krit.club logo

Areas Related to Circles - Area of Sector and Segment of a Circle

Grade 10CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

A Sector of a circle is the region enclosed by two radii and the corresponding arc connecting them. Visually, it resembles a 'slice of pizza' or a 'pie piece' originating from the center of the circle.

•

The Central Angle (θ\theta) is the angle formed at the center of the circle by the two radii that define the sector. This angle determines what fraction of the total circle the sector occupies (i.e., θ360∘\frac{\theta}{360^\circ}).

•

A Minor Sector is the smaller part of the circle where the central angle θ\theta is less than 180∘180^\circ. A Major Sector is the remaining larger part of the circle where the angle is 360∘−θ360^\circ - \theta. Visually, if you cut a small slice of cake, the slice is the minor sector and the rest of the cake is the major sector.

•

The Length of an Arc is the distance along the curved boundary of the sector. Visually, if you trace the 'crust' of a pizza slice, that curved length represents the arc length, which is a proportional part of the total circumference (2πr2\pi r).

•

A Segment of a circle is the region bounded by a chord and its corresponding arc. Visually, if you draw a straight line (chord) across a circle, the two regions created are segments. Unlike a sector, a segment does not necessarily include the center of the circle.

•

A Minor Segment is the smaller area between the chord and the minor arc, while the Major Segment is the larger area between the chord and the major arc. Visually, a minor segment looks like a flat 'cap' or 'sliver' cut off from the side of the circle.

•

The Area of a Minor Segment is calculated by finding the area of the corresponding sector and then subtracting the area of the triangle formed by the center and the two endpoints of the chord. Visually: Area of Segment = Area of Pizza Slice - Area of the Triangle formed by the radii and the chord.

📐Formulae

Area of a circle = πr2\pi r^2

Circumference of a circle = 2πr2\pi r

Length of an arc of a sector with angle θ\theta = θ360∘×2πr\frac{\theta}{360^\circ} \times 2\pi r

Area of a sector of a circle with radius rr and angle θ\theta = θ360∘×πr2\frac{\theta}{360^\circ} \times \pi r^2

Area of a major sector = πr2−Area of minor sector\pi r^2 - \text{Area of minor sector}

Area of a triangle with two sides as radii rr and included angle θ\theta = 12r2sin⁡θ\frac{1}{2} r^2 \sin \theta

Area of a minor segment = Area of sector - Area of triangle = (θ360∘×πr2)−12r2sin⁡θ\left( \frac{\theta}{360^\circ} \times \pi r^2 \right) - \frac{1}{2} r^2 \sin \theta

Area of a major segment = πr2−Area of minor segment\pi r^2 - \text{Area of minor segment}

💡Examples

Problem 1:

Find the area of a sector of a circle with radius 66 cm if the angle of the sector is 60∘60^\circ. (Use π=227\pi = \frac{22}{7})

Solution:

  1. Given: Radius r=6r = 6 cm, Angle θ=60∘\theta = 60^\circ.
  2. Use the formula: Area of sector = θ360∘×πr2\frac{\theta}{360^\circ} \times \pi r^2.
  3. Substitute the values: Area = 60360×227×6×6\frac{60}{360} \times \frac{22}{7} \times 6 \times 6.
  4. Simplify: Area = 16×227×36\frac{1}{6} \times \frac{22}{7} \times 36.
  5. Area = 22×67=1327\frac{22 \times 6}{7} = \frac{132}{7} cm2^2.
  6. In decimal form: Area ≈18.86\approx 18.86 cm2^2.

Explanation:

To find the area of the sector, we determine the fraction of the total circle area using the ratio of the central angle to 360∘360^\circ and multiply it by the full area πr2\pi r^2.

Problem 2:

A chord of a circle of radius 1010 cm subtends a right angle at the center. Find the area of the corresponding minor segment. (Use π=3.14\pi = 3.14)

Solution:

  1. Given: Radius r=10r = 10 cm, Central Angle θ=90∘\theta = 90^\circ.
  2. Step 1: Calculate Area of Sector = 90360×3.14×10×10=14×314=78.5\frac{90}{360} \times 3.14 \times 10 \times 10 = \frac{1}{4} \times 314 = 78.5 cm2^2.
  3. Step 2: Calculate Area of Triangle OAB=12×base×height=12×r×r=12×10×10=50OAB = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times r \times r = \frac{1}{2} \times 10 \times 10 = 50 cm2^2 (since the angle is 90∘90^\circ).
  4. Step 3: Area of Minor Segment = Area of Sector - Area of Triangle = 78.5−50=28.578.5 - 50 = 28.5 cm2^2.

Explanation:

The minor segment is the region between the chord and the arc. We find the area of the entire 'slice' (sector) and subtract the triangular part formed by the radii and the chord to leave only the segment area.