Eight-year-old Annie wants to go to a children’s carnival in a white lady costume, which will include a white cone-shaped hat. The parents took the opportunity to practice their geometric imagination with Anna and decided to make the hat with her instead of buying it.
Exercise. Anna and her mother found out with a tape measure that the circumference of Anna’s head is 52 cm. Together they further agreed that the hat would be 30 cm high. How will they create the hat?
Solution. The cap is formed by the lateral surface/face of a cone, where we know the perimeter of the base \(o\) (52 cm) and the height of the cone \(v\) (30 cm). The unrolled lateral face of the cone is then a circular sector with an unknown radius \(s\) (the size of the side of the cone) and an unknown central angle \(\varphi\). We need to calculate these data. Furthermore, we know that the arc length of a circular sector is equal to the perimeter of \(o\).
First, we calculate the radius of the base \(r\) from the perimeter of the base and then the length of the side \(s\) using the Pythagorean theorem.
\[ r = \frac{o}{2\pi} = \frac{52}{2\pi} \doteq 8{,}28\,\text{cm} \]
\[ s = \sqrt{v^2 + r^2} = \sqrt{30^2 + 8{,}28^2} \doteq 31{,}12\,\text{cm} \]
Now we determine the angle \(\varphi\). First, we calculate the perimeter \(O\) of the whole circle of radius \(s\), we get
\[ O = 2\pi s \doteq 195{,}53 \,\text{cm}. \]
Next, we use the direct proportionality between the arc length of this circle and the corresponding central angle to calculate the angle \(\varphi\):
\[ \varphi = \frac{o}{O}\cdot 360^{\circ} = \frac{52}{195{,}53}\cdot 360^{\circ} \doteq 95^{\circ}44'. \]
Annie and her parents create a hat out of a circular sector with an approximate radius of 31 cm and a central angle of approximately \(96^{\circ}\).