Hi! Now, we’re going to look at volume and surface area of composite figures. So on the problem, we need to find the surface area and the volume of the composite figure, and here is our figure. So in this figure, let’s start off by looking for the surface area.
All right, so for the surface area, we’re going to let r be the radius and l slant length height of the cone. So the surface are of a composite figure is the sum of the surface area of the hemisphere, which will stay as S, and the lateral cone, which we will say is l.
So the surface area is the sum of the hemisphere, so it’s half of the sphere, hemisphere is half of a sphere. So it’s the sum of the surface area of the hemisphere and the lateral cone. So to find the surface area of the hemisphere, we use the formula, one half times 4 pi r². So the surface area of a sphere is 4 pir². But since we are using a hemisphere, half of a sphere, we’re going to multiply that by half.
So we are going to take a half of a sphere. So now, we are going to go ahead and fill in our information we know. So the surface area of a hemisphere equals 1 ½ x 4 x pi, which is three 3.14 x radius which is 7². So then we’re going to continue and we get 1 1/2 x 4 x 3.14 and 7² is 49. And then if I compute the rest, I get 307.72 cm².
So the surface area of the hemisphere is 7.72 cm². So now, the lateral area of the cone. So we said, which stood for l. So when you find the lateral area of a cone, you use pi x r x l. Remember we said r is the radius and l is the height. So we’re going to go ahead and fill in our variables. So pi is 3.14 x 7 x 25, and then if we go ahead and compute that, we 549.5 cm².
So now, the surface area of our composite figure, so that would mean the surface area of the composite figure will be the sum of our hemisphere and our later area. So it would be the hemisphere which we know is 307.72 plus our lateral cone which is 549.5 and these are cm². So then when we add this up, we get 857.22 cm². So the surface area of the composite figure is 857.22 cm².
So let’s move down and let’s take a look at the volume. So when you are dealing with the volume of a figure, you are looking at the sum of the volumes, so you look at the sum of the volumes of the hemisphere and the cone. So to find the volume of the hemisphere, so I'm going to say volume of the hemisphere, you are going to use the formula, ½ x 4 pi / 3 x 3 cube.
So we’re going to go ahead and fill in our variables. So we have ½ x 4 1/3 x 3.14 x 7 cubed. If we multiply these together, we get 254.04 / 3 and that gives us 718.013 which is about, if we were to round it off, we’d 718.01. So the volume of the hemisphere is about 718.01.
Now, we need to find the volume of the cone. So the volume of the cone, we first need to find the height, because we know the radius, we know the slant but we don’t know the height of the cone. So we need to find the height. So to find the height, we are going to use the Pythagorean theorem. So the Pythagorean theorem is height squared plus the radius squared equals the slant height squared. So we’re looking for our height squared, we don’t know that. We do know our radius which is 7², and we know our slant height which is 25². So that means our height squared is equal to 49 and that equals 625. And then we’re going to go ahead and subtract 49 from both sides and we get 8² = 576, and then we’re going to take the square root of both sides, 576, and that means our height of 24. So now, we have height of 24 centimeters.
So now, we can go ahead and find the volume of our cone. So the volume of our cone, all right, we’ll put it right here. So the volume of the cone is equal to one-third times pi, times the radius squared times the height. So we’re going to continue and plug in our variables. So one-third times pi, which is 3.14 times our radius, which we know is 7² times our height which we now know is 24. And then if compute this, I get 1230.88.
So we know that the volume is 1230.88. So now, we have the volume of the hemisphere. So we have our volume of the hemisphere and we have volume of the cone. Now, to find the total volume, we have to add these two up, right? So let’s go ahead and add these up. So the total volume of the this figure will be our volume of the hemisphere, which is 718.01, and then we add that to the volume of the cone, which is 1230 and then we add those up, we get 148.89 cm². So the volume of the figure is 1948.89 cm².
So things to remember is the volume of the composite figure can be calculated by adding the volumes of the figures constituted in the composite figure. And the service area of the a composite figure is calculated by adding the surface areas of the individual figures constituting the composite figure. The areas of the common faces, not exposed to the outsides are not counted.
Transcription by:
Scribe4you Transcription Services