We need to know the basic measurements for distances. 12 inches = 1 foot; 3 feet = 1 yard; 5280 feet = 1 mile.
Now that we know our measurements we can start inserting them in formulas. We’ll start with the formula to find the area of a rectangle. This is length times width (L x W), OR base times height (B x H), OR front footage times depth (we’ll use FF x D). All three are the same formula. We’re just calling the sides or dimensions something different. If you multiply those dimensions, that will give you the AREA. If the dimensions are in feet, the area will be in square feet. If the dimensions are in inches, the area will be in square inches, and so on. When you are multiplying the dimensions, both sides must be using the same type of measurement, that is, you can’t multiply feet and inches, or yards by feet.
Sometimes, property may be sold by the front foot, instead of by the square foot. The front footage, or frontage, is the distance across the front of the property. This may be what makes a property valuable. For example, a waterfront property is made more valuable by having a greater frontage along the water, and not necessarily as much by how far back the property goes. The front of the property can be the longer or shorter dimension. If you are not told otherwise, it will normally be the dimension that is given first.
You may also get questions where you are given the area, and either the base or the height, and have to find the dimension that you are missing. All we have to do in this case is use the rectangle formula backwards. When we are using any formula backwards, we do ALL of the steps in the reverse order, and do everything the opposite. In the rectangle formula, when go forward we multiply and END with the area. If we are going backwards, we have to START with the area, and then DIVIDE. So, the area of the rectangle, divided by the dimension we know, will give us the dimension we don’t know.
Now let’s take a rectangle, and draw a line from one corner to the far corner.
Notice in the picture on the left, we have created two triangles with the same base and height as the rectangle. If we divide it into four triangles, like in the picture on the right, the height of the triangle is cut in half. Therefore, if we keep the same base and height, each triangle will have exactly half the area of the rectangle. This gives us the formula to find the area of a triangle, which is base times height, divided by two.
If we have to do the triangle formula backwards, again we have to start with the area. When we were going forward, the last thing we did was to divide by two. Therefore, if we are going backwards, the FIRST thing we have to do is MULTIPLY by two. THEN we divide by the dimension we know, that will give us the dimension we don’t know.
Let’s try a few questions to see how well we’re doing.
1. A rectangle has an area of 6500 square feet, and a depth of 130 feet, what is the front footage?
2. A triangle has a base of 240 inches, and a height of 30 feet, what is the area?
3. A triangle has an area of 20,000 square feet, and a base of 200 feet, what is the height?
The answers follow.
1. 6500 divided by 130 equals 50 feet.
2. 240 divided by 12 gives us 20 FEET. THEN, 20 FEET times 30 feet, divided by 2 equals 300 square feet.
3. 20,000 times 2 gives us 40,000, divided by 200 (the dimension we know), equals 200 feet.
The next shape we’ll learn about is a trapezoid. A trapezoid is a four-sided shape with two sides that are parallel (meaning they run in the same direction), and two sides that are not parallel (meaning they don’t run in the same direction).
Because two sides are not parallel, that means the two parallel sides will have a different length. These sides can be called Base 1 and Base 2, and are labeled in the drawing above. The first thing we need to do is average Base 1 and Base 2 together. Remember, to average things together, you have to add the things you are averaging, and divide by the number of things you are averaging. Here we are averaging TWO sides together, so we have to add them together and divide by TWO. Once we’ve done that, then we multiply by the height to get the area. The height in this case is NOT the length of either side. The height is the shortest distance between the two parallel sides, in other words, straight up. The height is shown in this drawing by the dotted line. When we put it all together, our total formula will look like this:
To give an example, if we had a four-sided shape with two parallel sides of 30 feet and 20 feet, and a height of 10 feet, we would add 20 + 30 to get 50, then divide by 2 to get 25, and then multiply by the height of 10, which would give us 250 Square Feet.
Now we’ll learn how to find the area of a circle. We need a few definitions to be able to accomplish this. The width of a circle is called the DIAMETER. This is the distance from one side of the circle to the other, through the center, and is indicated by the blue line on the circle below.
Half the width of the circle is called the RADIUS. This is the distance from the center of the circle to the outside, and is indicated on the circle above by the red line. The radius is one-half of the diameter. The symbol for radius is a small “r.” To satisfy your curiosity, the outside of the circle is called the circumference, which is in black on our circle above. If you divide the circumference by the diameter, that will give you π (pronounced like “pie”). (The symbol may not show up properly in all browsers, so if it doesn’t look right on your computer, sorry) π is a Greek letter, and when used in math is equivalent to approximately 3.14. For our purposes, that is what we will use, that is, π = 3.14. To “square” a number means to multiply a number by that same number. For example, five squared would be 5 x 5, or 25. Ten squared would be 10 x 10, or 100. When a number is squared, this is indicated by a small 2 in the upper right hand corner. You may occasionally see a small box in the upper right hand corner instead. The formula to find the area of a circle is:
This means your first step is to find the radius. If the question says, “The radius is 24 feet,” then congratulations, you’ve found the radius! If the question says, “The DIAMETER is 24 feet,” then you have to divide by two to find the radius, which would be 12. Once you know the radius, you have to square it. For example, 12 x 12 would be 144. After you do that, THEN multiply by π, in our example giving us 144 x 3.14 = 452.16.
You may also see the formula written as:
THIS IS THE SAME FORMULA! Whenever two symbols are placed next to each other in math, it is assumed that you are multiplying. Therefore it is not necessary to put the multiplication symbol in between them.
While we were going over basic Area problems, we did a few formulas backwards. When you are working on problems, don’t forget that the ORDER THAT YOU DO THINGS IS IMPORTANT! 1 + 7 x 2 + 3 is not the same as 1 + 7 + 3 x 2. (Try it if you don’t believe me.)
The last thing we need to deal with is that midnight train to Georgia (I did mention that it’s late when I do these, didn’t I?). Let’s try the question again. If you got it, you can move to the next page now. If Gladys puts three Pips on a midnight train, and one gets off every 80 minutes, how many will be on the train if it arrives in Georgia at 2:30AM? (Find an answer before you read further.) This is mostly just a matter of paying attention, which is another lesson that you really need to learn. Eighty minutes is one hour and twenty minutes. If the train leaves at midnight, then the first Pip gets off at 1:20AM. The next one would get off at 2:40AM, so at 2:30, there would still be 2 left on the train.