A circular garden plot has a radius of 10\text{ meters}. A sector of the garden with a cen… — ACT

A circular garden plot has a radius of $10\text{ meters}$ . A sector of the garden with a central angle of $72^\circ$ is planted with roses. What is the area, in square meters, of the sector planted with roses?

A
$4\pi$
B
$40\pi$
$20\pi$ Answer
D
$72\pi$
E
$100\pi$

Answer

The correct answer is

20\pi

, representing the area of the sector in square meters.

The area of a sector of a circle is calculated using the formula

\text{Area} = \frac{\theta}{360^\circ} \pi r^2

. Substituting

\theta = 72^\circ

and

r = 10\text{ meters}

gives

\frac{72^\circ}{360^\circ} \pi (10)^2 = \frac{1}{5} \cdot 100\pi = 20\pi

Step-by-Step Solution

Identify the formula for the area of a sector.

\text{Area} = \frac{\theta}{360^\circ} \pi r^2

The area of a sector is a proportional fraction of the total area of the circle.

Substitute the given values into the formula.

\text{Area} = \frac{72^\circ}{360^\circ} \pi (10)^2

The central angle is

72^\circ

and the radius is

10\text{ meters}

Simplify the fraction and the squared term.

\text{Area} = \frac{1}{5} \pi (100)

72/360

simplifies to

1/5

and

10^2 = 100

Perform the final multiplication.

20\pi

One-fifth of

100

20

Key Concept

Calculating the area of a circle sector given the radius and the central angle in degrees.

Estimated Time:1m 0s

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