Finding the arc length of a circle is one of the most practical skills in geometry, whether you are solving textbook problems, designing a curved garden path, or calculating the distance along a circular track. When teachers ask you to find each arc length round to the nearest hundredth, they are testing both your ability to apply the correct formula and your precision with decimal rounding. This guide walks you through everything you need to know, from the basic arc length formula to step-by-step examples and common pitfalls to avoid Less friction, more output..
What Is Arc Length?
Arc length is the distance measured along the curved line that makes up a portion of a circle’s circumference. That's why unlike a chord, which is a straight line connecting two points on a circle, an arc follows the actual curve. Every arc is defined by two endpoints and a central angle—the angle formed at the center of the circle by two radii that extend to those endpoints.
If you imagine slicing a pizza, the crust of a single slice represents an arc. The length of that crust depends on two things: the size of the pizza (its radius) and how wide the slice is (its central angle). To find each arc length round to the nearest hundredth, you need to know both measurements and apply the appropriate formula But it adds up..
The Arc Length Formula
There are two versions of the arc length formula, depending on whether the central angle is measured in degrees or radians.
When the Angle Is in Degrees
[ \text{Arc Length} = 2\pi r \times \frac{\theta}{360} ]
Where:
- (r) is the radius of the circle
- (\theta) (theta) is the central angle in degrees
- (\pi) is approximately 3.14159
This formula works because a full circle is 360 degrees. The fraction (\frac{\theta}{360}) tells you what portion of the full circumference the arc represents.
When the Angle Is in Radians
[ \text{Arc Length} = r\theta ]
Where:
- (r) is the radius
- (\theta) is the central angle in radians
Radians provide a more direct relationship between the radius and the arc, which is why this formula is shorter. Even so, most introductory geometry problems use degrees, so you will often need to convert or stick with the degree-based formula Less friction, more output..
Step-by-Step: How to Find Each Arc Length
Follow these steps whenever you need to find each arc length round to the nearest hundredth:
- Identify the given information. Write down the radius ((r)) and the central angle ((\theta)). Make sure you know whether the angle is in degrees or radians.
- Choose the correct formula. Use the degree formula if your angle is in degrees. Use (s = r\theta) if your angle is in radians.
- Substitute the values. Plug the radius and angle into the formula.
- Calculate using (\pi). Use your calculator’s (\pi) button for the most accurate result, or use 3.14159 if your teacher specifies.
- Round to the nearest hundredth. Look at the thousandths place (the third decimal digit). If it is 5 or greater, round up the hundredths place. If it is less than 5, leave the hundredths place as it is.
- Include units. Arc length is a distance, so your final answer should carry units such as centimeters, inches, meters, or feet.
Example Problems with Solutions
Below are several worked examples that show exactly how to find each arc length round to the nearest hundredth.
Example 1: A Minor Arc in Degrees
Problem: Find the arc length of a circle with a radius of 8 cm and a central angle of 45°. Round to the nearest hundredth.
Solution:
Use the degree formula:
[ s = 2\pi(8) \times \frac{45}{360} ]
[ s = 16\pi \times 0.125 ]
[ s = 2\pi ]
[ s \approx 6.283185307... ]
Rounding to the nearest hundredth: The third decimal digit is 3, so we keep the second digit as is Small thing, real impact. That alone is useful..
Answer: 6.28 cm
Example 2: A Larger Central Angle
Problem: Find the arc length for a circle with radius 12 inches and a central angle of 210°. Round to the nearest hundredth.
Solution:
[ s = 2\pi(12) \times \frac{210}{360} ]
[ s = 24\pi \times \frac{7}{12} ]
[ s = 14\pi ]
[ s \approx 43.98229715... ]
Rounding to the nearest hundredth: The third decimal digit is 2, so we do not round up That's the part that actually makes a difference..
Answer: 43.98 inches
Example 3: Radius Given as a Decimal
Problem: A circular running track has a radius of 35.5 meters. A runner covers an arc corresponding to a central angle of 72°. Find the distance run, rounded to the nearest hundredth.
Solution:
[ s = 2\pi(35.5) \times \frac{72}{360} ]
[ s = 71\pi \times 0.2 ]
[ s = 14.2\pi ]
[ s \approx 44.610456... ]
Rounding to the nearest hundredth: The third decimal digit is 0, so the hundredths digit stays at 1.
Answer: 44.61 meters
Example 4: Angle in Radians
Problem: Find the arc length when (r = 10) units and (\theta = 1.5) radians. Round to the nearest hundredth Simple as that..
Solution:
[ s = r\theta ]
[ s = 10 \times 1.5 ]
[ s = 15
