Areas Related to Circles: Sectors and Segments
Sector of a Circle: Definition and Properties
In geometry, a sector of a circle is a region bounded by two radii of the circle and the arc connecting their endpoints on the circumference.
Visually, a sector resembles a slice of a circular pie or pizza. It is defined by the central angle formed by the two radii.
In the diagram, O is the center of the circle, OA and OB are two radii, and $\overarc{AB}$ is the arc between points A and B. The region OAB enclosed by OA, OB, and $\overarc{AB}$ is a sector.
Components of a Sector
- Radii: The two straight line segments extending from the center of the circle to the circumference (OA and OB in the diagram). Their length is equal to the radius ($r$) of the circle.
- Arc: The portion of the circle's circumference between the endpoints of the two radii (the curved line $\overarc{AB}$).
- Central Angle ($\theta$): The angle formed by the two radii at the center of the circle ($\angle \text{AOB}$). This angle is typically measured in degrees or radians. The size of the central angle determines the size of the sector.
Types of Sectors
For any central angle less than $180^\circ$, there are two sectors associated with it:
-
Minor Sector:
This is the sector formed by the smaller central angle ($\theta < 180^\circ$) and the minor arc (the shorter arc connecting the two points on the circumference). When the term "sector" is used without further qualification, it generally refers to the minor sector.
-
Major Sector:
This is the larger sector formed by the same two radii but corresponding to the reflex angle ($360^\circ - \theta$ in degrees, or $2\pi - \theta$ in radians) and the major arc (the longer arc connecting the two points). The major sector completes the full circle when combined with the minor sector.
If the central angle is exactly $180^\circ$, both sectors are identical and are called semicircles.
Properties of Sectors
- Proportionality: The area of a sector and the length of its arc are directly proportional to the central angle. This means that if you double the central angle, you double the arc length and the sector area (as long as the angle remains less than $360^\circ$).
- Congruence: Sectors belonging to the same circle or to congruent circles are congruent if and only if their central angles are equal.
- Composition: The sum of the area of a minor sector and the area of its corresponding major sector equals the area of the entire circle.
Area(Minor Sector) + Area(Major Sector) = Area(Circle)
Similarly, the sum of the lengths of the minor arc and its corresponding major arc equals the circumference of the entire circle.
Length(Minor Arc) + Length(Major Arc) = Circumference(Circle)
- Boundary: The boundary of a sector consists of the two radii and the arc. The perimeter of a sector includes the lengths of the two radii and the length of the arc.
Perimeter(Sector) = Radius + Radius + Arc Length
Perimeter(Sector) = $2r + l$
where $l$ is the length of the arc of the sector.
Understanding these properties and the definition of a sector is essential for calculating its arc length and area.
Length of Arc of a Sector
The length of the arc ($l$) of a sector is the length of the curved part of the sector's boundary. It is a fraction of the total circumference of the circle, and this fraction is determined by the central angle of the sector.
The key relationship used to find the arc length is that the ratio of the sector's central angle to the total angle in a circle ($360^\circ$ or $2\pi$ radians) is equal to the ratio of the sector's arc length to the circle's total circumference ($C = 2\pi r$).
$\mathbf{\frac{Arc \$\$ Length}{Circumference} = \frac{Central \$\$ Angle}{Total \$\$ Angle \$\$ in \$\$ a \$\$ Circle}}$
... (1)
Formula Derivation
Let the radius of the circle be $r$, the circumference be $C = 2\pi r$, the central angle of the sector be $\theta$, and the arc length be $l$.
