Question

drag each tile to the correct box. consider the given circles. order the sectors from least to greatest according to their areas, in square units. sector a (169°, 4 units), sector b (82°, 9 units), sector c (117°, 6 units).

Explanation:

The formula for the area of a sector of a circle is \( A = \frac{\theta}{360^\circ} \times \pi r^2 \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle. We will calculate the area of each sector (Sector A, Sector B, and Sector C) using this formula and then compare the areas.

Step 1: Calculate the area of Sector A

- Radius of Sector A's circle, \( r_A = 4 \) units
- Central angle of Sector A, \( \theta_A = 169^\circ \)

Using the sector area formula:
\[ A_A = \frac{169^\circ}{360^\circ} \times \pi \times (4)^2 \]
\[ A_A = \frac{169}{360} \times \pi \times 16 \]
\[ A_A = \frac{169 \times 16}{360} \pi \]
\[ A_A = \frac{2704}{360} \pi \approx 7.511\pi \approx 23.59 \]

Step 2: Calculate the area of Sector B

- Radius of Sector B's circle, \( r_B = 9 \) units
- Central angle of Sector B, \( \theta_B = 82^\circ \)

Using the sector area formula:
\[ A_B = \frac{82^\circ}{360^\circ} \times \pi \times (9)^2 \]
\[ A_B = \frac{82}{360} \times \pi \times 81 \]
\[ A_B = \frac{82 \times 81}{360} \pi \]
\[ A_B = \frac{6642}{360} \pi \approx 18.45\pi \approx 57.97 \]

Wait, that can't be right. Wait, no, let's recalculate:

Wait, \( 82 \times 81 = 6642 \), and \( 6642 \div 360 = 18.45 \). Wait, but let's check Sector C.

Step 3: Calculate the area of Sector C

- Radius of Sector C's circle, \( r_C = 6 \) units
- Central angle of Sector C, \( \theta_C = 117^\circ \)

Using the sector area formula:
\[ A_C = \frac{117^\circ}{360^\circ} \times \pi \times (6)^2 \]
\[ A_C = \frac{117}{360} \times \pi \times 36 \]
\[ A_C = \frac{117 \times 36}{360} \pi \]
\[ A_C = \frac{4212}{360} \pi = 11.7\pi \approx 36.76 \]

Wait, now let's recheck Sector B:

\[ A_B = \frac{82}{360} \times \pi \times 81 = \frac{82 \times 81}{360} \pi = \frac{6642}{360} \pi \approx 18.45\pi \approx 57.97 \]

Wait, but Sector A was approximately \( 7.511\pi \approx 23.59 \), Sector C is \( 11.7\pi \approx 36.76 \), and Sector B is \( 18.45\pi \approx 57.97 \). Wait, that would mean the order from least to greatest is Sector A, Sector C, Sector B? Wait, no, wait, my calculation for Sector A must be wrong. Wait, \( 4^2 = 16 \), \( 169 \times 16 = 2704 \), \( 2704 \div 360 \approx 7.511 \), so \( 7.511\pi \approx 23.59 \). Sector C: \( 6^2 = 36 \), \( 117 \times 36 = 4212 \), \( 4212 \div 360 = 11.7 \), so \( 11.7\pi \approx 36.76 \). Sector B: \( 9^2 = 81 \), \( 82 \times 81 = 6642 \), \( 6642 \div 360 = 18.45 \), so \( 18.45\pi \approx 57.97 \). Wait, but that would mean Sector A (≈23.59) < Sector C (≈36.76) < Sector B (≈57.97). But let's check the calculations again.

Wait, maybe I made a mistake in the angle for Sector B? Wait, the diagram shows Sector B with 82 degrees? Wait, the original problem: Sector B has a central angle of 82 degrees? Wait, the user's image: "Sector B" with 82°? Wait, maybe I misread the angle. Wait, let me check again.

Wait, the user's image: Sector A: 169°, radius 4. Sector B: 82°, radius 9. Sector C: 117°, radius 6.

So recalculating:

Sector A:
\[ A_A = \frac{169}{360} \times \pi \times 4^2 = \frac{169}{360} \times 16\pi = \frac{2704}{360}\pi \approx 7.511\pi \approx 23.59 \]

Sector C:
\[ A_C = \frac{117}{360} \times \pi \times 6^2 = \frac{117}{360} \times 36\pi = \frac{4212}{360}\pi = 11.7\pi \approx 36.76 \]

Sector B:
\[ A_B = \frac{82}{360} \times \pi \times 9^2 = \frac{82}{360} \times 81\pi = \frac{6642}{360}\pi \approx 18.45\pi \approx 57.97 \]

Wait, but that would mean the order from least to greatest is Sector A, Sector C, Sector B? But that seems odd. Wait, maybe the angle for Sector B is 82 degrees? Wait, maybe I misread the angle. Wait, maybe the angle for Sector B is 82 degrees, but let's check the calculation again.

Wait, 82/360 * 81π = (82*81)/360 π = (6642)/360 π ≈ 18.45π ≈ 57.97. Sector C: 117/360 * 36π = 11.7π ≈ 36.76. Sector A: ~7.51π ≈23.59. So yes, Sector A (≈23.59) < Sector C (≈36.76) < Sector B (≈57.97). Wait, but that would mean the order from least to greatest is Sector A, Sector C, Sector B. But let's confirm with exact fractions.

Alternatively, we can compare the areas without calculating π, since π is a common factor. So we can compare \( \frac{\theta r^2}{360} \) for each sector.

For Sector A: \( \frac{169 \times 4^2}{360} = \frac{169 \times 16}{360} = \frac{2704}{360} \approx 7.511 \)

For Sector C: \( \frac{117 \times 6^2}{360} = \frac{117 \times 36}{360} = \frac{4212}{360} = 11.7 \)

For Sector B: \( \frac{82 \times 9^2}{360} = \frac{82 \times 81}{360} = \frac{6642}{360} \approx 18.45 \)

So comparing the values: 7.511 < 11.7 < 18.45. Therefore, the areas are in the order: Sector A < Sector C < Sector B.

Answer:

Sector A < Sector C < Sector B