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Formulas

Formulas

Author: Anthony Varela
Description:

Identify the correct formula for a given shape.

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Tutorial

Source: All images created by Anthony Varela

Video Transcription

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Hello. My name is Anthony Varela, and today I'd like to introduce some formulas. So first we're going to talk about what a formula is, and relate that to equations and variables. And then I'm going to introduce several common formulas that you'll want to commit to memory. And keep in mind-- the whole reason why formulas are so handy is that we can plug in given values-- so variables that we know-- and then solve for the variables that we don't know.

So let's just talk about some vocabulary here. So what is a formula? Well, a formula is a mathematical rule that relates two or more quantities. And we're going to see several examples of formulas today.

Formulas are oftentimes equations. So an equation is a mathematical statement that two expressions or quantities have the same value. They're equal to one another. And equations-- especially formulas-- have lots and lots of variables.

So a variable is a symbol-- which is usually a letter-- that is used to represent a value that can change. So our formulas are general mathematical rules containing variables in equations. And depending on our situation, we'll know a couple of values for those variables, and we'll need to solve for unknown values to those variables.

All right, so now let's talk about some common formulas. I'm going to go over a couple of formulas to find the area of different shapes. So let's talk about the area of a rectangle.

Here's a rectangle. And a rectangle has two dimensions. It has a height and a base. You might also know it as length and width. That's fine, too. And to find the area of a rectangle, we just multiply these two dimensions together.

So the formula is A equals bh, where A is the area, b is the base, and h is the height. So let's write that down as one of our area formulas. The area of a rectangle-- A equals bh.

Now let's think about a triangle. So what you've noticed here is that I've cut this rectangle in half using a diagonal, and I've created a triangle right here. So right now I might already be thinking, OK, the area of a triangle is very related to the area of a rectangle. Maybe it's just half of that. And that's actually true no matter what shape the triangle is.

A triangle still has a base, and it still has a height. Now, notice that when I'm talking about height, I'm talking about this vertical distance from the base to the top of the triangle. It's not the same as this side length right here.

And so to find the area of a triangle, we use this formula-- A equals 1/2 b times h-- or b times h divided by 2, however you'd like to think about that. So A is the area, b is the base, and h is the height. So we're going to write that down as one of our area formulas.

Now, how about a circle? So looking at a circle, a circle has what we call a radius. And this is the distance from the center of the circle out to the-- the circle itself. And the formula for the area of a circle is A equals pi r squared.

So it looks different than our other formulas so far. A still stands for area. Pi, remember, is a number that's approximately 3.14. And then r equals the radius of the circle. So we're going to write that down as one of our area formulas.

Now let's talk about volume formulas. So we're talking about three-dimensional shapes here. So here we have what looks like could be a kiddie pool out in your backyard, and this is a cylinder. Now, it has a circular base and a height.

So I have marked in the height of the cylinder, and then the radius is the radius of that circular base. And now to find the volume of a cylinder, we use the formula V equals pi r squared times h. So we actually see the area of a circle in our formula for the volume of a cylinder. So V stands for volume. Pi is approximately 3.14. r is the radius of that circular base, and then h is the height of the cylinder.

All right, so now let's talk about the volume of a rectangular prism. So this is similar to a rectangle, except it's three-dimensional. So this could be a box in your home. I'm sure you have several in your basement.

And a rectangular prism has three dimensions-- a height, a length, and a width. And to find the volume of a rectangular prism, we multiply these three dimensions together. So the formula is V equals lwh. And so V is the volume, l is the length, w is the width, and h is the height. We're just assigning a variable to each of these three dimensions, and multiplying them together. So that is the formula to find the volume of a rectangular prism.

All right, now, next, let's talk about a sphere. And so here I have a picture of the Earth. Now, the Earth isn't exactly spherical, but it's pretty close. And a sphere also has a radius. It's the distance from the center of the sphere out to the lines that define the sphere.

And the formula to find the volume of a sphere is V equals 4/3 times pi times r cubed. So it looks a little bit messy, but let's explain what all of this means. V equals the volume. Pi, remember, is approximately 3.14. And then r is the radius of the sphere. And then we just have this multiplier, 4/3, in our formula as well. So that's the formula for the volume of a sphere.

So now I'd like to talk about some other common formulas, one of which is the Pythagorean theorem. And you might have heard of this when learning about right triangles. So here's a right triangle. And the parts to a right triangle are as follows.

We have a horizontal and vertical leg, and then we have this diagonal hypotenuse, which is always opposite of that right angle that defines a right triangle. Now, the Pythagorean theorem-- the formula for this is a squared plus b squared equals c squared, where a and b are the legs of the right triangle, and c is the hypotenuse of the right triangle. So that is an important common formula that you'd like to remember.

Next we're going to talk about compound interest. And this all has to do with money-- so savings accounts, credit card accounts, investment accounts all are affected by compound interest. Now, here's the formula, and I'd like to explain what all of these mean, because right off the bat it looks very intimidating.

So A is the account balance. So that's how much money is in your checking account, how much money is in your credit account. P is the principal balance. And that can be thought of as the initial amount.

So if you put $500 in the bank and let it grow, $500 is the principal balance-- what you started out with. Then we have the variable r, which is the APR, or Annual Percentage Rate. So that's the interest, really, right there-- is that interest rate, r.

Then we see the variable n, that appears two times in this equation, and n talks about the number of times interest is compounded. So sometimes your interest is compounded once a month, so n would be 12 since there are 12 months in a year. If it's compounded just annually, n would equal 1-- for 1 time per year. So really, it's the number of times interest is compounded in a single year. And then t equals time in years. So we're going to write that down as a common formula that you'd like to remember.

All right, so what did we talk about today? Well, we introduced a couple of formulas. We have area formulas-- so area of a rectangle, area of a triangle, area of a circle. We talked about volume formulas-- so the volume of a cylinder, the volume of a rectangular prism, and the volume of a sphere. And we talked about the Pythagorean theorem that relates the legs of a right triangle to its hypotenuse, and we talked about compound interest, which is talking about investment accounts or other money accounts growing.

And so keep in mind, then, when dealing with these formulas, that depending on the situation, you'll know one or more of the variables, and you'll need to substitute that in to the formula to find the unknown variables. So thanks for watching this tutorial on formulas. Hope to catch you next time.

Terms to Know
Equation

A mathematical statement that two expressions or quantities have the same value.

Formula

A mathematical rule that relates two or more quantities.

Variable

A symbol (usually a letter) used to represent a value that can change.

Formulas to Know
Area of Circle

A subscript c i r c l e end subscript equals pi r squared

Area of Rectangle

A subscript r e c tan g l e end subscript equals b h

Area of Triangle

A subscript t r i a n g l e end subscript equals 1 half b h

Compound Interest

A equals P open parentheses 1 plus r over n close parentheses to the power of n t end exponent

Volume of Cylinder

V subscript c y l i n d e r end subscript equals pi r squared h

Volume of Rectangular Prism

V subscript r e c t. p r i s m end subscript equals l w h

Volume of Sphere

V subscript s p h e r e end subscript equals 4 over 3 pi r cubed