Arc (Circle)

$Arc (Circle): meaning, method and solved example (Geometry) – Maths PDF$

Sommaire

In short: An arc is a connected stretch of a circle’s edge between two of its points, and its measure in degrees equals the central angle that opens onto it.

GeometryGeometryG-C.A.2

What is Arc?

An arc is the curved stretch of a circle between two of its points, like the crust of a pizza slice with the cheese removed. Two points cut a circle into a shorter minor arc and a longer major arc; when they are opposite ends of a diameter, both pieces are semicircles.

Measure is an angle quantity, counted in degrees: it equals the central angle opening onto the piece. The whole circle is $360^{\circ}$, and neighbors that share only an endpoint add up, so $70^{\circ}$ followed by $55^{\circ}$ makes $125^{\circ}$. Measure in degrees is different from length in feet or centimeters.

How to spot it

A figure shows a curved piece of a circle highlighted between two labeled points.
The notation has a small curve over two letters, or the words minor arc and major arc appear.
A central angle is given and the matching piece of the rim must be named or measured.
Two neighboring pieces of a circle are known and the combined piece is requested.

The method

Find the central angle that opens onto the piece; its degrees are the measure.
For a major piece, subtract the minor piece from $360^{\circ}$.
For neighbors sharing an endpoint, add their measures; for a semicircle, the total is $180^{\circ}$.
If a length is wanted instead, convert the measure with the fraction $\frac{\text{measure}}{360}$ of the circumference.

The central angle of 112° gives a minor arc of 112°; the rest of the circle is 248°.

Arc: solved example

Problem

On a circular garden centered at $O$, two stakes $A$ and $B$ form a central angle $\angle AOB = 112^{\circ}$. A third stake $C$ lies next to $B$, and the rim between $B$ and $C$ measures $47^{\circ}$ with no overlap. Find the minor piece $AB$, the major piece $AB$ and the combined stretch from $A$ to $C$ through $B$.

Solution

The minor piece equals its central angle: $112^{\circ}$. The major piece is the remainder of the circle: $360 – 112 = 248^{\circ}$.

Neighbors add: $112 + 47 = 159$. The pieces measure $112^{\circ}$, $248^{\circ}$ and $159^{\circ}$.

Try it

Question 1

Two neighboring pieces of a circle measure $74^{\circ}$ and $(4x + 6)^{\circ}$, and together they form a semicircle. Find $x$.

Show the answer

$74 + 4x + 6 = 180$, so $4x = 100$ and $x = 25$.

Question 2

What does a piece measuring exactly $180^{\circ}$ look like, and what is its name?

Show the answer

It is half the circle, cut by a diameter: a semicircle.

Arc vs. Chord

A chord is a straight segment between two points of the circle, and it has a length. The curved path along the rim between the same two points has both a degree measure and a length. The chord cuts across; the curve follows the rim, so it is always longer than its chord.

Chord

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Related words

Whole dictionary: A–Z · Geometry · Geometry

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