A sector of a circle of radious 10cm and an angle 108° is bent to from a cone find the base radius
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Make two diagrams, one showing the sector in the original circle, and the second showing the cone
It should be clear to you that the arc of the sector becomes the circumference of the base of the cone.
arc/circumference = 108/360
arc/20π = 3/10
arc = 60π/10 = 6π cm
let the radius of the cone base be r
2πr = 6π
r = 3
the cone has a base radius of 3 cm
To find the base radius of the cone, we need to use the given information: the radius of the circle (10 cm) and the central angle of the sector (108°).
The circumference of the circle is given by the formula: C = 2πr, where r is the radius of the circle.
Since the sector is bent to form a cone, the arc length of the sector (s) corresponds to the circumference of the base of the cone. The formula to calculate the arc length is given by: s = (θ/360°) * C, where θ is the central angle of the sector in degrees.
So, we can calculate the arc length of the sector as follows:
s = (108/360) * (2π * 10)
s = (3/10) * (20π)
s = 6π cm
In a cone, the circumference of the base (C') is equal to 2π times the base radius (r'). Therefore, we can write the equation as:
C' = 2πr'
Now, we can equate the arc length of the sector with the circumference of the base of the cone:
s = C'
6π = 2πr'
Simplifying the equation by canceling out π from both sides, we get:
6 = 2r'
Dividing both sides of the equation by 2, we find:
r' = 6/2
r' = 3
Therefore, the base radius of the cone is 3 cm.
To find the base radius of the cone, we can use the relationship between the angle and arc length of the sector.
1. First, we need to find the arc length of the sector. The formula for the arc length of a sector is given by:
arc length = (angle/360°) * 2πr,
where angle is the measure of the central angle in degrees and r is the radius of the circle.
Substituting the known values, we have:
arc length = (108°/360°) * 2π(10cm)
= (3/9) * π(10cm)
= (1/3) * 10π cm
= 10π/3 cm
2. The arc length of the sector represents the circumference of the base circle of the cone. So, we can equate the arc length to 2π times the base radius of the cone.
10π/3 cm = 2π * base radius
3. Simplifying the equation, we can cancel out the π on both sides:
10/3 cm = 2 * base radius
4. Divide both sides by 2:
base radius = (10/3 cm) / 2
= 5/3 cm
≈ 1.67 cm
Therefore, the base radius of the cone is approximately 1.67 cm.