Geometry: Area of Semicircles and Sectors

Area of Semicircles and Sectors

• The area of a semicircle is calculated as half the area of a full circle.

Arc Length Calculations

• For a circle with an 8-inch radius, the length of a 40° arc is 16 π/9 inches.
• In a circle where a 30° arc is measured as 2 π inches long, the radius is determined to be 12 inches.
• An arc measuring 4 π ft in a circle with a radius of 10 ft corresponds to a degree measure of 72°.
• A bicycle wheel with a radius of 26 inches rotates through an 80° arc, resulting in an arc length of 11.56 π inches.

Arc Measure Derivations

• Given an arc length of 12 π inches and a circle radius of 10 inches, the degree measure of the arc is found to be 216 degrees.
• An arc measuring 30° accounts for 1/12 of the total circumference of the circle.

Area of Sectors

• The area of a sector with a radius of 10 inches and an arc measure of 45° is 12.5 π square inches.
• For a sector with a 90° arc and a radius of 1 foot, the area is calculated as 0.25 π square inches.
• A sector with a radius of 8 and an arc measure of 300° has an area of 53.3 π square inches.

Sector Area and Arc Measures

• In a circle where the radius is 10 cm, a sector area of 40 π square centimeters corresponds to a degree measure of 144°.
• A sector area of 7 π square feet with an arc measure of 70° indicates a radius of 6 feet.

Practical Applications

• For a 10" diameter pumpkin pie cut into six equal servings, the area of each piece is approximately 4.17 π square inches.
• An equilateral triangle inscribed in a circle with sides of 2√3 yields a sector area of 1.33 π square inches for one of its sectors.
• A square inscribed in a circle with sides of 3√2 results in a sector area of 2.25 π square inches.