Minor Arc

Q: What is the difference between a minor arc and a major arc?

A minor arc is the shorter arc between two points on a circle, with a central angle less than 180°. A major arc is the longer arc between the same two points, with a central angle greater than 180°. The two arcs together make up the entire circle, so their measures always add to 360°.

Q: How do you name a minor arc?

A minor arc is named using its two endpoints, such as arc AB (written $\overset{\frown}{AB}$). A major arc requires three letters — the two endpoints and one point in between — to distinguish it from the minor arc. For example, arc ACB specifies the longer path from A to B passing through C.

Q: Can a minor arc equal exactly 180°?

No. When the central angle is exactly 180°, the two arcs are equal in length and each one is called a semicircle, not a minor or major arc. A minor arc must have a central angle strictly less than 180°.

Minor Arc

The shorter of the two arcs between two points on a circle.

Circle with points A (top) and B (right), showing the minor arc—the shorter arc from A to B along the right side.

See also

Major arc

Key Formula

\text{Arc length} = \frac{\theta}{360°} \times 2\pi r

Where:

$\theta$ = The central angle (in degrees) that subtends the minor arc, where 0° < θ < 180°
$r$ = The radius of the circle
$2\pi r$ = The full circumference of the circle

Worked Example

Problem: A circle has a radius of 10 cm. Two points A and B on the circle create a central angle of 60°. Find the length of minor arc AB.

Step 1: Identify that the central angle is less than 180°, confirming arc AB is a minor arc.

\theta = 60° < 180°

Step 2: Write the arc length formula.

\text{Arc length} = \frac{\theta}{360°} \times 2\pi r

Step 3: Substitute the known values into the formula.

\text{Arc length} = \frac{60°}{360°} \times 2\pi(10)

Step 4: Simplify the fraction and compute.

\text{Arc length} = \frac{1}{6} \times 20\pi = \frac{20\pi}{6} = \frac{10\pi}{3} \approx 10.47 \text{ cm}

Answer: The minor arc AB has a length of

\frac{10\pi}{3} \approx 10.47

cm.

Another Example

This example works backward — given the arc length, you solve for the central angle, and then verify the arc is minor by checking that the angle is less than 180°.

Problem: In a circle with radius 12 cm, minor arc PQ has a length of 4π cm. Find the central angle that subtends this arc.

Step 1: Start with the arc length formula and solve for θ.

\text{Arc length} = \frac{\theta}{360°} \times 2\pi r

Step 2: Substitute the known values: arc length = 4π and r = 12.

4\pi = \frac{\theta}{360°} \times 2\pi(12)

Step 3: Simplify the right side.

4\pi = \frac{\theta}{360°} \times 24\pi

Step 4: Divide both sides by 24π to isolate the fraction containing θ.

\frac{4\pi}{24\pi} = \frac{\theta}{360°} \implies \frac{1}{6} = \frac{\theta}{360°}

Step 5: Multiply both sides by 360° to find θ.

\theta = \frac{360°}{6} = 60°

Answer: The central angle is 60°, which is less than 180°, confirming PQ is indeed a minor arc.

Frequently Asked Questions

What is the difference between a minor arc and a major arc?

A minor arc is the shorter arc between two points on a circle, with a central angle less than 180°. A major arc is the longer arc between the same two points, with a central angle greater than 180°. The two arcs together make up the entire circle, so their measures always add to 360°.

How do you name a minor arc?

A minor arc is named using its two endpoints, such as arc AB (written

\overset{\frown}{AB}

). A major arc requires three letters — the two endpoints and one point in between — to distinguish it from the minor arc. For example, arc ACB specifies the longer path from A to B passing through C.

Can a minor arc equal exactly 180°?

No. When the central angle is exactly 180°, the two arcs are equal in length and each one is called a semicircle, not a minor or major arc. A minor arc must have a central angle strictly less than 180°.

Minor Arc vs. Major Arc

	Minor Arc	Major Arc
Definition	The shorter arc between two points on a circle	The longer arc between two points on a circle
Central angle	Greater than 0° and less than 180°	Greater than 180° and less than 360°
Naming convention	Two letters (e.g., arc AB)	Three letters (e.g., arc ACB)
Relationship	Minor arc = 360° − Major arc	Major arc = 360° − Minor arc
Arc length formula	(θ / 360°) × 2πr	((360° − θ) / 360°) × 2πr, where θ is the minor arc's central angle

Why It Matters

Minor arcs appear throughout geometry whenever you work with circles — from finding arc lengths and sector areas to applying the inscribed angle theorem. In coordinate geometry and trigonometry, understanding which arc an angle subtends helps you correctly set up equations. Real-world applications include calculating distances along curved paths, designing circular gears, and determining how far a point travels along a wheel's rim.

Common Mistakes

Mistake: Confusing the measure of a minor arc with the measure of the major arc between the same two points.

Correction: Always check that your central angle is less than 180° for a minor arc. If you find an angle greater than 180°, you have the major arc. The two must sum to 360°.

Mistake: Using the arc's degree measure directly as its length.

Correction: The degree measure tells you the fraction of the circle the arc covers. To get the actual length, multiply the fraction (θ/360°) by the full circumference 2πr. An arc of 90° on a circle of radius 4 is not 90 units long — it is (90/360) × 2π(4) = 2π ≈ 6.28 units.

Related Terms

Arc of a Circle — General term for any portion of a circle's circumference
Major Arc — The longer arc complementing the minor arc
Circle — The shape on which arcs are defined
Central Angle — The angle whose measure equals the minor arc's degree measure
Sector — The region bounded by two radii and an arc
Circumference — The full perimeter of a circle used to compute arc length
Inscribed Angle — Equals half the intercepted arc's measure
Point — Endpoints that define the arc on the circle