Step 1a

Given,

The circumference of a circle = 31.4 cm

We know that the circumference(c) of a circle = 2πr.

Let the radius of the circle be r.

Explanation: We know that the circle’s circumference formula is \$2 \pi r\$. Here we have to find the radius(r) using the circumference value C = 31.4 cm.

Step 1b

Substitute the values in the formula

31.4 = 2πr

Divide both sides of the equation by 2π

\$(31.4) / (2π) = r\$

\$\cancel(31.4)^10 / (2 \times \cancel(3.14)^1) = r\$

\$\cancel(10)^5 / \cancel2^1 = r\$

r = 5 cm

Explanation: Here, we substitute the 'C' value in the formula. To find the circle’s radius, we substitute the π = 3.14. On cancellation, we get the circle’s radius is equal to 5cm.

Step 2a

The given area of a circle is 256 square units

We know that, the area of a circle is

\$"A" = π"r"^2\$

Rearrange the formula to solve for the radius:

\$"r" = sqrt("A"/π)\$

Explanation: We know that the area of the circle formula is \$"A" = π"r"^2\$. Here we have to find the radius(r) \$"r" = sqrt("A"/π)\$.

Step 2b

Substitute the given area into the formula:

\$"r" = sqrt(256/π)\$

\$"r" = sqrt(256/3.14159)\$

\$"r" = sqrt(81.4081)\$

r = 9.02

d = 2r

\$"d" = 2 times 9.02\$

d = 18.04

Explanation: Here, we substitute the π = 3.14 value in the radius formula. we get radius as r = 9.02, we get the diameter d =18.04.

Step 3a

The given arc length is 6 cm and the central angle is 40°

To find the radius of the circle, we can use the formula relating the arc length,

central angle, and radius:

arc length = radius \$times\$ central angle

Explanation: Here we have to find the radius(r) of the circle by using the arc length and central angle
arc length = radius \$times\$ central angle

Step 3b

Substituting these values into the formula, we have: 6 cm = r × 40°

The angle to be in radians rather than°. Since 1 radian is equal to

180° / π, we can convert the angle to radians.

⇒ \$6 "cm" = "r" \times ((40°) times (π / 180°))\$

⇒ \$6 = "r" \times ((2π) / 9) \$ radians

⇒ \$6 / ((2π) / 9) = "r"\$ radians

⇒ \$"r" = (54 "cm") / π\$ radians

⇒ \$"r" = (54 "cm") / 3.14\$ radians

Thus, the radius of the circle is 17.1828 cm.

Explanation: Here, we substitute the given values in the formula. To find the circle’s radius, we substitute the π = 3.14. On solving, we get the radius of the circle is 17.1828cm.