WEBVTT
00:00:01.130 --> 00:00:08.070
In this video, we will learn how to calculate the lateral and total surface areas of cones.
00:00:09.100 --> 00:00:16.930
We will begin by looking at the definition of a cone and the formulas for its lateral and total surface areas.
00:00:17.700 --> 00:00:23.230
We will then work through some example questions on surface areas of cones.
00:00:24.480 --> 00:00:27.680
Letβs firstly look at the definition of a cone.
00:00:28.990 --> 00:00:39.430
Cones are three-dimensional geometric shapes that have a circular base and a curved side that ends in a single vertex or apex.
00:00:40.590 --> 00:00:46.140
A right cone is a cone whose apex lies above the centroid of the base.
00:00:46.930 --> 00:00:49.980
The centroid is the centre of a circle.
00:00:51.160 --> 00:00:55.180
The height of a cone is the distance from the apex to the base.
00:00:55.570 --> 00:00:59.140
This is often known as the perpendicular height.
00:01:00.210 --> 00:01:07.430
The slant height of a cone is the distance from the apex to any point on the circumference of the base.
00:01:08.590 --> 00:01:15.220
This means that the radius perpendicular height and slant height form a right-angled triangle.
00:01:16.400 --> 00:01:25.580
This information will be useful as we can use Pythagorasβs theorem to help us solve problems involving the surface area of a cone.
00:01:26.550 --> 00:01:34.830
We will, now, look at the formulas we can use to calculate the lateral surface area and the total surface area of a cone.
00:01:36.460 --> 00:01:41.850
The lateral surface area of a cone is the area of a curved surface.
00:01:42.610 --> 00:01:47.320
This can be calculated using the formula πππ.
00:01:47.730 --> 00:01:52.420
This involves multiplying π by the radius by the slant height.
00:01:53.650 --> 00:02:00.020
The total surface area of a cone is the area of all the surfaces, including the base.
00:02:00.960 --> 00:02:11.090
As a cone only has two faces, the total surface area will be equal to the area of the curved surface plus the area of the base.
00:02:12.030 --> 00:02:17.240
The area of the curved surface, as already mentioned, is equal to πππ.
00:02:18.030 --> 00:02:24.080
As the base of a cone is a circle, its area will be equal to ππ squared.
00:02:25.080 --> 00:02:32.180
The total surface area of a cone is, therefore, equal to πππ plus ππ squared.
00:02:32.970 --> 00:02:37.930
The lateral surface area of a cone is just equal to πππ.
00:02:39.390 --> 00:02:52.140
Letβs look at the example where the radius of the base of the cone is five centimetres, the perpendicular height is 12 centimetres, and the slant height is 13 centimetres.
00:02:53.150 --> 00:03:00.570
We can calculate the lateral surface area of the cone by multiplying π by five by 13.
00:03:01.310 --> 00:03:04.930
Five multiplied by 13 is equal to 65.
00:03:05.250 --> 00:03:09.520
Therefore, the lateral surface area is 65π.
00:03:10.690 --> 00:03:18.170
Multiplying 65 by π gives us 204.2035 and so on.
00:03:19.130 --> 00:03:24.310
Rounding this to one decimal place gives us 204.2.
00:03:25.240 --> 00:03:32.280
The lateral surface area of the cone is 204.2 square centimetres.
00:03:32.980 --> 00:03:38.480
Note that our units are squared and not cubed, even though we have a 3D shape.
00:03:39.310 --> 00:03:50.660
The units for surface area are square centimetres, square metres, and so on, whereas the units for volume would be cubic centimetres and cubic metres.
00:03:52.040 --> 00:03:59.700
The total surface area of this cone will be equal to 65π plus π times five squared.
00:04:00.520 --> 00:04:06.270
We add the area of the curved or lateral surface to the area of the base.
00:04:06.890 --> 00:04:09.340
Five squared is equal to 25.
00:04:09.630 --> 00:04:13.170
So we have 65π plus 25π.
00:04:14.370 --> 00:04:16.990
This is equal to 90π.
00:04:18.130 --> 00:04:26.650
Once again, we can type this into the calculator, giving us 282.7433 and so on.
00:04:28.050 --> 00:04:37.350
Rounding this to one decimal place gives us a total surface area of 282.7 square centimetres.
