The Arc and Sector Lab

Section 7.4 -- Arc length and area of a sector

A 14-inch pizza cut into 8 slices. A 16-inch pizza cut into 8 slices. Both slices have the same angle. One has noticeably more crust and a lot more cheese. How much more? That is arc length and sector area, and the answers are not the same multiple.

Step 1 of 6

Picture a pizza with a 14-inch diameter, cut into 8 equal slices.

What fraction of the whole pizza is one slice?

What is the central angle of one slice, in degrees and radians?

Step 2 of 6

How long is the curved crust on one slice?

The whole circumference is \(2\pi r = 2\pi(7) = 14\pi \approx 43.98\) inches.

One slice is \(\frac{1}{8}\) of it.

Compute: \(\frac{1}{8} \times 14\pi = \,?\) (within 0.1 inches)

Arc length (inches):

Step 3 of 6

Now let's generalize. Replace \(\frac{1}{8}\) with \(\frac{\theta}{2\pi}\).

Arc length \(= \frac{\theta}{2\pi} \times 2\pi r\)

The \(2\pi\) cancels. What remains?

Step 4 of 6

Same move for area. One slice is \(\frac{1}{8}\) of \(\pi r^2\). Generalize with \(\frac{\theta}{2\pi}\).

Sector area \(= \frac{\theta}{2\pi} \times \pi r^2\)

The \(\pi\) cancels. What remains?

Step 5 of 6

Predict: if you double the angle (keep the radius the same), what happens?

Arc length:

Sector area:

Step 6 of 6

Now predict: if you double the RADIUS instead (keep the angle the same), what happens?

Arc length:

Sector area:

Explore Arc Length and Sector Area

Radius: 7

Angle (radians): π/4

Show degrees

Arc Length (s)

Sector Area (A)

Proportion of circle:

Arc Length Formula

\(s = r\theta\)

Sector Area Formula

\(A = \frac{1}{2}r^2\theta\)

Adjust the radius and angle sliders to see how arc length and sector area change.

Try This

A windshield wiper blade is 18 inches long and sweeps 110 degrees. How much glass area does it clear?

Hint: convert to radians first. Tolerance: within 5 square inches.

Area (square inches):

A sprinkler throws water 40 feet and rotates through 75 degrees. How many square feet does it water?

If you want to double the watered area without moving the sprinkler, which is more effective: doubling the angle or doubling the range?

Tolerance: within 10 square feet for the area.

Area (square feet):

To double the watered area, which is better?

A running track's curve is a semicircle. The inner lane has a radius of 36.5 m and the outer lane has a radius of 37.72 m. How much longer is the outer curve than the inner curve?

There are two curves per lap. How much longer is the outer lane for one full lap (curves only)?

Tolerance: within 0.5 m for one curve, within 1 m for both curves.

Difference per curve (meters):

Total difference per lap (meters):

Derive the arc length formula \(s = r\theta\) from the proportion of a circle's circumference
Derive the sector area formula \(A = \frac{1}{2}r^2\theta\) from the proportion of a circle's area
Explain why both formulas require \(\theta\) in radians
Distinguish how arc length scales linearly with radius while sector area scales quadratically
Apply both formulas to real-world measurement problems

Quick Check

A sector has radius 10 and central angle \(\frac{\pi}{3}\). What is its area?

Instructor Notes

Teaching Notes

The pizza framing is deliberate: students already have intuition about slices as fractions. The derivation of \(s = r\theta\) from \(\frac{\theta}{2\pi} \times 2\pi r\) lets students see the cancellation happen rather than memorize the result. The same move for area reinforces the pattern.

The key aha is step 6: doubling the radius doubles the arc but quadruples the area. This is where the pizza question from the hook resolves, and it connects to the broader principle that linear dimensions scale linearly while areas scale quadratically.

Common Student Errors

Using degrees directly in \(s = r\theta\). The formulas assume radians because the derivation divided by \(2\pi\), not by 360. Emphasize: if a problem gives degrees, convert first.
Assuming area scales the same way as arc length when radius changes. The \(r^2\) in the area formula means area grows faster than arc length. This is exactly the pizza insight.
Forgetting the \(\frac{1}{2}\) in the area formula. Students who remember \(s = r\theta\) sometimes write \(A = r^2\theta\) by analogy.

Discussion Questions

Why do these formulas only work when \(\theta\) is in radians? What would happen if you plugged in degrees?
A pizza shop charges by the slice. Is the 14-inch pizza or the 16-inch pizza a better deal per square inch, assuming the same price per slice?
If you double both the radius and the angle, what happens to the arc length? To the area?

Exam Connection

Arc length and sector area are standard exam topics. Common question formats: "Find the arc length given radius and angle," "Find the area given arc length and radius," and the reverse problems. The unit-conversion trap (degrees to radians) appears frequently. The running-track problem is a classic application.