Checkpoint G.PC.3.a

Answer explanation

The area of a sector is given by the formula A = (πr²θ)/360, where θ is in degrees. This formula derives from the proportion of the circle represented by the sector, making the correct choice A = (πr²θ)/360.

Tags

CCSS.HSG.C.B.5

A sector of a circle subtends a 60 degree central angle. What proportion of the entire circumference does the arc length of this sector represent?

• 1/2

• 1/6

• 1/4

• 1/3

Answer explanation

The arc length of a sector is proportional to its central angle. A full circle is 360 degrees, so the proportion is 60/360 = 1/6. Thus, the arc length represents 1/6 of the entire circumference.

Answer explanation

The proportion \( \frac{Area\ of\ Sector}{\pi r^2} \) represents the fraction of the circle's area, and \( \frac{Arc\ Measure}{2\pi r} \) relates the arc length to the circle's circumference, both equal to \( \frac{central\ \theta}{360\degree} \).

Tags

CCSS.HSG.C.B.5

A circular track is divided into 8 equal arcs. What fraction of the total circumference does each arc represent?

• 3/8

• 1/2

• 1/8

• 1/4

Answer explanation

The circular track is divided into 8 equal arcs. Each arc represents 1 out of the 8 parts of the total circumference, which is 1/8. Therefore, the correct answer is 1/8.

Tags

CCSS.3.NF.A.1

CCSS.3.NF.A.2A

• 1/4

• 3/8

• 2/3

• 5/12

Answer explanation

To find the fraction of the circumference, divide the angle of the arc by the total angle of the circle: 150° / 360° = 5/12. Thus, the arc represents 5/12 of the total circumference.

Tags

CCSS.4.MD.C.5B

CCSS.HSG.CO.A.1

• 5π/6

• π/2

• 3π/4

• 2π/3

Answer explanation

The arc length is calculated as (angle in radians) × (radius). Convert 150° to radians: 150° × (π/180) = 5π/6. Thus, the arc length as a proportion of the radius is 5π/6, which is the correct answer.

Tags

CCSS.HSG.C.B.5

• (15π)/16

• (3π)/4

• (7π)/16

• (5π)/8

Answer explanation

The total circumference of the track is 2πr = 100π meters. The arc length is (3/8) of the circumference: (3/8) * 100π = 37.5π meters. As a proportion of the radius (50 meters), this is (37.5π)/50 = (15π)/16.

Tags

CCSS.HSG.C.B.5

• 1/4

• 1/3

• 1/2

• 2/3

Answer explanation

A full circle has a central angle of 360 degrees. The sector with a 120-degree angle covers 120/360 = 1/3 of the circle's area. Therefore, the correct proportion is 1/3.

Tags

CCSS.4.MD.C.5B

CCSS.HSG.CO.A.1

• 1/2

• 1/8

• 3/4

• 1/4

Answer explanation

The total area of the circle is \(\pi r^2 = \pi (50^2) = 2500\pi\) square meters. The sector covering \(\frac{1}{4}\) of the circumference corresponds to \(\frac{1}{4}\) of the area, which is \(\frac{2500\pi}{4} = 625\pi\). Thus, the fraction is \(\frac{625\pi}{2500\pi} = \frac{1}{4}\).

Tags

CCSS.HSG.C.B.5

What happens to sector area if the radius is doubled?

• The sector area remains the same.

• The sector area doubles.

• The sector area decreases by half.

• The sector area quadruples.

Answer explanation

The area of a sector is given by the formula A = 1/2 * r^2 * θ. If the radius (r) is doubled, the new area becomes A' = 1/2 * (2r)^2 * θ = 2r^2 * θ, which is four times the original area. Thus, the sector area quadruples.

Tags

CCSS.HSG.C.B.5