00:04:38.140 --> 00:04:44.600
We will now look at some questions involving the lateral and total surface areas of a cone.
00:04:46.620 --> 00:04:57.490
Find, in terms of π, the lateral area of a right cone with base radius nine centimetres and height 13 centimetres.
00:04:58.740 --> 00:05:01.910
Letβs begin by drawing a diagram of the cone.
00:05:03.070 --> 00:05:07.330
Weβre told that the base radius is equal to nine centimetres.
00:05:08.290 --> 00:05:16.920
The height of the cone, which goes from the apex at the top to the centre or centroid of the base, is 13 centimetres.
00:05:18.030 --> 00:05:22.380
This creates a right-angled triangle with a slant height π.
00:05:23.630 --> 00:05:28.410
The lateral area of a cone is the area of its curved surface.
00:05:29.160 --> 00:05:31.590
This is equal to πππ.
00:05:31.760 --> 00:05:35.980
We multiplied π by the radius by the slant height.
00:05:37.150 --> 00:05:40.800
We know that the radius of the cone is nine centimetres.
00:05:41.070 --> 00:05:44.500
However, we donβt know the slant height at present.
00:05:45.480 --> 00:05:49.550
We can, however, calculate this by using Pythagorasβs theorem.
00:05:50.170 --> 00:05:59.860
This states that π squared plus π squared is equal to π squared, where π is the length of the hypotenuse in a right triangle.
00:06:00.650 --> 00:06:05.900
In this question, π squared is equal to nine squared plus 13 squared.
00:06:06.650 --> 00:06:09.080
Nine squared is equal to 81.
00:06:10.100 --> 00:06:13.860
13 squared is equal to 169.
00:06:15.130 --> 00:06:19.700
81 plus 169 is equal to 250.
00:06:20.010 --> 00:06:23.710
Therefore, π squared equals 250.
00:06:24.570 --> 00:06:30.570
Square-rooting both sides of this equation gives us π is equal to root 250.
00:06:31.810 --> 00:06:37.070
Root 250 is equal to root 25 multiplied by root 10.
00:06:38.050 --> 00:06:43.260
As root 25 is equal to five, this is equal to five root 10.
00:06:44.190 --> 00:06:48.990
The slant height of the cone is five root 10 centimetres.
00:06:49.990 --> 00:06:54.490
We can now substitute in this value to calculate the lateral area.
00:06:55.540 --> 00:07:01.970
The lateral area is equal to π multiplied by nine multiplied by five root 10.
00:07:02.830 --> 00:07:07.380
Nine multiplied by five root 10 is 45 root 10.
00:07:08.590 --> 00:07:15.960
As weβre asked to give our answer in terms of π, this is equal to 45 root 10π.
00:07:17.050 --> 00:07:29.470
The lateral area of a right cone with base radius nine centimetres and height 13 centimetres is 45 root 10π square centimetres.
00:07:30.360 --> 00:07:39.380
Remember that our units for any area or surface area are square centimetres, square metres, et cetera.
00:07:41.040 --> 00:07:46.690
We will now look at another question, calculating the total surface area of a cone.
00:07:48.070 --> 00:07:55.500
Find the total surface area of the right cone approximated to the nearest two decimal places.
00:07:56.950 --> 00:08:02.540
Weβre told on the diagram that the height of the cone is 14.5 centimetres.
00:08:03.000 --> 00:08:06.770
And its slant height is 16.5 centimetres.
00:08:07.240 --> 00:08:09.870
The radius is currently unknown.
00:08:11.170 --> 00:08:15.720
We can calculate the length of the radius by using Pythagorasβs theorem.
00:08:16.710 --> 00:08:27.390
This states that π squared plus π squared is equal to π squared, where π is the longest side of a right triangle, known as the hypotenuse.
00:08:28.420 --> 00:08:36.770
Substituting in our values gives us π squared plus 14.5 squared is equal to 16.5 squared.
00:08:37.850 --> 00:08:48.880
Subtracting 14.5 squared from both sides gives us π squared is equal to 16.5 squared minus 14.5 squared.
00:08:49.900 --> 00:08:58.920
Square-rooting both sides of this equation gives us π is equal to 16.5 squared minus 14.5 squared.
