A sector of a circle of radius 8cm is bent to form a cone. Find the radius of the cone and it's vertical angle if the angle substended at the centre by the sector is 280°
If you make a sketch you will see that the arc length of the
sector becomes the circumference of the the base of the cone
For the sector, arc = rØ , where Ø must be in radians.
280° = 280π/180 rads = 14π/9 rad
Then for the cone's circular base,
2πr = 14π/9
r = 7/9 cm
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To find the radius of the cone, we need to use the formula for the circumference of a circle:
C = 2πr
Since the sector of the circle is bent to form a cone, the circumference of the sector will be equal to the circumference of the base of the cone. In this case, the circumference of the sector is equal to the circumference of a circle with radius 8cm.
C = 2π(8)
C = 16π cm
We can now use the formula for the circumference of a circle to find the radius of the cone:
C = 2πr
16π = 2πr
Dividing both sides of the equation by 2π:
r = 8 cm
Therefore, the radius of the cone is 8cm.
To find the vertical angle of the cone, we can use the angle subtended at the center by the sector, which is 280°. The angle subtended at the center by the sector is equal to the angle subtended at the apex of the cone, as both the angle and the sector are bent together.
Since the angle at the apex is equal to twice the central angle, we can find the vertical angle of the cone by dividing the central angle by 2:
Vertical angle = 280° / 2
Vertical angle = 140°
Therefore, the radius of the cone is 8 cm, and its vertical angle is 140°.
To find the radius of the cone, we can use the formula for the circumference of a circle, which is given by:
C = 2πr
where C is the circumference and r is the radius.
Since the sector of the circle is bent to form the base of the cone, the circumference of the circle is equal to the circumference of the base of the cone. The angle subtended at the center by the sector is 280°, which means the circumference of the circle is 280/360 = 7/9 times the circumference of the entire circle.
Let's denote the radius of the cone as R. Since the radius of the circle is 8 cm, we can set up the following equation:
(7/9)(2π(8)) = 2πR
Simplifying the equation gives:
112π/9 = 2πR
Now, we can cancel out the π terms:
112/9 = 2R
Dividing both sides by 2 gives:
56/9 = R
Therefore, the radius of the cone is 56/9 cm.
To find the vertical angle of the cone, we can use the formula for the central angle of the cone's sector, which is equal to the vertical angle of the cone. The central angle is equal to the angle subtended at the center by the sector.
Therefore, the vertical angle of the cone is 280°.
In summary, the radius of the cone is 56/9 cm and its vertical angle is 280°.