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## High school geometry

### Course: High school geometry > Unit 9

Lesson 4: Density# Area density

Area or surface density is the amount of a quantity (often mass) per unit of area.
Density=Quantity/Area. Created by Sal Khan.

## Want to join the conversation?

- Quick question on one of the practice problems:

The city limits of Las Pythagoras form a perfect shape of an isosceles right triangle whose legs are both 25 kilometers long.

In the solution, it says that the area is (1/2)(25 squared). How is this possible if the base isn't given in the question?(4 votes)- The base is given, they just don't call it that. A right triangle can be oriented so one of the legs is the base and the other is the height. Because the lengths of both legs are 25 km, and they can serve as our base and height, the area would be (1/2)(25km)^2 or 312.50 square kilometers.(2 votes)

- What if it is finding the area?(2 votes)
- To find area from density and quantity, all you have to do is multiply the quantity divided by the density, and you have your area. Hope this helps!(2 votes)

- wait so how did that work y Andrew was wrong when my dog ate my homework like that's crazy.(2 votes)
- First question! How do you find the area of a cone? Cylinder? And sphere?(2 votes)
- As Kim said, you can find those in the search bar or some other sub category to hs geometry. Here's the link for cylinders (found off an EOC review guide) https://www.khanacademy.org/exercise/solid_geometry(1 vote)

- What can be a unit?(1 vote)
- The real takeaway from that situation is Tigersville animal control is underfunded.(0 votes)

## Video transcript

- [Narrator] In this video, we're going to talk about
density in the context of area. And the simplest way of
thinking about it is density is going to be some
quantity per unit area. So for example, let's say that I have a football
field right over here and I have another identical
football field right over here. Now they have the same area, but if I have let's say five
people on this football field actually six people on
this football field. And I only have three people
on this football field the density of people
per average unit area or the density of people I
should say per football field is going to be higher
in this left example. So it's always going to
be quantity per area. Now, with that out of the
way, let's do a worked example that helps us understand this
idea a little bit better. So here we're told the town of Tigersville has a population density of
13 cats per square kilometer. So they're giving us the density
let me write that 13 cats. So the quantity is quantity
and cats per square kilometer. That's the density right over there. The town is shaped like a
perfect isosceles trapezoid. So it looks something like this say perfect isosceles trapezoid. It's gonna look something like that, with two parallel boundaries,
12 kilometers apart. So this distance right
over here is 12 kilometers one measuring eight kilometers. So this side over here is eight kilometers the other is 16 that's
a longer one over there. How many cats are in Tigersville? So they give us the density
here and they give us, I think enough information
to figure out the area. And they want us to figure
out how many cats we have. So what is the quantity? So pause this video and see
if you can figure that out. Well, just as we said the density is equal to quantity divided by area. If we multiply both sides
of this equation by area you get area times density is
going to be equal to quantity. And we know the density. It's 13 cats per square
kilometer, and we can figure out the area and then just multiply the two. So what's the area of
this right over here? Well, the area of a
trapezoid is going to be, write here area is going
to be 12 kilometers. The height of the trapezoid times the average of
the two parallel sides I guess you could say. So the average of those it's
going to be eight kilometers plus 16 kilometers over two. So this is going to be
equal to 12 kilometers times eight plus 16 is
24 divided by two is 12. So times 12 kilometers. So this gives us 144 square kilometers. Now we know we have 13
cats per square kilometer. So let me do this year in another color. So if I multiply 13 cats
per kilometer squared and I multiply that times
this business right over here, times 144 square kilometers. And you might also notice
that the units cancel out the same way that variables might. So that cancels out with that. You're going to get 13 times 144. And the units that you're
left with is just cats. So 144 times 13, three times four is 12 that gives us to 13,
three times a hundred, 300 plus another hundred is 400. Now I'm just gonna multiply
144 essentially by 10 which is going to be 1440. And so if I add up all of that together I'm gonna jump down to here. I get 1872. So this is 1872 cats in
total and we are done.