00:08:59.940 --> 00:09:03.070
This is equal to the square root of 62.
00:09:04.040 --> 00:09:08.470
For accuracy, we will leave our answer in this form at present.
00:09:09.370 --> 00:09:13.640
We were asked to calculate the total surface area of the cone.
00:09:14.350 --> 00:09:18.930
A cone has two surfaces, a curved surface and a base.
00:09:19.280 --> 00:09:26.670
Therefore, the total surface area is equal to the area of the curved surface plus the area of the base.
00:09:27.440 --> 00:09:32.520
The area of the curved or lateral surface is equal to πππ.
00:09:32.780 --> 00:09:36.990
We multiply π by the radius by the slant height.
00:09:38.100 --> 00:09:45.060
As the base is a circle, we work out the area of the base by multiplying π by the radius squared.
00:09:46.270 --> 00:10:02.660
Substituting in our values for the radius and slant height gives us π multiplied by the square root of 62 multiplied by 16.5 plus π multiplied by the square root of 62 squared.
00:10:03.740 --> 00:10:08.320
The square root of 62 squared is just equal to 62.
00:10:09.410 --> 00:10:18.250
As we need to calculate this to two decimal places and not in terms of π, we can type this calculation into our calculator.
00:10:19.360 --> 00:10:26.390
This gives us an answer of 602.93801 and so on.
00:10:27.860 --> 00:10:30.960
The eight in the thousandths column is the deciding number.
00:10:31.560 --> 00:10:35.700
When this digit is greater than or equal to five, we round up.
00:10:36.980 --> 00:10:46.570
The total surface area of the cone to two decimal places is 602.94 square centimetres.
00:10:47.620 --> 00:10:51.790
Any surface area will be measured in square units.
00:10:52.800 --> 00:10:57.490
We will now look at the question involving surface area in context.
00:10:59.470 --> 00:11:09.050
A conical lampshade is 31 centimetres high and has a base circumference of 145.2 centimetres.
00:11:09.830 --> 00:11:14.110
Find the curved surface area of the outside of the lampshade.
00:11:14.890 --> 00:11:18.070
Give your answer to the nearest square centimetre.
00:11:19.290 --> 00:11:25.480
The lampshade is in the shape of a cone with a height of 31 centimetres as shown.
00:11:26.550 --> 00:11:32.200
The circumference of the base is equal to 145.2 centimetres.
00:11:33.120 --> 00:11:37.930
We have been asked to calculate the curved surface area of the lampshade.
00:11:38.800 --> 00:11:44.180
The curved or lateral surface area of a cone is equal to πππ.
00:11:44.510 --> 00:11:49.400
We multiply π by the radius by the slant height π.
00:11:50.750 --> 00:11:53.650
Currently, we donβt know either of these values.
00:11:53.820 --> 00:11:57.150
We donβt know the slant height, and we donβt know the radius.
00:11:58.380 --> 00:12:03.740
The circumference of a circle can be calculated using the formula two ππ.
00:12:04.200 --> 00:12:08.220
We can use this to calculate the radius in this question.
00:12:09.170 --> 00:12:13.840
145.2 is equal to two ππ.
00:12:14.930 --> 00:12:24.050
Dividing both sides of this equation by two π gives us π is equal to 145.2 divided by two π.
00:12:25.270 --> 00:12:29.950
This means that π is equal to 23.1092 and so on.
00:12:29.950 --> 00:12:35.100
For accuracy, weβll not round this answer at this point.
00:12:35.100 --> 00:12:49.930
As we know that the radius of the cone is 23.10 and so on centimetres and that the height is 31 centimetres, we can now calculate the slant height.
00:12:50.860 --> 00:13:04.400
We will do this using Pythagorasβs theorem, which states that π squared plus π squared is equal to π squared, where π is the longest side of a right triangle known as the hypotenuse.
00:13:05.200 --> 00:13:16.540
Substituting in our values gives us π squared is equal to 31 squared plus 23.1092 and so on squared.
00:13:17.350 --> 00:13:26.290
Typing this into our calculator gives us π squared is equal to 1495.0396 and so on.