Using Degrees:
If the central angle $\theta$ is measured in degrees, the total angle in a circle is $360^\circ$. Substitute these values into the proportionality relationship (1):
$\frac{l}{2\pi r} = \frac{\theta}{360^\circ}$
[Using (1) with angle in degrees]
To find $l$, multiply both sides of the equation by $2\pi r$:
"$l = \frac{\theta}{360^\circ} \times 2\pi r$"
Rearranging for clarity:
$\textbf{Length \$\$ of \$\$ Arc} (l) = \mathbf{\frac{\theta}{360^\circ} \times 2\pi r}$
[Formula for arc length, $\theta$ in degrees] ... (2)
Using Radians:
If the central angle $\theta$ is measured in radians, the total angle in a circle is $2\pi$ radians. Substitute these values into the proportionality relationship (1):
$\frac{l}{2\pi r} = \frac{\theta}{2\pi}$
[Using (1) with angle in radians]
To find $l$, multiply both sides of the equation by $2\pi r$:
"$l = \frac{\theta}{2\pi} \times 2\pi r$"
Simplify the expression by cancelling out $2\pi$ from the numerator and denominator:
"$l = \frac{\theta}{\cancel{2\pi}} \times \cancel{2\pi} r$"
$\textbf{Length \$\$ of \$\$ Arc} (l) = \mathbf{r \theta}$
[Formula for arc length, $\theta$ in radians] ... (3)
This simpler formula ($l = r\theta$) is one of the main reasons why radians are the standard unit for angles in many areas of mathematics and physics, particularly when dealing with circular motion or sectors.
Examples
Example 1. Find the length of the arc of a sector with a central angle of $60^\circ$ in a circle of radius $21$ cm. (Use $\pi = \frac{22}{7}$)
Answer:
Given:
Central angle, $\theta = 60^\circ$.
Radius of the circle, $r = 21$ cm.
Value of $\pi = \frac{22}{7}$.
To Find:
Length of the arc, $l$.
Solution:
Since the angle is given in degrees, we use the formula for arc length in degrees:
"$l = \frac{\theta}{360^\circ} \times 2\pi r$"
[Formula (2)]
Substitute the given values for $\theta$, $\pi$, and $r$:
"$l = \frac{60^\circ}{360^\circ} \times 2 \times \frac{22}{7} \times 21 \$ \text{cm}$"
[Substituting values]
Simplify the fraction involving the angle:
"$l = \frac{\cancel{60}^{1}}{\cancel{360}_6} \times 2 \times \frac{22}{7} \times 21 \$ \text{cm}$"
[Simplifying $\frac{60}{360}$]
"$l = \frac{1}{6} \times 2 \times \frac{22}{7} \times 21 \$ \text{cm}$"
Now perform the multiplication, cancelling common factors:
"$l = \frac{1}{\cancel{6}_3} \times \cancel{2}_1 \times \frac{22}{\cancel{7}_1} \times \cancel{21}^3 \$ \text{cm}$"
[Cancel 6 with 2 and 3, Cancel 7 with 21]
"$l = \frac{1}{3} \times 1 \times 22 \times 3 \$ \text{cm}$"
"$l = \frac{22 \times \cancel{3}}{\cancel{3}} \$ \text{cm}$"
"$\mathbf{l = 22 \$\$ cm}$"
Therefore, the length of the arc of the sector is 22 centimetres (cm).
Example 2. Find the length of the arc of a sector with a central angle of $1.5$ radians in a circle of radius $8$ cm.
Answer:
Given:
Central angle, $\theta = 1.5$ radians.
Radius of the circle, $r = 8$ cm.
To Find:
Length of the arc, $l$.
Solution:
Since the angle is given in radians, we use the simpler formula for arc length in radians:
"$l = r \theta$"
[Formula (3)]
Substitute the given values for $r$ and $\theta$:
"$l = 8 \$ \text{cm} \times 1.5$"
[Substituting values]
Perform the multiplication:
"$8 \times 1.5 = 8 \times \frac{15}{10} = 8 \times \frac{3}{2} = \frac{24}{2} = 12$"
"$\mathbf{l = 12 \$\$ cm}$"
Therefore, the length of the arc is 12 centimetres (cm).
