3.6.2 Sector
A sector is a region enclosed by two radii of a circle and an arc that connects the endpoints of the radii. A sector is essentially a portion of the circle bounded by two radii and an arc.
The measure of a sector is usually given in degrees or radians and is equal to the measure of the central angle that subtends the arc of the sector.
The area of a sector can be calculated using the formula
\( A =\frac{θ}{360}πr^2 \)
where \( A \) is the area of the sector, \( θ \)is the central angle (in degrees) subtended by the arc of the sector, \( r \) is the radius of the circle, and \( \pi \) is the constant pi (approximately equal to \( 3.14159 \)).
Worked example:
The diagram shows a sector of a circle of radius \( 3.8 cm \).
The arc length is \( 7.7 \) cm.
Calculate the value of \( y \).
Arc length is given by
\( l=\frac{θ}{360}\times 2πr \)
Substitute values from the diagram
\( 7.7=\frac{y}{360}\times 2\times \pi \times 3.8 \)
\( \frac{7.7\times 360}{2π×3.8}=y \)
\( y=116.0993427 \)
Rounding to one decimal place
\( y=116.1 \)
\( y=116.1 \)
Calculate the area of the sector.
Area of the sector is given by the formula
Area of sector \( =\frac{θ}{360}\times r^2 \)
Substitute the values from the diagram
Area of sector\( =\frac{116.1}{360}\times π \times 3.8^2 \)
Area of sector\( =14.63 \)
Area of sector \( =14.63 \)
Test yourself
Question 1:
\( OAB \) is a sector of a circle with center \( O \) and radius \( 7 \)cm.
The area of the sector is \( 40 \) \( cm^2 \).
Calculate the perimeter of the sector.
Give your answer correct to \( 3 \) significant figures.
[4]
Question 2:
A circle center \( O \) has radius \( 9 cm \).
Calculate the perimeter of the shaded sector of the circle.
Give your answer correct to \( 3 \) significant figures.
[4]
Question 3:
\( OAC \)is a sector of a circle , center \( O \), radius \( 10 m \).
\( BA \) is a tangent to the circle at point \( A \).
\( BC \) is a tangent to the circle at point \( C \).
Angle \( AOC=120° \)
Calculate area of the shaded region.
Give your answer correct to three significant figures.
[5]
Question 4:
The diagram shows a circle, center \( O \).
The straight line \( ABC \) is a tangent to the circle at \( B \).
\( OB=8 \) cm, \( AB=15 \) cm and \( BC=22.4 \) cm.
\( AO \) crosses the circle at \( X \) and \( OC \) crosses the circle at \( Y \).
Calculate angle \( XOY \).
[5]
Calculate the length of the arc \( XBY \).
[2]
Calculate the total area of the two shaded regions.
[4]
Question 5:
The diagram shows a design made from a triangle \( AOC \) joined to a sector \( OCB \).
\( AC=8 \) cm, \( OB=OC=7 \) cm and angle \( ACO=78° \) .
Use the cosine rule to show that \( OA=9.47 \) cm, correct to \( 2 \) decimal places.
[4]
Calculate angle \( OAC \).
[3]
The perimeter of the design is \( 29.5 \) cm.
Show that the angle \( COB=41.2° \), correct to \( 1 \) decimal place.
[5]
Calculate the total area of the design.
[4]