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Please please help!! I'm not sure if I have the right answer!!

In a circle with center O, central angle AOB has a measure of (5pi)/4 radians. The area of the sector formed by central angle AOB is what fraction of the circle?

Please please help Im not sure if I have the right answer In a circle with center O central angle AOB has a measure of 5pi4 radians The area of the sector forme class=

Respuesta :

Answer: The area of the sector formed by central angle AOB is 5/8 th of the total area of the circle.

Step-by-step explanation:

Let r be the radius of the circle having the center O,

⇒ The area of the circle = [tex]\pi r^2[/tex]  square unit.

And, the central angle AOB = [tex]\frac{5\pi}{4}\text{ radian}[/tex]

[tex]=(\frac{5\pi}{4}\times \frac{180}{\pi})^{\circ}[/tex]  ( since, [tex]\pi[/tex] = 180° )

[tex]=(\frac{900}{4})^{\circ}[/tex]

[tex]=225^{\circ}[/tex]

Hence, the area of sector AOB

[tex]=\frac{225^{\circ}}{360^{\circ}}\times \pi r^2[/tex]

[tex]=\frac{5}{8}\times \pi r^2[/tex] square unit.

Now,

[tex]\frac{\text{Area of sector AOB}}{\text{Area of circle}}=\frac{\frac{5}{8}\times \pi r^2}{\pi r^2}[/tex]

[tex]=\frac{5}{8}[/tex]

⇒ Area of sector AOB = 5/8 × Area of the circle.

Hence, the area of the sector formed by central angle AOB is 5/8 th of the total area of the circle.

johanrusli

The area of the sector formed by central angle AOB is 5/8 of the area of the circle

Further explanation

The basic formula that need to be recalled is:

Circular Area = π × R²

Circle Circumference = 2 × π × R

where:

R = radius of circle

The area of sector:

[tex]\text{Area of Sector} = \frac{\text{Central Angle}}{2 \pi} \times \text{Area of Circle}[/tex]

The length of arc:

[tex]\text{Length of Arc} = \frac{\text{Central Angle}}{2 \pi} \times \text{Circumference of Circle}[/tex]

Let us now tackle the problem!

This problem is about finding the area of the sector.

We can find a comparison of the area of the sector with the area of a circle with the following formula.

[tex]\text{Area of Sector} = \frac{\text{Central Angle}}{2 \pi} \times \text{Area of Circle}[/tex]

[tex]\frac{\text{Area of Sector}}{\text{Area of Circle}} = \frac{\text{Central Angle}}{2 \pi}[/tex]

[tex]\frac{\text{Area of Sector}}{\text{Area of Circle}} = \frac{5 \pi / 4}{2 \pi}[/tex]

[tex]\frac{\text{Area of Sector}}{\text{Area of Circle}} = \frac{5 / 4}{2}[/tex]

[tex]\frac{\text{Area of Sector}}{\text{Area of Circle}} = \large {\boxed {\frac{5}{8}} }[/tex]

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Answer details

Grade: College

Subject: Mathematics

Chapter: Trigonometry

Keywords: Sine , Cosine , Tangent , Opposite , Adjacent , Hypotenuse, Circle , Arc , Sector , Area

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