Area of a Sector of a Circle
The area of a sector is the measure of the two-dimensional region enclosed by the two radii and the arc of the sector. Similar to the arc length, the area of a sector is a fraction of the total area of the circle, determined by the size of the central angle.
The key relationship used to find the area of a sector is that the ratio of the sector's area to the total area of the circle ($A_{\text{circle}} = \pi r^2$) is equal to the ratio of the sector's central angle ($\theta$) to the total angle in a circle ($360^\circ$ or $2\pi$ radians).
$\mathbf{\frac{Area \$\$ of \$\$ Sector}{Area \$\$ of \$\$ Circle} = \frac{Central \$\$ Angle}{Total \$\$ Angle \$\$ in \$\$ a \$\$ Circle}}$
... (1)
Formula Derivation
Let the radius of the circle be $r$, the area of the circle be $A_{\text{circle}} = \pi r^2$, the central angle of the sector be $\theta$, and the area of the sector be $A_{\text{sector}}$.
Using Degrees:
If the central angle $\theta$ is measured in degrees, the total angle in a circle is $360^\circ$. Substitute these values into the proportionality relationship (1):
$\frac{A_{\text{sector}}}{\pi r^2} = \frac{\theta}{360^\circ}$
[Using (1) with angle in degrees]
To find $A_{\text{sector}}$, multiply both sides of the equation by $\pi r^2$:
"$A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2$"
Rearranging for clarity:
$\textbf{Area \$\$ of \$\$ Sector} (A_{\text{sector}}) = \mathbf{\frac{\theta}{360^\circ} \times \pi r^2}$
[Formula for sector area, $\theta$ in degrees] ... (2)
Using Radians:
If the central angle $\theta$ is measured in radians, the total angle in a circle is $2\pi$ radians. Substitute these values into the proportionality relationship (1):
$\frac{A_{\text{sector}}}{\pi r^2} = \frac{\theta}{2\pi}$
[Using (1) with angle in radians]
To find $A_{\text{sector}}$, multiply both sides of the equation by $\pi r^2$:
"$A_{\text{sector}} = \frac{\theta}{2\pi} \times \pi r^2$"
Simplify the expression by cancelling out $\pi$ from the numerator and denominator:
"$A_{\text{sector}} = \frac{\theta}{2\cancel{\pi}} \times \cancel{\pi} r^2$"
$\textbf{Area \$\$ of \$\$ Sector} (A_{\text{sector}}) = \mathbf{\frac{1}{2} r^2 \theta}$
[Formula for sector area, $\theta$ in radians] ... (3)
This formula using radians is particularly elegant.
Formula using Arc Length:
We can also express the area of a sector in terms of its arc length ($l$) and radius ($r$). From the formula for arc length in radians, $l = r\theta$ (or $\theta = l/r$), assuming $\theta$ is in radians. Substitute this expression for $\theta$ into the sector area formula in radians (3):
"$A_{\text{sector}} = \frac{1}{2} r^2 \theta$"
[From (3)]
"$A_{\text{sector}} = \frac{1}{2} r^2 \left(\frac{l}{r}\right)$"
[Substitute $\theta = l/r$]
Simplify the expression:
"$A_{\text{sector}} = \frac{1}{2} r^{\cancel{2}} \frac{l}{\cancel{r}}$"
$\textbf{Area \$\$ of \$\$ Sector} (A_{\text{sector}}) = \mathbf{\frac{1}{2} l r}$
[Formula for sector area, using arc length and radius] ... (4)
This formula resembles the area of a triangle ($\frac{1}{2} \times \text{base} \times \text{height}$), where the arc length $l$ serves as the 'base' and the radius $r$ as the 'height'.
Examples
Example 1. Find the area of a sector of a circle with radius $4$ cm and an angle of $30^\circ$. Also, find the area of the corresponding major sector. (Use $\pi = 3.14$)
Answer:
Given:
Radius of the circle, $r = 4$ cm.
Central angle of the minor sector, $\theta = 30^\circ$.
