To find the radius of the base of the cone and its vertical angle formed when a sector of a circle is bent, we start from the details provided. We have a sector of a circle with a central angle of \(216^\circ\) and a radius of \(5 , \text{cm}\).
Step 1: Calculate the arc length of the sector
The arc length \(L\) of the sector is calculated using the formula:
\[ L = \frac{\theta}{360^\circ} \times 2\pi r \]
where:
- \(\theta = 216^\circ\) is the angle of the sector,
- \(r = 5 , \text{cm}\) is the radius of the circle.
Substituting the values:
\[ L = \frac{216}{360} \times 2\pi \times 5 \]
Calculating the fraction first:
\[ \frac{216}{360} = \frac{3}{5} \]
Now substituting this back into the arc length formula:
\[ L = \frac{3}{5} \times 10\pi = 6\pi , \text{cm} \]
Step 2: Determine the radius of the base of the cone
When the sector is bent to form a cone, the arc length \(L\) becomes the circumference \(C\) of the base of the cone. The circumference of the base of the cone is given by:
\[ C = 2\pi R \]
where \(R\) is the radius of the base of the cone.
Setting the two expressions for circumference equal gives:
\[ 6\pi = 2\pi R \]
Dividing both sides by \(2\pi\):
\[ R = \frac{6\pi}{2\pi} = 3 , \text{cm} \]
Step 3: Calculate the vertical angle of the cone
To find the vertical angle \(\alpha\) of the cone, we use the relationship between the slant height \(l\) (which is the radius of the original circle) and the radius \(R\) of the base.
Using the right triangle formed by the radius of the circle, the radius of the base, and the height of the cone, we know that:
\[ l = 5 , \text{cm} \quad \text{and} \quad R = 3 , \text{cm} \]
Using the Pythagorean theorem, let \(h\) be the height of the cone:
\[ l^2 = R^2 + h^2 \] \[ 5^2 = 3^2 + h^2 \] \[ 25 = 9 + h^2 \] \[ h^2 = 16 \implies h = 4 , \text{cm} \]
Now, the vertical angle \(\alpha\) can be found using the tangent function:
\[ \tan\left(\frac{\alpha}{2}\right) = \frac{R}{h} = \frac{3}{4} \]
To find \(\frac{\alpha}{2}\):
\[ \frac{\alpha}{2} = \tan^{-1}\left(\frac{3}{4}\right) \]
Calculating \(\frac{\alpha}{2}\):
\[ \frac{\alpha}{2} \approx 36.87^\circ \quad \text{(approximately)} \]
Thus, the total angle is:
\[ \alpha \approx 2 \times 36.87^\circ \approx 73.74^\circ \]
Final Answers
- The radius of the base of the cone is \(3 , \text{cm}\).
- The vertical angle of the cone is approximately \(73.74^\circ\).
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