Area of a sector of a circle is the region inside a part of a circle bounded by an arc and two radii. But how exactly do we calculate this area? What formula stands behind this idea? And is it possible to find the sector’s area when the angle is given in both radians and degrees? Let’s look closer and learn how to find the area of a sector using clear and efficient calculation methods.
Formula for the Area of a Sector of a Circle: Step-by-Step Explanation
A sector of a circle (often simply called a sector) is the part of a circle bounded by an arc and the two radii that connect the arc’s endpoints to the center. The arc that bounds the sector is called the sector’s arc. In the figure below, two sectors with arcs ALB and AMB are shown; the first of these sectors is shaded.
Let’s derive the formula for the area of a sector of a circle of radius R, bounded by an arc whose central angle is α degrees. Since the area of the entire circle is π⋅R2, the area of a 1° sector is (π⋅R2)/360. Therefore, the area of a sector with an angle α (in degrees) is:
If α is given in radians, then the angle in degrees equals α⋅180/π. Substituting this into the previous formula gives:
Thus, we obtain formulas that let us compute the area of a sector of a circle whether the angle is expressed in degrees or in radians.
Area of a Sector of a Circle: Practical Problems and Their Solutions
To better understand how to find the area of a sector of a circle, let’s look at a few specific examples. Although each problem includes a final answer, isn’t it more interesting to try solving them on your own before checking the results?
Example 1. If the central angle of a sector is 60°, and the circle’s radius is 7 cm, what is the area of a sector?
According to the conditions, the central angle is α=60° and the radius is R=7 cm. Therefore, using formula (1), we have:
Hence, the area of a sector of a circle is 25.643 cm2.
Example 2. Find the area of a sector if the circle’s radius is 6 cm and the central angle is (2⋅π)/3
In this case, α=(2⋅π)/3 radians and R=6 cm. Therefore, using formula (2), we get:
Thus, the area of a sector of a circle is 37.68 cm2.
Example 3. The side of the square shown in the figure below is 10 cm. Compute the area of the shaded figure RFGH
The area of a square equals the square of its side, so AABCD=AB2=102=100 cm2. Four circular sectors are highlighted inside the square. The radius of each sector equals half of the square’s side, that is, R=AB/2=10/2=5 cm.
Since we have a square, the degree measure α of each sector equals 90°. Therefore, the area of each sector is:
Next, subtracting the areas of the four circular sectors from the area of the square, we determine the area of the shaded figure EFGH: AEFGH=AABCD-4⋅A=100-4⋅19.625=21.5 см2.
See Also: Explore Other Important Aspects of Circle Geometry!
Want to expand your knowledge of geometry? Let’s take a look at a few more engaging details related to studying the circle!
- What Is a Circle: Definition and Components — Learn the basic concepts and elements that define a circle’s structure, and how they influence its characteristics.
- Properties of a Circle in Action: Example Problems with Answers — Deepen your understanding of a circle’s geometric properties through practical tasks and their solutions.
- Area of a Circle: From Definition to Practical Problems — Find out how to determine a circle’s area and apply this measure in different situations to solve practical problems.
Area of a Sector of a Circle: From Flowchart to Code — Build Your Own Sector Area Calculator
If you enjoy programming and like seeing how logic turns into action, this is your next creative challenge! Take the flowchart that outlines the algorithm for calculating the area of a sector of a circle and transform it into working code. Step by step, convert each block of the diagram into real program instructions that take the radius and angle as inputs and return the area as output. You can use any language you prefer — Pascal, Python, or JavaScript — and give your program a touch of personality with clear prompts and friendly messages. It’s a great way to combine mathematical thinking with coding skills and watch geometry come to life on your screen!
