Question

For her 16th birthday party Patriot got a birthday cake with a radius of 10 centimeters. her dad cut her a slice of cake with a central angle of 20 degrees.

A. convert the angle measure of Patty's slice to radians.

B. find the area of Patty's slice of cake Leave your answer interms of pie.

C. Find the perimeter of Patty's slice of cake. ​

Answer 1

Part A

To convert degrees to radians, multiply the given value by π/180°.

`\begin{aligned}\implies 20^(\circ) & =20^(\circ) * (\pi)/(180^(\circ)) \: \sf rad\\ & =(1)/(9) \pi \: \sf rad\end{aligned}`

Part B

Given:

• r = 10 cm

• 
  `\theta=(1)/(9) \pi`

`\begin{aligned}\textsf{Area of a sector of a circle} & =\frac12 r^2 \theta\\\\\implies \textsf{Area of slice of cake} & =(1)/(2) \cdot 10^2 \cdot (1)/(9) \pi\\\\& = (50)/(9) \pi \: \sf cm^2\end{aligned}`

Part C

Given:

• r = 10 cm

• 
  `\theta=(1)/(9) \pi`

`\begin{aligned}\textsf{Perimeter} & = 2r + \textsf{arc length}\\& = 2r + r \theta \\& = 2(10)+10\left((1)/(9) \pi \right) \\& = 20 + (10)/(9) \pi \\& = 23.4906585\\& = 23.5\: \sf cm \:(1 \:dp)\end{aligned}`