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[Slide 1] 
[Dr. Shijin Kozhumal] Welcome to the second lecture on math review. In this lecture we'll look at area and volume calculations. This mostly you’ve already studied in high school and some of you in middle school. 
[Slide 2]
[Kozhumal] Area of a rectangle, it's nothing but the number of square units to cover the surface of figure. So if you have a box, if this is one unit- so if one small box is one unit, then all we need to do is take the number of units along the length and number of units along the width. So the total area is here 4 times 3, that's 12 square units. Always the area is measured in square units. You can do an example, area of rectangle with a length of 14 inch and a width of 9 inches, so you can get area of 126 square inches. 
[Slide 3]
[Kozhumal] You can do one more example. Here we have a width of 10 feet, 4 inches and a length of 25 feet, 6 inches. The first thing to do here is convert this to a decimal system. We know four inches should be 10 plus 4 by 12. That is 10.333 feet. Here it is 25 feet 6 inches, is one half feet, and then just multiply it. So you should get 263.5 square feet. 
[Slide 4]
[Kozhumal] And [Unintelligible] do an example with area of the rectangle with a width of 6.2 meter and length of 12.8 meter, all you need to do is just multiply length times width. So that should be 79.36 meters squared. 
[Slide 5]
[Kozhumal] We might need to calculate the area of circles. So typically like in pool fires, say if have a pool file for a given diameter, so how do we calculate the surface area of the pool fire? It's nothing but the area of the circle with the diameter. So the equations we need are: if diameter is known, this is the diameter, if this is known, the area of circle is pi (which is roughly 3.14) times diameter squared, divided by 4. Or if radius, that's half the diameter, if that is directly known you can use pi times radius squared. Whichever is convenient to you. 
[Slide 6]
[Kozhumal] Let's do some examples. Use pi is equal to 3.14 to find the area of the following circles. I would recommend pause the lecture here and try to work it out yourself and come back and look at the answers. So in case a, the radius is given here. The radius is given here also, the radius is given in inches. Here the radius is given in feet. Here the diameter is given in centimeters. Here the diameter is given in meters. So here the area is pi times radius squared, should be able to get this number. And here also, if you are using the equation radius squared, we need to convert this diameter into radius. Radius is diameter divided by two. So if diameter is 10 centimeters, radius is 5 centimeters, so area is 78.5 centimeters squared. Here radius is directly available, so 3.14 multiplied by the radius squared. Here the radius is in inches, so the area will be in square inches. Here the diameter is given in meters, so radius is 1.5 divided by 2, that's 0.75, so square of 0.75 multiplied by 3.14 and you will get the answer in square meters. Hope this is good enough. 
[Slide 7]
[Kozhumal] Let's look at area conversions. This might seem very obvious and straightforward, but many times people mess up. We discussed in the last lecture one meter is 100 centimeters, so one meter squared is not 100 centimeters squared, it's 100 centimeters times 100 centimeters or 10,000 centimeters squared. Say for example, look at pot width, which is 1 foot times 1 foot. We know that there are 12 inches in a foot. Assume there are like 12 such divisions. So up to 12. Same way along the width. There are twelve such divisions. So this whole pot is one square foot, and this more rectangle. Let me use a different color. This small rectangle is one square inch. So there are 12 of those in the first row and 12 in the second row, so total we will have 144 or 12 times 12. So we have 144 square inches. Same way if you are using centimeters. There will be 100 centimeters in a meter for the lengthwise and widthwise, so one meter squared is not 100 centimeters square: it is 10,000 centimeters squared. Same way, we can do the conversions with meter to feet. We know one meter is 3.28 feet. This you can find from conversion tables. So one meter squared should be 3.28 squared, that is 10.76 square feet. You can do the same conversion otherwise from square feet to square meters. As we discussed here, one foot is 12 inches, so one square foot is 12: 12 along this direction, 12 along this direction, so total 12 times 12, 144 square inches. 
