Learn the formulas for the area and arc length of circles and sectors, as well as how to solve problems involving composite figures.
Formulas for Sectors
Arc length = 2π r × (central angle/360)
Area = π r² × (central angle/360)
Example 1
Find the area and arc length of a sector with a radius of 6 cm and a central angle of 90°. (π＝3.14)
Solution:
Area: 3.14×6×6×(90÷360)＝3.14×36×0.25＝28.26cm²
Arc length: 2×3.14×6×(90÷360)＝2×3.14×6×0.25＝9.42cm
Find the area and arc length of a sector with a radius of 6 cm and a central angle of 90°. (π＝3.14)
Solution:
Area: 3.14×6×6×(90÷360)＝3.14×36×0.25＝28.26cm²
Arc length: 2×3.14×6×(90÷360)＝2×3.14×6×0.25＝9.42cm
Composite Figures
For complex figures, think in terms of “adding” or “subtracting.”
Example 2 (Semicircle and Triangle)
A circle (with a diameter of 10 cm) is inscribed in a square with sides of 10 cm. Find the area of the part left after subtracting the circle from the square. (π＝3.14)
Solution:
Area of the square: 10×10＝100cm²
Area of the circle: 3.14×5×5＝78.5cm²
Remaining area: 100－78.5＝21.5cm²
A circle (with a diameter of 10 cm) is inscribed in a square with sides of 10 cm. Find the area of the part left after subtracting the circle from the square. (π＝3.14)
Solution:
Area of the square: 10×10＝100cm²
Area of the circle: 3.14×5×5＝78.5cm²
Remaining area: 100－78.5＝21.5cm²
Key Points
- Sector: multiply the area or circumference of the whole circle by “central angle/360”
- Composite figures have three patterns: “subtract,” “add,” and “rearrange”
- Area-preserving transformations: use transformations that change the shape without changing the area
Practice Problems
- Find the area of a sector with a radius of 4 cm and a central angle of 120°. (π＝3.14)
- Find the area of a figure formed by cutting out quarter-circles with a radius of 1 cm from the four corners of a rectangle measuring 6 cm vertically and 8 cm horizontally. (π＝3.14)
Answers and Explanations
- Answer: Approximately 16.75cm²
Explanation: Area of a sector＝πr²×（central angle÷360）＝3.14×4×4×（120÷360）＝3.14×16×(1/3)＝50.24÷3≈16.75cm²（Exactly, 3.14×16/3≒16.75cm²）. - Answer: 44.86cm²
Explanation: Area of the rectangle: 6×8＝48cm². The total area of the four quarter-circles＝one circle (radius 1 cm): 3.14×1×1＝3.14cm². Remaining area: 48－3.14＝44.86cm².