In this section, we're gonna learn how to find the surface area of
some three-dimensional shapes, of which are rectangular solids,
spheres, and right circular cylinders.
Now for the cylinders, we're also going to find the surface area
of those cylinders that are closed at both ends, of cylinders that are only closed at one end,
open at the other end, and cylinders that are open at both ends.
So, to make this process easier, we'll provide you with some formulas
in order to find the surface area. So for starters, let's take a look at this
formula for the rectangular solid. To find this,
we need to find the areas of all six sides of the solid,
and to do that, we would need measurements for our
length, our width, and our height.
So you take a look at what we have written here,
the surface area of this rectangular solid is found by
finding the areas of the two sides, the front, and the back,
the top and the base. So we add all those areas together in order to find
our total surface area. So using our measurements here with length,
width, and height, we can find our formula
in order to find the surface area. So, the surface area for a rectangular solid is
found by finding 2(length)(height),
so that's for this side here and the other side in the back;
2(height)(width)
which is for these ends here, h(w);
and 2(length)(width), which is for our top
and our base. So, moving on, one of the other formulas that you'll work with
is for a sphere. Now, we find that
by multiplying π(r²); "r" represents the radius
of this figure, and multiplying that by 4.
Now for our cylinders, we have three separate formulas which you'll work with
and so to do this first one,
where we have the top and bottom enclosed,
we need to find the area of that circular base.
So the area of a circle, π(r²), and you can see here in this formula
we have two circle areas so we have the base
which is π(r²), and the top, another π(r²) and so
that's why we have this 2 here to multiply times this
area. Now the last part in the formula is to find the surface area
of the side of the cylinder, and we do that
by multiplying 2(π)(r)(h), (h) = height of the figure.
It's also important to remember that for
right circular cylinders
the sides of our cylinder are perpendicular to the base, and that's why
these formulas work.
So the other two formulas for the cylinders--this one is for when we have a
base that's enclosed on one side
and it's open at the top, so you'll notice that π(r²) here
is singular. We don't double it as we did in the previous formula
because we only need to find the area of one end, then we add it to the area
the whole entire side of that cylinder,
and we get our total surface area. Lastly,
we find the surface area of a cylinder that is open
on both ends, so we don't need to find the area that circle at all
we just find the surface area the entire side just 2(π)(r)(h).
So let's work through some examples and see how we can put these formulas to the
test.
When finding the surface area of a rectangular solid,
we use the formula:
Surface area = 2(length)(height) + 2(height)(width) + 2(length)(width).
So we have these measurements here for this
solid, where our length (l) = 10 feet,
our width (w) = 7 feet,
and our height (h) = 7 feet as well.
So now that we have our values for length, width, and height,
let's plug them into our formula and find our surface area. So,
let's bring down our "S" which represents surface area, and that's equal to
2 times our length, which is 10, times our height,
which is 7,
then we need 2(height)(width), so that's 2(7)(7).
Lastly, we need 2(length)(width),
so we have 2(10)(7)
So now let's just do the math:
2(10)(7) = 140, plus
2(7)(7) = 98, plus
2(10)(7) = 140 once more.
And then finding this sum:
140 + 98 + 140 = 378.
Now, we can't forget
to add our units here, and we have feet in our original figure.
So since we're finding the surface area, feet(feet)
gives us square feet (feet²), so it's important to remember
the correct units
when finishing this problem here. So, the surface area
of this rectangular solid that has a length of 10 feet,
a width of 7 feet, and a height of 7 feet,
we get 378 ft²
for the surface area. So, let's move on to right circular cylinders.
Okay, let's find the surface area of a right circular cylinder that's closed at
both ends.
So, we use the formula:
surface area = 2(π)(radius)² + 2(π)(radius)(height).
So for this example, we have a right circular cylinder
closed at both ends that has a radius of 2 feet.
This is a radius of the circular base; and it also has a height
of 8 feet. So let's go ahead and write out our
known values here: radius of 2 feet,
and height of 8 feet.
So we plug these values into our formula to get our surface area;
we have S = 2(π),
let's use π = 3.14,
and then our radius is 2,
and we square that value. Then we add to it
2(π), we're using again π = 3.14,
times the radius, which is 2
and the height, which is 8.
So, from here all we need to do is work on math to get our answer.
2(3.14)(2²), 2² = 4,
this entire product here comes out to be 25.12,
and we add to that the product of
2(3.14)(2)(8),
which is a 100.48.
Finding the sum, 25.12 + 100.48 = 125.6.
Again, we need to remember our units: we have feet again, so feet(feet) = ft²
those are the correct units for our answer.
So the surface area for this right circular cylinder
that is closed at both ends, so we used the correct formula,
that has a radius of 2 feet and a height of 8 feet
gives us a surface area 125.6 ft²
Lastly, let's take a look at surface area of a sphere.
To find the surface area of a sphere,
we use the formula S = 4(π)(radius)².
For this example, we have a sphere has a radius of 2 miles,
so let's go ahead and use this value, plug it into our formula,
and find our surface area. So we have
S = 4(π),
we'll use π = 3.14,
multiply this times the radius squared, which is 2 miles.
Simplifying this a bit: 4(3.14),
2² = 4,
doing our math: 4(3.14)(4) gives us a surface area of 50.24.
Now again, we need to remember our units. This time we're
working with miles, and we have to square those miles for our answer.
So the total surface area of a sphere with a radius
of 2 miles, is 50.24 mi².
So it's really important when you're working with surface area to
remember the correct units
and make sure you are using the correct formula to get your answer.