Podcast
Questions and Answers
How is the area of a semicircle found?
Take half the area of the whole circle.
Find the length of an arc of 40° in a circle with an 8 inch radius.
16 π/9 inches
What is the radius of a circle in which a 30° arc is 2 π inches long?
12 inches
What is the degree measure of an arc 4 π ft long in a circle of radius 10 ft?
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A bicycle wheel with radius 26" rotates through an arc that measures 80°. What is the length of the arc of the tire that touched the ground?
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If the length of an arc is 12 π inches and the radius of the circle is 10 inches, what is the measure of the arc?
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What part of the circumference is an arc whose measure is 30°?
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What is the area of a sector with radius 10" and measure of arc equal to 45°?
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What is the area of a sector with measure of arc equal to 90° and radius equal to 1 foot?
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What is the area of a sector with radius equal to 8 and measure of arc equal to 300°?
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In a circle of radius 10 cm, a sector has an area of 40 π sq.centimeters. What is the degree measure of the arc of the sector?
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What is the radius of a circle with a sector area of 7 π sq.ft and an arc whose measure is 70°?
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A 10" diameter pumpkin pie is cut into six equal servings. What is the area of the top of each piece of pie?
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An equilateral triangle with a side of 2√3 is inscribed in a circle. What is the area of one of the sectors formed by the radii to the vertices of the triangle?
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A square with sides of 3√2 is inscribed in a circle. What is the area of one of the sectors formed by the radii to the vertices of the square?
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Study Notes
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.
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Description
This quiz covers the concepts of area calculations for semicircles and sectors, as well as arc length computations. Test your understanding of these geometric principles through a series of problems involving circles' measurements and properties. Challenge your skills with real-world applications related to arcs and sectors.
