A circular sector of radius 135 and angle 120 degrees is folded into a right circular cone. What is the height of this cone?
Let’s begin by sketching what is happening in this situation. We are told firstly that we have a circular sector with a radius of 135 and an angle of 120 degrees. Notice that we aren’t given any units for the radius, but we know that this will be a length unit. We are told that this circular sector is then folded into a right circular cone, which might look something like this.
In the question, we need to work out the height of this cone. As we are given that this is a right circular cone, then that means that the vertex of the cone lies above the center of the base, and the height will be the perpendicular height. Let’s define the height of the cone to be ℎ and observe that we can create a right triangle within the cone. Now, we could work out the value of ℎ if we knew both of the lengths of the other two sides in this right triangle. Let’s look firstly at this lateral edge on the cone. In fact, we already know the length of this edge length. We remember that when we folded up this circular sector to create the cone, then the edge length will be the same as the radius of the circular sector. So that will also be 135 length units.
Next, we know that this third side at the base of the right triangle is a line segment formed from the center of the circle to the outside edge. In fact, it would be a radius of this circle at the base of this cone. Although we aren’t given this radius, we can work it out. Remember that it’s the outside edge of the circular sector which forms the circumference at the base of the cone. And therefore, the circumference of this circle at the base of the cone is the same length as the arc length of the circular sector.
Recall that if we have a circular sector with an angle of 𝜃 degrees, then the arc length of that sector is found by 𝜃 over 360 degrees multiplied by two 𝜋𝑟. The value of two 𝜋𝑟 is the circumference of the whole circle, so we’re really taking a fraction of the circumference of the entire circle. We know that in the circular sector, the angle 𝜃 is 120 degrees and the radius is 135. Therefore, the arc length of the sector can be given as 120 degrees over 360 degrees multiplied by two times 𝜋 times 135. We can then simplify this to two-thirds 𝜋 times 135. And this in turn can be given as 90𝜋. And those would be in the same length units as the radius.
Now we’ve worked out that the arc length is 90𝜋, then this means that the circumference of the circle at the base of the cone is also 90𝜋 length units. We can recall that the circumference and radius are related in the formula that the circumference 𝐶 is equal to two 𝜋𝑟. Note that this radius value 𝑟 is not the same as the radius value in the previous formula. In this circle at the base of the cone, we know that the circumference is 90𝜋, so we have 90𝜋 is equal to two 𝜋𝑟.
We can take out a common factor of 𝜋 from both sides, giving us 90 is equal to two 𝑟. Dividing through by two then gives us that 𝑟 is equal to 45. As the radius of this circle is 45 length units, then we can note that we now have enough information to find the third side in this right triangle created in the cone. We can do this by applying the Pythagorean theorem. This theorem states that in any right triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides. In our triangle then, the two shorter sides of ℎ and 45 can be defined as the lengths 𝑎 and 𝑏, and the hypotenuse is 135. So we define this as 𝐶.
Substituting these into the Pythagorean theorem gives ℎ squared plus 45 squared equals 135 squared. Evaluating the squares gives ℎ squared plus 2025 equals 18225. We can then subtract 2025 from both sides, giving us that ℎ squared is equal to 16200. Taking the square root of both sides gives us that ℎ is equal to the square root of 16200. At this point, it can be useful to check if we need to give our answer as a decimal. And as we’re not told to give the answer to the nearest hundredth or tenth, for example, then let’s see if we can simplify this square root.
16200 can be written as the product of 8100 and two, where 8100 is a square number. Simplifying the square root of 8100 multiplied by the square root of two gives us 90 multiplied by the square root of two. Therefore, we have worked out the perpendicular height ℎ of this cone. So we can give the answer that the height of the cone is 90 root two. If we needed to give a unit, we could give this as length units.