Radians and measurement applications

Radians

Definitions

Radian

A way of measuring the circle in terms of $\pi$

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

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

$$2\pi r=2\pi (1)=2\pi$$

So a full circle ($360^{\circ }$) corresponds to $2\pi$ radians.

That means:

• Half a circle ($180^{\circ }$) is $\pi$
• One fourth of a circle ($90^{\circ }$) is $\frac{\pi }{2}$

This works for any circle, no matter the size.

These relationships let us convert between degrees and radians. Since $180^{\circ }=\pi$, we can use this to switch between the two units.

Converting degrees to radians

How to convert degrees to radians

Multiply by

$$\frac{\pi }{180}$$

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

Think of it like a unit swap: the degrees cancel out, and you’re left with radians. Always multiply the degree value by $\frac{\pi }{180}$, then simplify your fraction if possible.

Example 1:

What is $120^{\circ }$ in radians?

Start with the key idea:

$$180^{\circ }=\pi \text{ radians}$$

To convert degrees → radians, multiply by $\frac{\pi }{180}$.

Now apply it:

$$\begin{array}{rl}120\cdot \frac{\pi }{180}& =\frac{120\pi }{180}\end{array}$$

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

$$\begin{array}{rl}\frac{120\pi }{180}& =\frac{2\pi }{3}\end{array}$$

So, $120^{\circ }=\frac{2\pi }{3}$ radians.

Example 2:

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

Start with the same idea:

$$\pi \text{ radians}=180^{\circ }$$

To convert radians → degrees, multiply by $\frac{180}{\pi }$.

Now apply it:

$$\begin{array}{rl}\frac{5\pi }{4}\cdot \frac{180}{\pi }& =\frac{900\pi }{4\pi }\end{array}$$

The $\pi$ cancels out:

$$\begin{array}{rl}\frac{900\pi }{4\pi }& =\frac{900}{4}\end{array}$$

Simplify:

$$\begin{array}{rl}\frac{900}{4}& =225\end{array}$$

So, $\frac{5\pi }{4}$ radians = $225^{\circ }$.

Steps taken:

• Multiply by the correct conversion factor
  • $\frac{\pi }{180}$ for degrees → radians
  • $\frac{180}{\pi }$ for radians → degrees
• Cancel out units (degrees or $\pi$)
• Simplify the fraction

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:

$$volume=length\times width\times height$$

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

To find the volume, we first find the area of the base using length × width, then multiply by the height. This works because you are finding the area of the base, then stacking that area upward to fill the shape.

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

Cylinders

To find the volume of a cylinder:

$$Volume=\pi \times r^2\times height$$

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, $\pi r^2$ 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 $5$.

Use the formula:

$$V=\pi r^{2}h$$

Start by identifying the values:

• $r=3$
• $h=5$

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

$$\begin{array}{rl}V& =\pi r^{2}h\\ & =\pi (3{)}^2(5)\\ & =\pi (9)(5)\\ & =45\pi \end{array}$$

The volume is $45\pi$.

Sidenote

Common mistake!

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 cube with side length $2$ cm?

A. $4$ cm²
B. $8$ cm²
C. $16$ cm²
D. $24$ cm²

Each face is a square, so use the formula for area: side × side.

$$\begin{array}{rl}\text{Area of one face}& =2^2=4\\ \text{Total surface area}& =4\times 6=24\end{array}$$

The surface area is $24$ cm².

Density

Definitions

Density

The proportion of mass to volume of an object

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:

$$Density=\frac{mass}{volume}$$

Sidenote

Important!

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.

$$\begin{array}{rl}V& =4\times 3\times 5\\ & =60{\text{ cm}}^3\end{array}$$

Now divide mass by volume.

$$\begin{array}{rl}Density& =\frac{180}{60}\\ & =3\frac{g}{c{m}^3}\end{array}$$

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

Tip!

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?

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

Split the figure into smaller rectangles.

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 $46{\text{ cm}}^2$, so the answer is C.

Knowledge check: A cylindrical water tank has a radius of $2$ meters and a height of $6$ meters. The tank is made of metal with a mass of $96\pi$ kilograms.

What is the density of the tank?

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

(spoiler)

C. $4\frac{kg}{m^3}$