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This essential trains for: SAT-I, GMAT, AMC-8, Math Kangaroo 9-10, Math Kangaroo 7-8, SAT-II.

Two different radii delimit two circular sectors, one that is associated with the minor arc and one that is associated with the major arc.

figure

A segment of a circle is the figure delimited by a chord and the circle.

figure

Computing the length of an arc and the area of a sector

In a circle, the arc length and the area of a sector are directly proportional with the central angle that generates them.

This means that the formulas to calculate them are easily derived using the length and the area of the whole circle.

Do not memorize the formulas - instead, use direct proportionality:

Central angle in degrees Arc length
equation equation
equation equation

Find the unknown arc length using the properties of proportions:

equation

Similarly,

Central angle in radians Arc length
equation equation
equation equation

Find the unknown arc length using the properties of proportions:

equation

Very often, easy fractions of the circle are used in problems, making the use of a formula an overkill.

It is better to think it terms of what fraction of a circle we have in mind.

Example:

In a circle of radius 5 units, what is the length of the arc intercepted by an angle of 24°?

Notice that 360 divides evenly into 24:

equation

Thus, the arc is the 15th part of the whole circumference:

equation

Central angle in degrees Sector Area
equation equation
equation equation

Find the unknown sector area using the properties of proportions:

equation

Similarly,

Central angle in radians Sector Area
equation equation
equation equation

Find the unknown sector area using the properties of proportions:

equation

Area and perimeter of a segment

Use the corresponding circle sector and the isosceles triangle formed by the radii that are incident to the ends of the chord to find the perimeter and the area of a sector.

figure

Example:

A chord that has the same length as the radius of a circle will always form an equilateral triangle with the radii that are incident to its ends.

Therefore, the length of the minor arc, plus the length of the chord will form the perimeter of the sector:

equation

The area of such a segment is the difference between the area of the circular sector and the area of the equilateral triangle inscribed in it:

equation

Of course, this example has been much simplified by the fact that the chord corresponds to a central angle of 60o.