[Slide 8]
[Kozhumal] Same way with volume. It’s the amount of space occupied by a three-dimensional object. We calculate them by multiplying length, width, and height. There's a third element coming into picture when it comes to volume. Formula is simple: length times width times height. Make sure everything is in the same units. Always you would end up with cubic units because it's multiplied by length, width, and height, three. This example, luckily here the length, width, height, everything is given in inches so all we need to do is multiply these three numbers: 5 times 5 times 2, so it should be 50 cubic inches or inch cubed. 
[Slide 9]
[Kozhumal] Let’s do one more example here in meters. All units are meters. If not, convert to meters first then multiply those numbers and you should get 273.6 cubic meters or meters cubed. 
[Slide 10]
[Kozhumal] Let's look at volume of cylinder. We know here it's slightly different from a rectangular prism. The top surface area of a cylinder is pi-r-squared, or pi divided by 4, D squared. So then we need to multiply that area by the height to calculate the volume. So the equation is pi times square of the radius times height. Or if the diameter is directly known, pi divided by 4 times diameter squared times the height. 
[Slide 11]
[Kozhumal] Let's do an example. The tank on the fuel tanker shown below is in the shape of a cylinder 2 meters in diameter, which is shown here, and 6 meters long. Find the volume of the tank in cubic meters. So luckily both the diameter and the length are given in meters, so we can use the equation pi divided by 4 times diameter squared times height. If you plug the numbers in the calculator, you'll get a volume of 18.84 cubic meters. And in many problems related to fire, we will need to use area of circle and cylinders very often. 
[Slide 12]
[Kozhumal] Same way as we discussed during the conversion of area, volume conversion there will be a third direction. We can divide the units along the length, along the width, it's very hard to draw in a two-dimensional surface the third direction. Also it should have the same divisions. So one cubic meter will be 100 centimeter for the length times 100 centimeter for the width times 100 centimeter for the height, so total 1 million centimeters cubed. We can use the same conversions for meters to feet. So 1 meter is 3.28 feet, roughly, so one cubic meter will be 3.28 cubic feet. That's 35.3147 cubic feet. You can do the same conversion backwards or from cubic feet to cubic meters. Same way, you know, one foot is 12 inch so one cubic feet is 1728. This is 12 cubed cubic inches. They seem very obvious but many times we end up making mistakes in area and volume conversion, so be careful. 
[Slide 13]
[Kozhumal] Sometimes when we deal with fire, we also need to calculate the surface areas. Say for example, what is the surface area of all surfaces enclosing a compartment? So how you do that is we can simplify things because most of the time if a room or compartment is in a rectangular shape, the top and bottom should be exactly the same. Same way the front and back should be exactly the same. Similarly, the left side and the right side both should be exactly the same, as long as they use rectangular shape. So we can simplify the equation. So one side should be length times width, we have the two of the same. See for example here if we use this as the length and this as the height and this as the width. So the top side is length times width and bottom is also length times width, so we have 2 times length times width. And the front is height times the length, and same with the back so we have 2 times length times height. And both the left and right, we have this is height and this is width, so width times height, same with the right side so we have 2 times width times height. So all we need to remember is this formula. 
[Slide 14]
[Kozhumal] So in this example, height is shown as 4 meters, the width of 5 meters, and length of 10 meters. It's a large classroom. Just plug in the numbers, we should get a surface area of the room, of the area of the walls, floor, and ceiling as 220 square meters. 
[Slide 15]
[Kozhumal] Let's look at a similar example problem, this time with a ventilation opening or door. So calculate the volume and surface area, excluding the ventilation opening. Volume is the same: length, 2 meter; times width, 7 meter; times height, 2.5 meter. That straightforward. And when it comes to the area, we need to calculate the area as we did in the last example: 2 times length times width, plus 2 times length times height, plus 2 times width times height, minus the area of the door. Here it is 2 meter times 0.8 meter. That is 1.6 square meters. So the calculated value minus 1.6 meter squared. You can do the calculation yourself and then look at the numbers. So you should get volume of 210 cubic meters and surface area of 261.4 square meters. Volume in cubic meters and area in square meters. Thank you. Hope this was useful. 
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