Value of $\pi = 3.14$.
To Find:
1. Area of the minor sector ($A_{\text{minor}}$).
2. Area of the corresponding major sector ($A_{\text{major}}$).
Solution:
Part 1: Area of the minor sector ($A_{\text{minor}}$)
Since the angle is given in degrees, we use the formula for the area of a sector in degrees:
"$A_{\text{minor}} = \frac{\theta}{360^\circ} \times \pi r^2$"
[Formula (2)]
Substitute the given values for $\theta$, $\pi$, and $r$:
"$A_{\text{minor}} = \frac{30^\circ}{360^\circ} \times 3.14 \times (4 \$ \text{cm})^2$"
[Substituting values]
Simplify the fraction involving the angle:
"$A_{\text{minor}} = \frac{\cancel{30}^{1}}{\cancel{360}_{12}} \times 3.14 \times 16 \$ \text{cm}^2$"
[Simplifying $\frac{30}{360}$ and $4^2$]
"$A_{\text{minor}} = \frac{1}{12} \times 3.14 \times 16 \$ \text{cm}^2$"
Perform the multiplication and simplification:
"$A_{\text{minor}} = \frac{3.14 \times 16}{12} \$ \text{cm}^2$"
"$A_{\text{minor}} = \frac{3.14 \times \cancel{16}^4}{\cancel{12}_3} \$ \text{cm}^2$"
[Simplifying $\frac{16}{12}$ by dividing by 4]
"$A_{\text{minor}} = \frac{3.14 \times 4}{3} \$ \text{cm}^2 = \frac{12.56}{3} \$ \text{cm}^2$"
Calculate the decimal value:
"$12.56 \div 3 \approx 4.1866...$"
"$\mathbf{A_{\text{minor}} \approx 4.19 \$\$ cm^2}$"
[Rounding to two decimal places]
The area of the minor sector is approximately 4.19 square centimetres ($\text{cm}^2$).
Part 2: Area of the corresponding major sector ($A_{\text{major}}$)
Method 1: Using the angle of the major sector
The central angle of the major sector is the remaining angle in the circle:
Angle of Major Sector $= 360^\circ - \theta = 360^\circ - 30^\circ = 330^\circ$
Using the formula for sector area with this angle:
"$A_{\text{major}} = \frac{330^\circ}{360^\circ} \times \pi r^2$"
[Using Formula (2)]
Substitute the values:
"$A_{\text{major}} = \frac{\cancel{330}^{11}}{\cancel{360}_{12}} \times 3.14 \times (4 \$ \text{cm})^2$"
[Simplifying $\frac{330}{360}$ and $4^2$]
"$A_{\text{major}} = \frac{11}{12} \times 3.14 \times 16 \$ \text{cm}^2$"
"$A_{\text{major}} = \frac{11 \times 3.14 \times \cancel{16}^4}{\cancel{12}_3} \$ \text{cm}^2$"
[Simplifying $\frac{16}{12}$ by dividing by 4]
"$A_{\text{major}} = \frac{11 \times 12.56}{3} \$ \text{cm}^2 = \frac{138.16}{3} \$ \text{cm}^2$"
"$138.16 \div 3 \approx 46.0533...$"
"$\mathbf{A_{\text{major}} \approx 46.05 \$\$ cm^2}$"
[Rounding to two decimal places]
Method 2: Subtracting minor sector area from total circle area
Calculate the total area of the circle:
Area of Circle ($A_{\text{circle}}$) $= \pi r^2 = 3.14 \times (4 \$ \text{cm})^2$"
"$A_{\text{circle}} = 3.14 \times 16 \$ \text{cm}^2$"
"$A_{\text{circle}} = 50.24 \$ \text{cm}^2$"
The area of the major sector is the total area minus the minor sector area:
"$A_{\text{major}} = A_{\text{circle}} - A_{\text{minor}}$"
"$\approx 50.24 \$ \text{cm}^2 - 4.1867 \$ \text{cm}^2$"
[Using the more precise value for $A_{\text{minor}}$]
"$\approx 46.0533 \$ \text{cm}^2$"
"$\mathbf{A_{\text{major}} \approx 46.05 \$\$ cm^2}$"
[Rounding to two decimal places]
Both methods give approximately the same result. The area of the corresponding major sector is approximately 46.05 square centimetres ($\text{cm}^2$).
