Eight equally spaced holes (purple dots) are to be drilled into a wooden disc.  The green dot is the center of the disc.  The disc is 20 inches in diameter and the holes are to be drilled 1 inch from the edge of the disc, measured to the center of each hole.  Find the distance between two adjacent holes measured from the center each hole.

To solve this problem, we need to find the distance between two adjacent holes that are equally spaced around a circular path within a disc. Given the information:

- The disc has a diameter of 20 inches, so the radius of the disc is:
  \[
  R = \frac{20}{2} = 10 \text{ inches}
  \]
  
- The holes are drilled 1 inch from the edge of the disc, so the radius of the circle that passes through the center of these holes is:
  \[
  r = R - 1 = 10 - 1 = 9 \text{ inches}
  \]

This means the centers of the holes lie on a circle of radius 9 inches.

### Step 1: Calculate the angle between adjacent holes
Since there are 8 equally spaced holes around the circle, the angle between each adjacent hole, measured in degrees, is:
\[
\theta = \frac{360^\circ}{8} = 45^\circ
\]

### Step 2: Use the chord length formula to find the distance between adjacent holes
The distance \(d\) between two adjacent holes (chord length) can be calculated using the formula for the chord length:
\[
d = 2r \sin\left(\frac{\theta}{2}\right)
\]

Substituting the known values:
\[
d = 2 \times 9 \times \sin\left(\frac{45^\circ}{2}\right)
\]
\[
d = 18 \times \sin(22.5^\circ)
\]

### Step 3: Calculate the sine of 22.5 degrees
Using a calculator or known sine value:
\[
\sin(22.5^\circ) \approx 0.3827
\]

So the distance between two adjacent holes is:
\[
d = 18 \times 0.3827 \approx 6.8886 \text{ inches}
\]

### Answer
The distance between two adjacent holes, measured from the center of each hole, is approximately \(6.89\) inches.

Would you like more details on any of these steps, or do you have any questions?

Here are five related questions to expand on this topic:
1. What is the formula for finding the distance between points on a circle when they are not equally spaced?
2. How do you calculate the circumference of a circle, and what role does it play in equally spaced points?
3. How would the distance between adjacent holes change if the number of holes increased?
4. What is the relationship between chord length and arc length in a circle?
5. How do you convert angles from degrees to radians in calculations involving circles?

**Tip:** When dealing with problems involving circular distances, using radian measure for angles can often simplify the calculations.

Eight equally spaced holes (purple dots) are to be drilled into a wooden disc. The green dot is the center of the disc. The disc is 20 inches in diameter, and the holes are to be drilled 1 inch from the edge of the disc, measured to the center of each hole. Find the distance between two adjacent holes measured from the center of each hole.

Learn how to calculate the distance between two adjacent holes equally spaced around a wooden disc with a 20-inch diameter. Using geometry and trigonometry concepts such as the chord length formula, this problem offers a step-by-step solution. Ideal for students in Grades 9-11.

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