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Length Of An Arc And Area Of A Sector

Derive and apply the formulas for the length of an arc and the area of a sector using the unitary method.

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Defining Sectors and Arcs

Concept introduction for sectors and corresponding arcs.

A sector of a circle is the portion (or part) of the circular region enclosed by two radii and their corresponding arc. Think of it like a slice of pizza cut from the center.

When we describe a sector, we measure the central angle it creates, usually represented by the symbol $\theta$ .

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Length of an Arc and Area of a Sector Visual

Must visually connect the angle $\theta$ to the fraction of the total circular area/perimeter.

Clean scientific diagram of a circle with a shaded minor sector enclosed by two radii and an arc, clearly labeling the r…

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Deriving the Formulas

Using the unitary method to find sector area and arc length.

To calculate the area of a sector, we use the Unitary Method, comparing it to the full circle.

We know that the area of a full circle (which forms a 360° angle at the center) is $\pi r^2$ .

When the degree measure is 360, the area is $\pi r^2$ .
So, when the degree measure is 1, the area is $\frac{\pi r^2}{360}$ .
Therefore, when the degree measure is $\theta$ , the area is $\frac{\theta}{360} \times \pi r^2$ .

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Formula: Area of a Sector

Formal statement of the sector area formula.

Area of a Sector

\frac{θ}{360} \times π r^{2}

Symbol Legend

$r$ = radius of the circle
$θ$ = angle in degrees

Major Sector Area

To find the major sector area, subtract the minor sector from the total circle area:

π r^{2} - Minor Sector Area

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Formula: Length of an Arc

Formal statement of the arc length formula.

Length of an Arc

\frac{θ}{360} \times 2 π r

Symbol Legend

$r$ = radius of the circle
$θ$ = angle in degrees

Concept Check

Just like sector area is a fraction of the total area, the arc length is the exact same fraction

(\frac{θ}{360})

of the total circumference

(2 π r)

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Example 1: Finding Sector Areas

Step-by-step solution for minor and major sector area.

Problem. Find the area of the sector of a circle with radius 4 cm and of angle 30°. Also, find the area of the corresponding major sector (Use $\pi = 3.14$ ).

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Faded Example: Apply the Formula

Partially solved problem finding sector area.

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To find the area of the sector with radius $r = 6$ cm and angle $\theta = 60^\circ$ , we must determine what fraction of the full circle is covered. We start with the formula: Area = $(\theta/360) \times \pi r^2$ . By substituting the central angle, we find that the fraction of the circle is . We also need to square the radius, which gives us an $r^2$ value of . Next, we substitute these into our formula along with $\pi = 22/7$ . This translates to $(1/6) \times (22/7) \times 36$ . Carefully multiplying these numbers together yields our final simplified fractional answer. The area of the sector is cm².

← backNCERT X - Mathematics next →Area Of A Segment Of A Circle

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Defining Sectors and Arcs

Concept introduction for sectors and corresponding arcs.

A sector of a circle is the portion (or part) of the circular region enclosed by two radii and their corresponding arc. Think of it like a slice of pizza cut from the center.

When we describe a sector, we measure the central angle it creates, usually represented by the symbol $\theta$ .

1 / 2

IMAGE

Length of an Arc and Area of a Sector Visual

Must visually connect the angle $\theta$ to the fraction of the total circular area/perimeter.

Clean scientific diagram of a circle with a shaded minor sector enclosed by two radii and an arc, clearly labeling the r…

content

Deriving the Formulas

Using the unitary method to find sector area and arc length.

To calculate the area of a sector, we use the Unitary Method, comparing it to the full circle.

We know that the area of a full circle (which forms a 360° angle at the center) is $\pi r^2$ .

When the degree measure is 360, the area is $\pi r^2$ .
So, when the degree measure is 1, the area is $\frac{\pi r^2}{360}$ .
Therefore, when the degree measure is $\theta$ , the area is $\frac{\theta}{360} \times \pi r^2$ .

1 / 2

content

Example 1: Finding Sector Areas

Step-by-step solution for minor and major sector area.

Problem. Find the area of the sector of a circle with radius 4 cm and of angle 30°. Also, find the area of the corresponding major sector (Use $\pi = 3.14$ ).

1 / 4

reading

Faded Example: Apply the Formula

Partially solved problem finding sector area.

0 of 3 blanks filled