Video Transcript
A sheet of paper in the shape of a sector of radius 29 centimeters and area 203𝜋 square centimeters is folded into a right cone by gluing together the radii of line segment 𝐴𝐵 and line segment 𝐴𝐶. What is the height of the cone? Recall that the sector area is given by half the product of its radius and the length of its arc.
We are very helpfully given a diagram here, and notice that it’s the curved surface of the cone which has an area of 203𝜋 square centimeters. We are asked here to find the height of the cone. The height of the cone can be seen by this line here in orange, and we know that it’s going to go to the center of the circle at the base of the cone because we’re told that this is a right cone. Let’s define the height of the cone to be ℎ centimeters.
It might be very tempting to think that because the radius of this sector is 29 centimeters, then the height of the cone must also be 29 centimeters. But in fact, it’s going to be the slant height of the cone that’s 29 centimeters. So in order to work out the height ℎ centimeters, we can create this triangle within the cone. This triangle will be a right triangle because the height is the perpendicular height. So because we have a right triangle and we have one length that we know and one length that we want to find out, then we need to work out this third length in the triangle if we want to then apply the Pythagorean theorem.
So what is this length? Well, we know that it will be the radius of this circle at the base of the cone. We can define it with the letter 𝑟. So before we can apply the Pythagorean theorem, we’ll need to work out this value, the radius of this circle at the base of the cone. Notice that this circle is different to the circle that created this sector. It has a smaller radius that will not be equal to 29 centimeters. We could determine the radius of this circle if we knew the circumference of the circle, that’s the distance around the outside edge.
And the length of the circumference is going to be the same as the length of the arc in this sector. That’s because remember that to create this cone, we rolled up or folded this sector to create the cone. So if we know the arc length of this sector, that will be the same as the circumference of the smaller circle. We are very helpfully reminded that the sector area is given by half the product of its radius and the length of its arc. We can use the letter 𝐿 to represent the length of the arc in this formula.
We are given that the area is 203𝜋 square centimeters. We know that the radius of this larger circle is 29 centimeters. And we’re trying to determine the length of the arc 𝐿. We can then multiply both sides by two to get 406𝜋 equals 29𝐿. And then dividing both sides by 29, we have 406 over 29𝜋 equals 𝐿. When we actually work out 406 over 29, we get a value of 14. And then of course we can include the units so that the arc length is equal to 14𝜋 centimeters.
Now we know that the arc length of this large sector is 14𝜋 centimeters, it’s the same as the circumference of the smaller circle. And that will allow us to work out the radius of the smaller circle. Let’s clear some space for some more workings. We can recall that the circumference and the radius of a circle are related by the formula circumference 𝐶 equals two times 𝜋 times the radius. Plugging in the value that the circumference is 14𝜋 into this formula, we get 14𝜋 is equal to two 𝜋𝑟. Dividing both sides by 𝜋 would give us 14 is equal to two 𝑟. We can then divide both sides by two, working out that the radius 𝑟 is equal to seven centimeters.
But remember we haven’t finished this question yet. We have worked out that the radius is seven centimeters so that we can finally calculate the height of this cone. We need one last formula, the Pythagorean theorem. This theorem states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares on the other two sides. In the triangle within the cone, we have got the value of the hypotenuse 𝑐 as 29 centimeters. The values of 𝑎 and 𝑏 are the other two sides. One of those is the height ℎ centimeters, and one of them is seven centimeters.
Substituting these values into the Pythagorean theorem, we have ℎ squared plus seven squared equals 29 squared. Evaluating the squares, seven squared is 49 and 29 squared is 841. We can then subtract 49 from both sides of the equation, giving us ℎ squared is equal to 792. We then take the square root of both sides, ignoring the negative value, since the height ℎ is a length.
At this point, we can either find this value to a decimal approximation or we can leave it in the square root form. However, if we’re leaving it in this form, we can simplify it a little further. We can then recognize that we have a square factor of 36 which is multiplied by 22 to get 792. Evaluating the square root of 36, we have six multiplied by root 22. We can therefore give the answer that the height of the cone is six root 22 centimeters. Or if we wanted to give this as a decimal value, it would be approximately 28.14 centimeters to the nearest hundredth.