00:13:26.320 --> 00:13:35.560
Square-rooting both sides gives us π is equal to 38.6657 and so on.
00:13:36.640 --> 00:13:44.280
We can now substitute the value of the radius and the slant height into the formula for the curved surface area.
00:13:44.900 --> 00:13:49.080
We multiply π by the radius by the slant height.
00:13:50.300 --> 00:13:58.660
Typing this into the calculator gives us a curved surface area of 2807.132.
00:13:59.530 --> 00:14:06.220
We need to round this to the nearest square centimetre, which means we need to round to the nearest whole number.
00:14:07.350 --> 00:14:15.190
The curved surface area of the lampshade is, therefore, equal to 2807 square centimetres.
00:14:16.270 --> 00:14:21.930
We will now look at one final example involving surface areas of cones.
00:14:23.900 --> 00:14:33.150
A right cone has slant height 35 centimetres and surface area 450π square centimetres.
00:14:33.740 --> 00:14:36.290
What is the radius of its base?
00:14:37.710 --> 00:14:45.000
We recall here that the surface area of a cone is equal to πππ plus ππ squared.
00:14:45.870 --> 00:14:51.320
πππ is equal to the curved or lateral surface area of a cone.
00:14:52.450 --> 00:14:58.080
ππ squared is equal to the base area, as the base of a cone is a circle.
00:14:59.110 --> 00:15:04.420
Our value for π is the radius of the base, and π is the slant height.
00:15:05.420 --> 00:15:09.670
We know that the total surface area is 450π.
00:15:10.650 --> 00:15:13.460
The slant height is 35 centimetres.
00:15:13.730 --> 00:15:17.980
Therefore, the curved surface area is 35ππ.
00:15:18.890 --> 00:15:21.740
The area of the base is ππ squared.
00:15:22.380 --> 00:15:28.300
As π is common to all three terms, we can divide both sides of the equation by π.
00:15:29.290 --> 00:15:35.380
This gives us 450 is equal to 35π plus π squared.
00:15:36.240 --> 00:15:43.830
Subtracting 450 from both sides of this equation will give us a quadratic equation equal to zero.
00:15:44.660 --> 00:15:50.490
π squared plus 35π minus 450 is equal to zero.
00:15:51.510 --> 00:15:54.890
We can solve this by factoring or factorising.
00:15:55.890 --> 00:16:03.080
We need to find two numbers that have a product of negative 450 and a sum of 35.
00:16:03.840 --> 00:16:08.890
45 multiplied by negative 10 is negative 450.
00:16:09.720 --> 00:16:14.640
And 45 plus negative 10 is equal to 35.
00:16:15.490 --> 00:16:23.140
This means that our two brackets, or parentheses, will be π plus 45 and π minus 10.
00:16:24.410 --> 00:16:30.590
In order to solve this equation equal to zero, one of the parentheses must be equal to zero.
00:16:31.330 --> 00:16:37.670
Either π plus 45 equals zero or π minus 10 equals zero.
00:16:38.410 --> 00:16:45.330
Solving these two equations gives us π equals negative 45 or π equals 10.
00:16:46.320 --> 00:16:50.160
The radius is a length and, therefore, cannot be negative.
00:16:51.280 --> 00:17:05.730
We can, therefore, conclude that a right cone with slant height 35 centimetres and surface area 450π square centimetres has a base radius of 10 centimetres.
00:17:06.600 --> 00:17:10.070
We will now summarise the key points from this video.
00:17:11.580 --> 00:17:15.310
A cone is a 3D shape with two surfaces.
00:17:15.960 --> 00:17:18.840
We have a lateral or curved surface.
00:17:19.530 --> 00:17:25.020
We can calculate the area of this surface using the formula πππ.
00:17:25.300 --> 00:17:29.520
We multiply π by the radius by the slant height.
00:17:30.810 --> 00:17:33.640
A cone also has a circular base.
00:17:34.430 --> 00:17:37.080
This has an area of ππ squared.
00:17:37.310 --> 00:17:40.450
We multiply π by the radius squared.
00:17:41.400 --> 00:17:49.310
The total surface area of any cone is, therefore, equal to πππ plus ππ squared.
00:17:50.330 --> 00:17:57.070
Surface area is measured in square units such as square centimetres or square metres.