Segment of a Circle: Definition and Properties
A segment of a circle is a region of the circle bounded by a chord and the arc that the chord subtends. In simpler terms, it's the area cut from a circle by a straight line (the chord).
Think of a circular pizza or cake. If you make a straight cut across the cake (this is the chord), the smaller piece you separate (bounded by the cut and the curved crust) is a segment of the circle.
In the diagram, AB is a chord. The region enclosed by the chord AB and the arc $\overarc{AB}$ is a segment of the circle.
Components of a Segment
- Chord: The straight line segment that connects two points on the circumference of the circle (AB in the diagram).
- Arc: The continuous portion of the circle's circumference between the two endpoints of the chord (the curved line $\overarc{AB}$). The segment is bounded by this arc and the chord.
Types of Segments
Any chord that is not a diameter divides the circle into two distinct segments:
-
Minor Segment:
This is the smaller region bounded by the chord and the minor arc (the shorter arc). When the term "segment" is used alone, it usually refers to the minor segment.
-
Major Segment:
This is the larger region bounded by the same chord and the major arc (the longer arc). The major segment complements the minor segment to form the entire circle.
If the chord is a diameter, it passes through the center and divides the circle into two equal parts. Each of these equal segments is called a semicircle.
Relationship with Sector and Triangle
A segment is closely related to a sector that shares the same arc and endpoints of the chord. Consider a circle with center O, and a chord AB. The radii OA and OB form a sector OAB.
The segment bounded by chord AB and arc AB is the region between the chord AB and the arc AB. This region can be seen as the difference between the sector OAB and the triangle OAB formed by the two radii and the chord.
-
Minor Segment:
The minor segment is the area of the minor sector OAB minus the area of the triangle $\triangle \text{OAB}$.
Area(Minor Segment) = Area(Minor Sector OAB) - Area($\triangle$ OAB)
-
Major Segment:
The major segment is the area of the major sector OAB plus the area of the triangle $\triangle \text{OAB}$. Alternatively, and more commonly, it is the area of the entire circle minus the area of the minor segment.
Area(Major Segment) = Area(Major Sector OAB) + Area($\triangle$ OAB)
Area(Major Segment) = Area(Circle) - Area(Minor Segment)
This relationship is crucial for calculating the area of a segment, as shown in the next section.
Area of a Segment of a Circle
The area of a segment of a circle is calculated by subtracting the area of the triangle formed by the two radii and the chord from the area of the corresponding sector. This method applies to both minor and major segments, though it's usually easier to calculate the minor segment and subtract from the total circle area for the major segment.
Formula Derivation (for Minor Segment)
Consider the minor segment formed by chord AB and minor arc AB in a circle with center O and radius $r$. Let the central angle subtended by the arc (and chord) at the center be $\theta$.
From the previous section, we know that the area of the minor segment is:
$\mathbf{A_{\text{segment}} = Area(Minor \$\$ Sector \$\$ OAB) - Area(\triangle OAB)}$
... (1)
We need the formulas for the area of the sector and the area of the triangle $\triangle \text{OAB}$ in terms of $r$ and $\theta$.
Area of Sector OAB:
The area of a sector with radius $r$ and central angle $\theta$ is:
- If $\theta$ is in degrees:
$\text{Area(Sector)} = \frac{\theta}{360^\circ} \times \pi r^2$
... (2a)
- If $\theta$ is in radians:
$\text{Area(Sector)} = \frac{1}{2} r^2 \theta$
... (2b)
Area of Triangle OAB:
In $\triangle \text{OAB}$, we have two sides equal to the radius ($OA=OB=r$), and the included angle is $\theta$. The area of a triangle given two sides and the included angle is $\frac{1}{2} \times \text{side}_1 \times \text{side}_2 \times \sin(\text{included angle})$.
