A tent is in this shape of a right circular cone of height 126 centimeters. Given that its base has circumference 527.52 centimeters, what is its slant height, if we use 3.14 for 𝜋?
So, let’s have a look at our cone. We’re told that the height is 126 centimeters. And we need to work out the slant height, which is the distance along the slanted edge of the cone. Let’s see if we could calculate it using the triangle inside the cone. We know that this must have a right angle, since it’s formed with the horizontal line on the ground and the vertical line going to the top of the tent. The height of our triangle will be 126 centimeters.
We need to work out the slant height of our triangle. In this case, it would be the hypotenuse. So, let’s define a letter 𝑙 which will represent our slant height, the hypotenuse of our triangle. As we know I have a right-angle triangle, it would be possible to calculate our hypotenuse if we knew the size of the base of our triangle. If we return to our cone, we could see that this length would represent the radius of the circle.
In the question, we weren’t given the radius, but we were told the circumference of the circle. The circumference is the distance all way round the outside of the circle. The formula for the circumference of a circle is equal to 𝜋 times the diameter, or two times 𝜋 times the radius. In this case, we have the circumference and we want to find the radius. So, we can write the formula the circumference equals two 𝜋𝑟.
Substituting in our values, we would have 527.52 equals two times 3.14 times 𝑟. And we’re using 3.14 for 𝜋 as we were asked in the question. Simplifying the values on the right-hand side, we would have 527.52 equals 6.28 times 𝑟. And now, to find 𝑟 by itself, we need to divide both sides of our equation by 6.28. So, 527.52 divided by 6.28 will be 𝑟, which we can evaluate to give us the radius 𝑟 equals 84 centimeters.
So, now, looking at our triangle formed in our cone, we have two lengths, the base of 84 centimeters, the height of 126 centimeters. We can now find 𝑙, our slant height. We can use the Pythagorean theorem, which says that the square on hypotenuse is equal to the sum of the squares on the other two sides. So, applying this to our triangle then, we can fill in the values 𝑙 squared, which is our hypotenuse, equals 126 squared plus 84 squared. And it doesn’t matter which way round our 126 and our 84 are written.
And using our calculator to evaluate this will give us 𝑙 squared equals 15876 plus 7056. So, 𝑙 squared equals 22932. To find 𝑙 by itself then, we take the square root of both sides. Therefore, 𝑙 equals the square root of 22932, which is 42 root 13 centimeters. As a decimal, this is also equal to 151.4331536 centimeters. And that’s also a perfectly valid answer.
The reason we get an answer like 42 root 13 is by the fact that the square root of 22932 is also equal to the square root of 42 times 42 times 13, which simplifies to 42 root 13. The question didn’t specify an answer in decimal form or in root form, but we prefer the one in root form since it’s neater and more accurate as it doesn’t require rounding. So, our final answer for the slant height is 42 root 13 centimeters.