder has a radius of $2$ meters, a height of $6$ meters, and a mass of $96 π$ kilograms. What is its density?

A. $1 \frac{k g}{m ^{3}}$
B. $2 \frac{k g}{m ^{3}}$
C. $4 \frac{k g}{m ^{3}}$
D. $8 \frac{k g}{m ^{3}}$

(spoiler)

C. $4 \frac{k g}{m ^{3}}$

Radians

Radian: angle where arc length equals radius; full circle = $2 π$ radians = $360°$
Key equivalences: $180° = π$ , $90° = \frac{π}{2}$
Degrees→radians: multiply by $\frac{π}{180}$ ; radians→degrees: multiply by $\frac{180}{π}$
Simplify/cancel fractions before multiplying to save time

Volume

Volume = area of base × height
Rectangular prism: $V = l e n g t h \times w i d t h \times h e i g h t$
Cylinder: $V = π r^{2} h$ (remember to square the radius first)
Exam tip: leave answers as multiples of $π$ unless choices are decimals

Surface area

Total area of all outside faces; add area of every face together
Cube: 6 equal square faces; rectangular prism: 6 rectangular faces (3 pairs)
Strategy: draw the shape, label each face, compute opposite-face pairs, then sum (e.g., $16 + 12 + 24 = 52 c m^{2}$ )

Density

Formula: $D e n s i t y = \frac{ma ss}{v o l u m e}$
Higher density = more mass packed in same space
Ensure units match (commonly $\frac{g}{c m ^{3}}$ ); find volume first, then divide mass by it

Area of irregular shapes

Break complex figure into familiar shapes (rectangles, squares, triangles)
Calculate area of each piece separately, then sum for total area
Look for straight lines to split shape; label pieces to avoid missing any area

Next | 2.7 Math strategies

Radians and measurement applications

Radians

Definitions

Radian: The angle at the center of a circle that cuts off an arc as long as the radius.

Another way to measure angles in a circle is in radians. Radians measure a circle as fractions of $2 π$ .

Here’s why $2 π$ shows up: if a circle has radius $1$ , then its circumference is

$2 π r = 2 π (1) = 2 π$

A full turn sweeps the whole circumference, which is $2 π$ radii long, so $36 0^{\circ} = 2 π$ radians.

That means:

Half a circle ( $18 0^{\circ}$ ) is $π$
One fourth of a circle ( $9 0^{\circ}$ ) is $\frac{π}{2}$

This works for any circle, no matter the size.

These relationships let us convert between degrees and radians. Since $18 0^{\circ} = π$ , we can use this to switch between the two units.