Area($\triangle$ OAB) $= \frac{1}{2} \times OA \times OB \times \sin(\theta)$
$\mathbf{Area(\triangle OAB) = \frac{1}{2} r^2 \sin(\theta)}$
... (3)
This formula for the area of the triangle works whether $\theta$ is in degrees or radians, as long as the $\sin(\theta)$ function is evaluated using the correct angle unit.
Combining to find Segment Area:
Substitute the formulas for the area of the sector and the area of the triangle into equation (1).
- If $\theta$ is in degrees:
$\mathbf{A_{\text{segment}} = \left( \frac{\theta}{360^\circ} \times \pi r^2 \right) - \left( \frac{1}{2} r^2 \sin(\theta) \right)}$
[Formula for minor segment area, $\theta$ in degrees] ... (4a)
We can factor out $r^2$ from this expression:
$\mathbf{A_{\text{segment}} = r^2 \left( \frac{\pi \theta}{360^\circ} - \frac{1}{2} \sin(\theta) \right)}$
- If $\theta$ is in radians:
$\mathbf{A_{\text{segment}} = \left( \frac{1}{2} r^2 \theta \right) - \left( \frac{1}{2} r^2 \sin(\theta) \right)}$
[Formula for minor segment area, $\theta$ in radians] ... (4b)
Factor out $\frac{1}{2} r^2$ from this expression:
$\mathbf{A_{\text{segment}} = \frac{1}{2} r^2 (\theta - \sin(\theta))}$
Note: When using the radian formula $A_{\text{segment}} = \frac{1}{2} r^2 (\theta - \sin(\theta))$, $\theta$ must be in radians, and $\sin(\theta)$ must be calculated with $\theta$ in radians. For the degree formula, $\theta$ is in degrees, and $\sin(\theta)$ is calculated with $\theta$ in degrees.
Area of Major Segment:
The area of the major segment is the area of the entire circle minus the area of the corresponding minor segment:
$\mathbf{A_{\text{major segment}} = Area(Circle) - Area(Minor \$\$ Segment)}$
... (5)
$\mathbf{A_{\text{major segment}} = \pi r^2 - A_{\text{minor segment}}}$
[Using Formula (5)]
Example
Example 1. Find the area of the segment corresponding to a chord of a circle of radius $15$ cm which subtends an angle of $120^\circ$ at the center. (Use $\pi = 3.14$ and $\sqrt{3} = 1.73$)
Answer:
Given:
Radius of the circle, $r = 15$ cm.
Central angle subtended by the chord, $\theta = 120^\circ$.
Approximate values: $\pi \approx 3.14$, $\sqrt{3} \approx 1.73$.
To Find:
Area of the corresponding minor segment.
Solution:
The area of the minor segment is the difference between the area of the sector formed by the angle $\theta$ and the area of the triangle formed by the two radii and the chord.
