How to Calculate the Area of a Segment of a Circle with Radius and Angle?
The concept of area of segment of a circle plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios.
What Is Area of Segment of a Circle?
A segment of a circle is defined as the region enclosed by a chord and the corresponding arc lying between the chord’s endpoints. You’ll find this concept applied in areas such as finding pizza slice areas, designing round gardens, and calculating segments in engineering drawings.
Key Formula for Area of Segment of a Circle
Here’s the standard formula: \( \text{Area of Segment} = \text{Area of Sector} - \text{Area of Triangle} \)
In radians, the formula commonly used is: \( \text{Area} = \dfrac{1}{2} r^2 (\theta - \sin\theta) \) where \( r \) is the radius and \( \theta \) is the angle in radians.
In degrees, substitute \( \theta \) in radians as \( \theta = \dfrac{\pi}{180} \times \text{angle in degrees} \).
Cross-Disciplinary Usage
Area of segment of a circle is not only useful in Maths but also plays an important role in Physics, Computer Science, and daily logical reasoning. Students preparing for JEE or NEET will see its relevance in various questions, such as calculating fields, parts of wheels, or circular data segments.
Step-by-Step Illustration
Let’s see an example: Find the area of a segment of a circle with radius 10 cm and a central angle of 60°.
\( 60^\circ = \dfrac{\pi}{3} \) radians
2. Use the segment formula:
\( \text{Area} = \dfrac{1}{2} \times 10^2 \times \left( \dfrac{\pi}{3} - \sin \dfrac{\pi}{3} \right) \)
\( \sin \dfrac{\pi}{3} = \dfrac{\sqrt{3}}{2} \approx 0.866 \)
\( \text{Area} = 50 \times \left( 1.047 - 0.866 \right) = 50 \times 0.181 = 9.05 \) cm²
Final Answer: The segment area is about 9.05 cm²
Speed Trick or Vedic Shortcut
Here’s a quick shortcut for the area of minor segment in exams: If the question uses a small angle (like 30°, 45°, 60°), remember that \(\sin\theta\) values are standard and easy to recall. Substitute these directly for fast calculation in the formula. This helps speed up MCQs and time-bound tests.
- For 60°, \(\sin 60° = 0.866\), for 30°, \(\sin 30° = 0.5\), for 45°, \(\sin 45° = 0.707\).
- Plug directly into the formula: \( \text{Area} = \frac{1}{2} r^2 (\theta - \sin\theta) \) where \( \theta \) is in radians.
Tricks like this aren’t just cool — they’re practical in competitive exams like NTSE, Olympiads, and even JEE. Vedantu’s live sessions include more such shortcuts to help you build speed and accuracy.
Try These Yourself
- Find the area of the segment of a circle with radius 5 cm and angle 90°.
- If a chord is 8 cm long in a circle of radius 5 cm, find the area of the minor segment (use formula with chord length).
- Calculate the area of the major segment if the central angle is 120° and radius is 7 cm.
- Find the area of a semicircular segment (hint: angle is 180°).
Frequent Errors and Misunderstandings
- Mixing up sector and segment area formulas.
- Forgetting to convert degrees to radians before using the standard formula.
- Confusing minor segment (smaller part) with major segment (larger part).
- Missing units in the final answer (cm² or m²).
- Not subtracting the triangle area correctly when using the sector minus triangle formula.
Relation to Other Concepts
The idea of area of segment of a circle connects closely with topics such as Area of a Circle and Area of Sector. Mastering this helps with understanding more advanced chapters in Mensuration and Geometry, especially in classes 9, 10, and beyond.
Classroom Tip
A quick way to remember the area of segment formula is: “Sector area minus triangle area.” Build the diagram, shade the segment, and always double-check if you need the minor or major segment. Vedantu’s teachers often use circle cut-outs and colored sections to help students visualize and retain these ideas easily in live classes.
We explored area of segment of a circle—from definition, formula, example calculation, mistakes, and how it connects to other subjects. Continue practicing with Vedantu to become confident in solving problems using this important Maths concept.
Keep Learning: Internal Links
- Area of a Circle — Understand circles before segments.
- Chord of a Circle — Explore how chords relate to segments.
- Area of Sector and Segment of a Circle — For combined exam sum practice.
- Arc of a Circle — Learn about arcs, essential in segment area problems.
- Properties of Circle — Master all-round circle concepts.
FAQs on Area of Segment of a Circle Explained with Formulas, Examples & Tips
1. What is a segment of a circle?
A segment of a circle is a region bounded by a chord and the arc of the circle. It's the area enclosed between a chord and the arc it cuts off. There are two types: major segments (larger area) and minor segments (smaller area).
2. How do I find the area of a segment of a circle?
The area of a segment is calculated by subtracting the area of a triangle from the area of a sector. The formula depends on whether you know the central angle (θ) or the height (h) of the segment. If θ is known (in radians): Area = (1/2)r²(θ - sin θ). If the height and radius are known, a more complex calculation is required involving trigonometric functions.
3. What's the difference between a sector and a segment of a circle?
A sector is a pie-slice shaped region bounded by two radii and an arc. A segment is bounded by a chord and an arc. Think of it this way: a sector is defined by angles and radii, while a segment is defined by a chord and an arc.
4. How do I find the area of a major segment?
The area of a major segment is found by subtracting the area of the minor segment from the total area of the circle (πr²). First, calculate the area of the minor segment using the appropriate formula, then subtract this value from the total area of the circle.
5. Can I use the segment area formula with angles in degrees?
Yes, but you need to convert the angle from degrees to radians first. Use the conversion: θ (radians) = (π/180) × θ (degrees). Then, substitute the radian measure into the area formula: Area = (1/2)r²(θ - sin θ).
6. What are some common mistakes students make when calculating segment area?
Common errors include: Using the wrong formula, forgetting to convert degrees to radians, incorrectly calculating the area of the triangle within the segment, and misinterpreting whether they are dealing with a major or minor segment. Careful attention to units is also crucial.
7. How to find the area of a segment when only the radius and height are given?
When only the radius (r) and height (h) of the segment are provided, the calculation is more involved. It requires using the formula: Area = r²cos⁻¹((r-h)/r) - (r-h)√(2rh - h²).
8. What is the relationship between the chord and the central angle of a segment?
The length of the chord is directly related to the central angle (θ) subtended by the arc. A larger central angle will generally result in a longer chord. The relationship is expressed trigonometrically: chord length = 2r sin(θ/2), where r is the radius.
9. Are there any online calculators to help with segment area calculations?
Yes, many online calculators are available. Searching for “circle segment area calculator” will provide several options. These tools can help verify your calculations and save time, especially during exam preparation.
10. What are some real-world applications of calculating segment area?
Calculating segment area is useful in various fields, including engineering (e.g., calculating the cross-sectional area of a pipe), architecture (e.g., designing circular windows or arches), and even in some aspects of landscaping (e.g., calculating the area of a curved flowerbed).
11. How do I determine if a segment is major or minor?
The major segment is the larger portion of the circle divided by the chord, while the minor segment is the smaller area. If the central angle is less than 180 degrees, the segment is minor; otherwise, it's major.
12. What if the chord is a diameter?
If the chord is the diameter, then the segment becomes a semicircle, simplifying the calculation. The area of the semicircle is simply half the area of the entire circle (1/2 πr²).
