An arc AB of length 5cm is marked an a circle of radius 3cm. Find the area of the sector bounded by this arc and the radii from A and B.
θ = 5/3, so
a = 1/2 r^2 θ = 1/2 * 3^2 * 5/3 = ?
Well, let's see here. To find the area of the sector, we need to find the central angle first. Since the arc AB has a length of 5cm, and the radius of the circle is 3cm, we can use the formula:
arc length = radius * central angle
So, 5cm = 3cm * central angle. Solving for the central angle, we divide both sides by 3cm, giving us:
central angle = 5cm / 3cm
Now, let's convert this angle to radians, because circles love radians. Since there are 2π radians in a complete circle, we can set up the following proportion:
2π radians = 360 degrees
x radians = central angle in degrees
Cross-multiplying, we get:
2π * x = 360 degrees
Solving for x, we divide both sides by 2π, giving us:
x = 360 degrees / 2π
Now that we have the central angle in radians, we can finally find the area of the sector. The formula for the area of a sector is:
area = (central angle / 2π) * π * r^2
In this case, the radius is 3cm and the central angle is 360 degrees / 2π. Plugging in the values, we get:
area = (360 degrees / 2π) * π * 3cm^2
But wait a second! We got degrees creeping in there again. Silly degrees! Since we need to work with radians, let's convert 360 degrees to radians. We know that 180 degrees is equal to π radians, so:
360 degrees = 2π radians
So now, the formula becomes:
area = (2π / 2π) * π * 3cm^2
Canceling out the π and the 2π, we are left with:
area = 3cm^2
Hooray! The area of the sector bounded by the arc AB and the radii from A and B is simply 3cm^2.
To find the area of the sector bounded by the arc AB and the radii from A and B, we can use the formula:
Area of sector = (θ/360) * π * r^2
Where:
- θ is the central angle (in degrees) formed by the arc AB at the center of the circle
- r is the radius of the circle
First, let's find the central angle θ:
We know that the length of the arc AB is 5 cm, and the radius of the circle is 3 cm. The circumference (C) of a circle is given by the formula C = 2πr.
We can use the formula:
Arc length = (θ/360) * C
Substituting the values, we have:
5 = (θ/360) * (2π * 3)
5 = (θ/360) * (6π)
5 = (θ/60) * π
θ/60 = 5/π
θ ≈ (5/π) * 60
Now that we have the value of θ, we can substitute it into the formula for the area of the sector:
Area of sector = (θ/360) * π * r^2
Area of sector = ((5/π) * 60/360) * π * (3^2)
Area of sector = (1/6) * π * 9
Area of sector ≈ (π/2) cm^2
Therefore, the area of the sector bounded by the arc AB and the radii from A and B is approximately (π/2) cm^2.
To find the area of the sector, we need to know the central angle of the sector.
The central angle can be found by using the formula:
θ = (Arc Length / Radius)
In this case, the arc length is given as 5 cm and the radius is given as 3 cm.
Plugging these values into the formula, we have:
θ = (5 cm / 3 cm)
Therefore, the central angle is approximately 1.67 radians.
The area of the sector can be calculated using the formula:
Area = (θ/2) * r^2
Where θ is the central angle and r is the radius of the circle.
Plugging in the values, we get:
Area = (1.67 radians / 2) * (3 cm)^2
= (1.67/2) * 9 cm^2
= 8.01 cm^2
Therefore, the area of the sector bounded by the arc AB and the radii from A and B is approximately 8.01 cm^2.