Step 1: Calculate the Area of the Sector
Using the formula for the area of a sector with angle in degrees:
"$A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2$"
[Using Formula (2a)]
Substitute the given values:
"$A_{\text{sector}} = \frac{120^\circ}{360^\circ} \times 3.14 \times (15 \$ \text{cm})^2$"
[Substituting values]
Simplify the fraction $\frac{120}{360} = \frac{1}{3}$ and calculate $(15)^2 = 225$:
"$A_{\text{sector}} = \frac{1}{3} \times 3.14 \times 225 \$ \text{cm}^2$"
"$A_{\text{sector}} = 3.14 \times \frac{225}{3} \$ \text{cm}^2 = 3.14 \times 75 \$ \text{cm}^2$"
[Simplifying $\frac{225}{3}$]
Calculate the product:
$\begin{array}{cc}& & 3\ . & 1 & 4 \\ \times & & & 7 & 5 \\ \hline && 1\ 5 & 7 & 0 \\ & 219 & 8 & \times \\ \hline & 235\ . & 5 & 0 \\ \hline \end{array}$"$A_{\text{sector}} = 235.50 \$ \text{cm}^2$"
Step 2: Calculate the Area of the Triangle (OAB)
Using the formula for the area of $\triangle \text{OAB}$:
"$A_{\triangle OAB} = \frac{1}{2} r^2 \sin(\theta)$"
[Using Formula (3)]
Substitute the given values for $r$ and $\theta$:
"$A_{\triangle OAB} = \frac{1}{2} \times (15 \$ \text{cm})^2 \times \sin(120^\circ)$"
[Substituting values]
Calculate $(15)^2 = 225$ and find the value of $\sin(120^\circ)$. We know that $\sin(120^\circ)$ lies in the second quadrant, where sine is positive. $\sin(120^\circ) = \sin(180^\circ - 60^\circ) = \sin(60^\circ) = \frac{\sqrt{3}}{2}$.
"$A_{\triangle OAB} = \frac{1}{2} \times 225 \$ \text{cm}^2 \times \frac{\sqrt{3}}{2}$"
[Using $\sin(120^\circ) = \frac{\sqrt{3}}{2}$]
"$A_{\triangle OAB} = \frac{225 \sqrt{3}}{4} \$ \text{cm}^2$"
Substitute the approximate value of $\sqrt{3} = 1.73$:
"$A_{\triangle OAB} = \frac{225 \times 1.73}{4} \$ \text{cm}^2$"
Calculate the numerator: $225 \times 1.73$
$\begin{array}{cc}& & 2 & 2 & 5 \\ \times & & 1\ . & 7 & 3 \\ \hline && 6 & 7 & 5 \\ & 157 & 5 & \times \\ 225 & \times & \times \\ \hline 389\ . & 2 & 5 \\ \hline \end{array}$"$A_{\triangle OAB} = \frac{389.25}{4} \$ \text{cm}^2$"
Perform the division:
$\begin{array}{r} 97\ . \ 3\ 1\ 2\ 5\phantom{)} \\ 4{\overline{\smash{\big)}\,389\ . \ 2500\phantom{)}}} \\ \underline{-~\phantom{(}36\phantom{.0000)}} \\ 29\phantom{.0000)} \\ \underline{-~\phantom{(}28\phantom{.0000)}} \\ 1\ 2\phantom{000)} \\ \underline{-~\phantom{()1}2\phantom{000)}} \\ 0\ 5\phantom{00)} \\ \underline{-~\phantom{()0}4\phantom{00)}} \\ 10\phantom{0)} \\ \underline{-~\phantom{()0}8\phantom{0)}} \\ 20\phantom{)} \\ \underline{-~\phantom{()}20\phantom{)}}\\ 0\phantom{)} \end{array}$"$A_{\triangle OAB} = 97.3125 \$ \text{cm}^2$"
Step 3: Calculate the Area of the Segment
Using formula (1):
"$A_{\text{segment}} = A_{\text{sector}} - A_{\triangle OAB}$"
Substitute the calculated areas:
"$A_{\text{segment}} = 235.50 \$ \text{cm}^2 - 97.3125 \$ \text{cm}^2$"
[Substituting areas]
Perform the subtraction:
$\begin{array}{ccccccc} & 2 & 3 & 5\ . & 5 & 0 & 0 & 0 \\ - & & 9 & 7\ . & 3 & 1 & 2 & 5 \\ \hline & 1 & 3 & 8\ . & 1 & 8 & 7 & 5 \\ \hline \end{array}$"$A_{\text{segment}} = 138.1875 \$ \text{cm}^2$"
Rounding to two decimal places as the input values for $\pi$ and $\sqrt{3}$ suggest approximate calculation:
"$\mathbf{A_{\text{segment}} \approx 138.19 \$\$ cm^2}$"
Therefore, the area of the segment is approximately 138.19 square centimetres ($\text{cm}^2$).
