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Arc: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">ArcArc: Definition and ExamplesTable of ContentsArc - Definition and Examples in Mathematics Definition of Arc in Math

In mathematics, an arc is defined as a portion of the boundary of a circle or a curve. It can also be referred to as an open curve. The arc is the distance between any two points traced along the circumference of a circle. We denote an arc using the symbol "⌢". For example, arc ABABAB can be written as AB^\widehat{AB}AB or BA^\widehat{BA}BA. The order of points does not matter when naming an arc.

Arcs can be classified into different types based on their length. When an arc divides a circle into two parts, the shorter distance between the two endpoints is called a minor arc, while the longer distance is called a major arc. Unless specified, an arc is always considered a minor arc. To specify the major arc, we can take a third point on the arc and use three letters in the name. Additionally, a semicircle is an arc that has its endpoints on the diameter of a circle.

Examples of Arc in Math Example 1: Calculating Arc Length with a 60-Degree Angle Problem:

Calculate the length of an arc that subtends an angle of 606060 degrees at the center of a circle with a radius of 5 cm.

Step-by-step solution:

Step 1, Recall the arc length formula: y360×2πr\frac{y}{360} \times 2\pi r360y​×2πr, where yyy is the angle and rrr is the radius.

Step 2, Fill in the known values. In this problem, y=60y = 60y=60 degrees and r=5r = 5r=5 cm.

Step 3, Substitute these values into the formula:

Arc length = 60360×2×3.14×5\frac{60}{360} \times 2 \times 3.14 \times 536060​×2×3.14×5

Step 4, Calculate the value:

Arc length = 60360×2×3.14×5=5.23\frac{60}{360} \times 2 \times 3.14 \times 5 = 5.2336060​×2×3.14×5=5.23 cm Example 2: Finding Arc Length with a 40-Degree Angle Problem:

Calculate the length of the arc that subtends an angle of 404040 degrees at the center of a circle with a radius of 666 cm.

Step-by-step solution:

Step 1, Remember the arc length formula: y360×2πr\frac{y}{360} \times 2\pi r360y​×2πr, where yyy is the angle and rrr is the radius.

Step 2, Identify the known values. Here, y=40y = 40y=40 degrees and r=6r = 6r=6 cm.

Step 3, Put these values into the formula:

Arc length = 40360×2×3.14×6\frac{40}{360} \times 2 \times 3.14 \times 636040​×2×3.14×6

Step 4, Solve the equation:

Arc length = 40360×2×3.14×6=4.186\frac{40}{360} \times 2 \times 3.14 \times 6 = 4.18636040​×2×3.14×6=4.186 cm Example 3: Identifying Major Arcs in a Circle Problem:

Identify the major arc in this circle.

Arc

Step-by-step solution:

Step 1, Remember that a major arc is the longer distance between two endpoints on a circle.

Step 2, Look at the circle with points AAA, BBB, CCC, and DDD marked on it.

Step 3, Notice that there are two possible paths between points AAA and CCC: path ABCABCABC and path ADCADCADC.

Step 4, Compare the two paths. Path ADCADCADC covers a greater portion of the circle's circumference.

Step 5, Conclusion: ADC^\widehat{ADC}ADC is the major arc, and ABC^\widehat{ABC}ABC is the minor arc.

Comments(3)FFigureSkatingDevoteeZoeNovember 6, 2025I've used this arc definition with my students. It's super clear! The examples really helped them grasp the concept. Thanks!

BBaseballFanaticScarlettNovember 5, 2025This arc glossary page is great! I've used it to explain arcs to my students, and the examples really helped them grasp the concept.

PPlantParentHankNovember 4, 2025I've used this arc definition with my students. It's clear and the examples really help them grasp the concept. Great resource!

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