Question 1: The area of a sector of a circle is given by the formula: (θ/360) × πr^2, where θ is the central angle in degrees. Explain in your own words what this formula means for calculating the sector area.

Answer: The formula calculates the part of the circle's area proportional to the angle θ compared to the full 360°, with πr^2 being the total area of the circle.

Question 1: Calculate the area of a sector with a central angle of 60° in a circle of radius 10 cm.

Answer: 16.67 cm^2

Question 2: Find the fraction of the circle's area represented by a sector with a 90° angle.

Answer: 1/4

Question 3: In a circle with radius 15 cm, what is the sector area when the central angle is 120°?

Answer: 47.12 cm^2

Question 1: A sector with a central angle of 45° is cut from a circle of radius 8 cm. Calculate the sector area and express it as a fraction of the entire circle. Explain each step.

Answer: The sector area is (45/360)×π×8^2 = (1/8)×π×64 = 8π ≈ 25.13 cm^2. The total circle area is π×64 ≈ 201.06 cm^2. Fraction = 25.13/201.06 ≈ 1/8.

Question 2: A sector appears to have a central angle of 150°, but the actual measured angle is 135°. The radius is 12 cm. Identify the mistake in the calculation of the sector area and correct it.

Answer: Incorrect: used 150° in calculation. Correct: use 135°, so area = (135/360)×π×12^2 = (3/8)×π×144 = 54π ≈ 169.65 cm^2.

Question 1: A pizza is cut into 8 equal slices. What fraction of the pizza does each slice represent? If the pizza has a radius of 20 inches, what is the area of one slice?

Answer: Each slice is 1/8 of the pizza. Area of pizza = π×20^2 = 400π ≈ 1256.64 in^2. Area of one slice = (1/8)×400π ≈ 157 in^2.

Question 1: A sector of a circle has an area of 50 cm^2. The radius of the circle is 10 cm. Find the central angle θ of the sector in degrees.

Answer: θ = (50 / (π×10^2)) × 360 = (50 / 100π) × 360 ≈ (1/2π)×360 ≈ 57.3°

Question 2: Construct a sector with a radius of 7 cm and a central angle of 100°. Calculate the sector area and verify your answer with the formula.

Answer: Area = (100/360)×π×7^2 ≈ (5/18)×π×49 ≈ 42.78 cm^2.

Question 1: A student calculates the area of a sector with a 90° angle and radius 10 cm as 100π cm^2. Is this correct? If not, identify the mistake and find the correct area.

Answer: Incorrect: used full circle area instead of sector. Correct area = (90/360)×π×10^2 = (1/4)×π×100 = 25π ≈ 78.54 cm^2.

Question 2: A student claims that the fraction of the circle's area for a 180° sector is 1/2. Is this correct? Explain and justify your answer.

Answer: Yes, since (180/360) = 1/2, the sector is half the circle.