Perimeter and Area of Sector and Segment (Consolidated Formulas)
This section provides a summary of the key formulas for calculating the perimeter and area of sectors and segments of a circle with radius $r$ and central angle $\theta$. It's important to use consistent units for the angle ($\theta$) – either degrees or radians – when applying these formulas.
Consolidated Formulas
| Quantity | Formula ( $\theta$ in degrees) | Formula ( $\theta$ in radians) | Notes / Additional Information |
|---|---|---|---|
| Arc Length ($l$) (Length of the curved boundary) |
$l = \frac{\theta}{360^\circ} \times 2\pi r$ | $l = r \theta$ | $\theta$ must be in the specified unit (degrees or radians) for each formula. |
| Area of Sector (Region bounded by two radii and the arc) |
$A_{\text{sector}} = \frac{\theta}{360^\circ} \times \pi r^2$ | $A_{\text{sector}} = \frac{1}{2} r^2 \theta$ | Both degree and radian formulas are consistent. The area can also be found using the arc length: $A_{\text{sector}} = \frac{1}{2} l r$. |
| Area of Triangle formed by Radii and Chord ($\triangle$ OAB) | $A_{\triangle} = \frac{1}{2} r^2 \sin(\theta)$ | This triangle is formed by the two radii OA, OB and the chord AB. Formula is the same for $\theta$ in degrees or radians, but ensure $\sin(\theta)$ is evaluated using the correct mode for $\theta$. | |
| Area of Minor Segment (Region bounded by chord and minor arc) |
$A_{\text{minor seg}} = A_{\text{sector}} - A_{\triangle}$ $A_{\text{minor seg}} = \left( \frac{\theta}{360^\circ} \times \pi r^2 \right) - \left( \frac{1}{2} r^2 \sin(\theta) \right)$ |
$A_{\text{minor seg}} = A_{\text{sector}} - A_{\triangle}$ $A_{\text{minor seg}} = \frac{1}{2} r^2 \theta - \frac{1}{2} r^2 \sin(\theta)$ |
Difference between the sector area and the triangle area. Use the $\theta$ in radians formula for the $\frac{1}{2}r^2(\theta - \sin\theta)$ form. |
| Area of Major Segment (Region bounded by chord and major arc) |
$A_{\text{major seg}} = \text{Area of Circle} - A_{\text{minor seg}}$ $A_{\text{major seg}} = \pi r^2 - A_{\text{minor seg}}$ |
Sum of Area(Major Sector) + Area(Triangle). Usually easier to calculate via subtraction. | |
| Perimeter of Sector (Boundary length of sector) |
$P_{\text{sector}} = \text{Length of Arc} (l) + 2 \times \text{Radius} (r)$ $P_{\text{sector}} = l + 2r$ |
Boundary consists of the arc and the two radii. Calculate arc length using appropriate formula ($\theta$ in deg or rad). | |
| Perimeter of Segment (Boundary length of segment) |
$P_{\text{segment}} = \text{Length of Arc} (l) + \text{Length of Chord AB}$ | Boundary consists of the arc and the chord. Chord length AB = $2r \sin(\theta/2)$. Calculate arc length $l$ using appropriate formula. Use consistent units for $\theta$ for both $l$ and $\sin(\theta/2)$. | |
Important Reminders:
- Always check the units of the central angle ($\theta$). If it's given in degrees, use the degree formulas. If it's given in radians, use the radian formulas. Do not mix them.
- When using the $\sin(\theta)$ function to find the area of the triangle or the length of the chord, ensure that your calculator is in the correct mode (degrees or radians) corresponding to the unit of $\theta$.
- $\pi \approx \frac{22}{7}$ or $3.14$ are common approximations. Use the value specified in the problem.