I can't seem to remember the formula for this geometry problem...?

Yes. Let me explain.

The radius is given to you as r=12.

The total circumference of the pie should be:

2πr = 24π ≈ 75.4

Now, what you have done is subtracted 16 from that.

What percentage have you subtracted?

16 / 75.4 = 21.22%

You have also removed that amount of the 360° angle around the center of the pie. In other words, the angle you removed is:

(21.22%) × 360° = 0.2212 × 360° = 76.4°

That's your answer.

Note: The similarity between circumference (75.4) and the angle (76.4°) is a complete coincidence.

The formula for the circumference of a circle (the "crust" as you call it) is:

C = 2π r

r is the radius, which in your case is 12 inches. So the total circumference would be:

C = 2π * 12

= 24π

≈ 75.4 inches

You have a section of the circumference that is only 16 inches.

Similarly, your angle will only be part of the full 360°

These two ratios will be the same:

16 to 75.4 (arc to full circumference)

x to 360° (angle to full circle)

Equate them:

16/75.4 = x/360°

Simplify the left hand side:

0.2122 = x/360°

Multiply both sides by 360°

x = 0.2122 * 360°

x ≈ 76.4°

What you are describing is a sector of the complete circle, and the info given to you are:

radius = r = 12

arc = s= 16

The formula to find the angle is

s= r * tetha

Being tetha the angle given in radians. So lets find that:

tetha= s/r= 16/12= 1.33 radians.

Now we use conversions to find the angle in degrees:

1.33 radians * ( 360 degrees/ 2 pi radians) = 76.4 degrees.

Thats all

ANSWER

I think the "sides" you are referring to are the radii (singular form: radius). The 16 inches you are talking about is the arc length (the circumference) excluding the radii.

The basic foruma for getting the circumference (C) is

2(pi)(r); pi is the ratio of the circumference to the diameter and is approximately equal to 3.14.

The arc length is like a pro-rated circumference. If x is the angle of the arc, then

C of arc = 2(pi)(r)(x/360)

16 = 2 (pi) (12) (x/360)

x=((8)(360))/((12)(pi))

x=((360)2)/((3)(pi))

x=720/((3)(pi))

x=240/pi

If we convert it to decimal, the angle is approximately 76.43 degrees.

Crust length = Side length * Θ

where

Crust length = 16

Side length = 12

Θ = angle of pie cut (in radians)

Substituting values,

16 = 12(Θ)

Θ = 16/12 = 4/3 radians

and since there are 180 degrees per π radians, then

Θ = (4/3)(180/π)

Θ = 76.39 degrees

Hope this helps.

I'll answer the question because it is too hard to explain...

You take the side measured 12 (Radius) and find the circumference.

C=2•3.14•12= 75.36

Here is the formula: arc length = angle/360 • C

16 = angle/360 • 75.36

Divide 75.36 to each side to get:

0.2123 = angle/360

Multiply 360 to each side to get:

76.4 = angle

There's your answer.

ii

Circumference of whole pie = 2 π r = 24 π ins

Length of arc = 16 ins

24 π<------------>360°

16 <-------------->16/(24 π) x 360°

2 / (3π) x 360° = 76.4°