WEBVTT
00:00:01.400 --> 00:00:08.760
A right circular cone has base diameter 10 centimeters and height 12 centimeters.
00:00:09.120 --> 00:00:12.760
Determine the total surface area to the nearest tenth.
00:00:13.400 --> 00:00:18.760
First letβs draw a sketch of this cone so we can visualize it more clearly.
00:00:19.160 --> 00:00:26.920
Weβre told that the cone has a base diameter of 10 centimeters and a height of 12 centimeters.
00:00:27.480 --> 00:00:31.360
Weβre asked to calculate the total surface area of the cone.
00:00:31.680 --> 00:00:38.240
The total surface area of a cone is the area of its base plus the lateral surface area.
00:00:38.720 --> 00:00:48.560
This is ππ squared plus πππ, where π is the radius of the base of the cone and π is the slant height.
00:00:49.160 --> 00:00:51.600
We know the radius of the base of the cone.
00:00:51.840 --> 00:00:59.880
If the diameter is 10 centimeters, then the radius is half of this; itβs five centimeters.
00:01:00.480 --> 00:01:13.040
So our calculation becomes π multiplied by five squared, for the area of the base, plus π multiplied by five multiplied by π, for the lateral surface area.
00:01:13.840 --> 00:01:16.840
Now we havenβt been given the value of π in the question.
00:01:17.360 --> 00:01:19.920
Remember, π is the slant height of the cone.
00:01:20.560 --> 00:01:24.840
Weβve been given the perpendicular height, 12.
00:01:25.440 --> 00:01:35.720
Itβs really important that you read questions like this carefully and determine whether youβve been given the height or the slant height because they arenβt the same as each other.
00:01:36.080 --> 00:01:43.040
Letβs think about how we can calculate the slant height from the information we have.
00:01:43.600 --> 00:01:50.680
The slant height, the vertical height, and the radius of the cone form a right-angled triangle.
00:01:51.280 --> 00:02:01.800
This means that we can apply the Pythagorean theorem to calculate the slant height as we know the other two sides of this right-angled triangle.
00:02:02.600 --> 00:02:09.840
Applying the Pythagorean theorem tells us that π squared is equal to five squared plus 12 squared.
00:02:10.320 --> 00:02:19.320
Evaluating each of these numbers gives us π squared is equal to 25 plus 144.
00:02:20.040 --> 00:02:24.760
The sum of these two values is 169.
00:02:25.280 --> 00:02:33.080
Therefore π is equal to the square root of 169, which is 13.
00:02:33.840 --> 00:02:39.040
So now we know the slant height of the cone, 13 centimeters.
00:02:39.760 --> 00:02:47.520
You may have spotted this a little bit earlier as the values of five, 12, and 13 form a Pythagorean triple.
00:02:48.000 --> 00:02:51.200
At any case, we now know the value of π.
00:02:51.880 --> 00:03:03.080
Substituting into our calculation for the total surface area, we now have π multiplied by five squared plus π multiplied by five multiplied by 13.
00:03:03.520 --> 00:03:10.680
Evaluating each of the constants gives 25π plus 65π.
00:03:11.040 --> 00:03:13.280
This gives a total of 90π.
00:03:13.840 --> 00:03:22.480
Now we weβre asked for our answer not as a multiple of π, but as a decimal to the nearest tenth.
00:03:23.120 --> 00:03:25.640
So we need to evaluate this.
00:03:26.200 --> 00:03:37.520
As a decimal, this is 282.7433 and the decimal continues.
00:03:38.280 --> 00:03:51.800
Remember we need to round it to the nearest tenth, and so we have our answer for the total surface area of the cone: 282.7 centimeters squared.