Today we're going to talk about formulas. A formula is just a mathematical rule that relates two or more different variables. A lot of times, you'll see a formula in the form of an equation, which is showing that two different quantities, or expressions, are equal to each other. And we often use formulas to figure out variables that we don't know by substituting values for variables that we do.
So today, we're going to go over a couple of different common formulas and the different variables that each of those formulas use. So we're going to start with three different formulas for finding the area of different shapes. So remember, the area is the amount of space that's enclosed in a two dimensional object. So first, let's look at the area of a rectangle.
We have the formula a equals b times h, where a is our area, b is our base, and h is the height. And if we're looking at our picture of a rectangle, again the area is the enclosed space, the base is the bottom horizontal side of our rectangle, and the height is the vertical side of our rectangle.
Pretty similarly, we have a formula for the area of a triangle. So again, our formula is a equals 1/2 times b times h. Our a is, again, the area. b is our base. And h is our height. And when we're looking at these three variables in the picture of our triangle, again, area is the enclosed space, b is still the base of our triangle, the horizontal side, but our height is the vertical distance. So not the diagonal distance of our triangle, but the vertical distance of our triangle.
Lastly, we have a formula for the area of a circle. So again, in our formula, area is equal to pi times r squared. So a stands for our area. Pi, which can be approximated with the decimal 3.14, is the ratio of the circumference to the diameter of a circle. And r stands for our radius.
So again, area is the enclosed space, and our radius is the distance from the center of our circle to the edge of our circle. So next, let's look at some formulas for volume.
All right. So let's look at some common formulas for calculating the volume of a three dimensional object. So volume is how much space is enclosed in a three dimensional object. So make sure as we go through these formulas, you get them into your notes, so you can refer to them later. All right. So first is the volume of a rectangular prism. And a rectangular prism is like a block or cereal box.
So in our formula, V is going to stand for our volume. l stands for the length, w stands for the width, and h stands for the height. So if we're looking at our picture, the width of our rectangular prism will be this side here. The length is the side that forms the other side of the base, and the height is the vertical distance. All right. Let's look at a formula for the volume of a cylinder, a cylinder being something that looks like a pop can.
So again, our V is going to stand for our volume. Pi can be approximated with 3.14. r is going to be the radius of our circular base, so the top or the bottom of our cylinder. And h, again, is our height. So looking at our picture, we've got our radius, which is going to be again from the center of the circular base to the edge of the circle, and our height is going to be our vertical distance of the cylinder. All right.
Our last formula for volume would be the volume of a sphere. A sphere being like a 3D circle, like a globe. And in this formula, our V stands for our volume. Pi, again, is going to be approximated with 3.14. And r, again, is our radius. So in our circle-- or sorry, in our sphere-- the radius is going to be from the center of the circle, which you actually wouldn't be able to see. It's in the inside of the circle. To the outside of the circle, that would be our radius.
All right. Lastly, we have two unrelated but very common formulas. The first one is the Pythagorean theorem. The Pythagorean Theorem is used for calculating the side length of a right triangle. And in this formula, a and b are side lengths. The horizontal and vertical lengths of our triangle. And it doesn't matter which leg is a and which one is b. However, c is the hypotenuse of our right triangle. And so that's this diagonal side of the triangle. c always has to be the hypotenuse.
Let's look at another formula, which is called the compound interest formula, which can be used to find information about the balance in an account that uses compound interest. So again, a very common and useful formula. So this formula has quite a few variables. a stands for our account balance. Our p is called the principal, or our initial amount that we would have in an account.
The r in this formula stands for the interest rate. And this value that you put in for r should be a decimal, not the percentage form of the interest rate. Then our n, we have n in two parts of our formula. That stands for the number of times that our interest rate is compounded in a year, number of times our interest is compounded in a year. So that could be something like compounded monthly, compounded daily, sometimes compounded yearly. And our last variable is t, which stands for our time.
All right. Let's go over our key points from today. So we first talked about the fact that formulas are used to relate two or more different quantities. And we most often use formulas when we are given one or more of those variables, and we need to solve for an unknown variable, something that we want to find. And this is done by substituting values for known variables into our formula and then evaluating the formula to find the unknown variable.
So I didn't include any of the eight formulas that we went over today into our key points, but they are definitely important. So if you have not written those formulas down into your notes, make sure that you go back and get those into your notes so that you can refer to them later. So I hope that these key points and examples helped you understand a little bit more about using formulas. Keep using these notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.
(00:00 - 00:35) Introduction
(00:36 - 02:49) Formulas for Area
(02:50 - 05:22) Formulas for Volume
(05:23 - 07:41) Formulas for Pythagorean Theorem and Compound Interest
(07:42 - 08:46) Summary
A mathematical rule that relates two or more quantities.
A mathematical statement that two expressions or quantities have the same value.
A symbol (usually a letter) used to represent a value that can change.