 Hi, I'm Zor. Welcome to Unisorification. Today I will talk about area, area of different simple shapes in plane geometry. This is kind of continuation of the topics which I started with introduction of what is basically the measurement, what is the unit of measurement, etc. So we will concentrate on plane objects on the two-dimensional plane. Alright, first of all, let me just remind very briefly what I was talking in the introduction, which is basically that we do need a unit of measurement, and unit of measurement of the area on the plane is some kind of a square, the side of which is equal to one, and so the area of which we just establish as being equal to one. Well, establish is not really the right word, probably we set it to be one. One square something, if this is the one meter, it will be square meter, or any other unit of measurement. So this is basically the definition of the unit of measurement. And the second thing which I was talking about was that if you have any rectangle and linear measurements of its sides, r, a, and b, then using certain logic, basically I show that the area is equal to the result of multiplication of these two numbers. You can say this is the length, this is the width, or whatever other names you want to call these two sides, but anyway, two sides of rectangle multiplied in their linear measure will give you the area of this rectangle. So I will use this particular fact and I will also use something which I again mentioned before. If you have congruent figures, then their area is the same. Now, using these two facts, the fact about the area of a rectangle and the fact that congruent figures have the same measure, the same area, we will talk about specific objects in plane geometry and we will talk about the kind of area they actually have. The object number one is parallelogram. Now, how to calculate the area of parallelogram? It's not a rectangle which we know how to put our squares. However, what we can do is we can use the fact that congruent figures have the same area. So what I will do is the following. I will drop a couple of certain decolours. So this is a, b, c and b. This will be x and this will be y. So I dropped two perpendiculars from b and c. What's important right now is to notice that these two triangles, a, b, x and b, c, y are congruent. Now, y is very simple. First of all, these are perpendiculars which means that both triangles are right triangles. Now, they have the same leg which is basically the distance between two parallel lines. This distance is always the same wherever I measure this. And also they have congruent hypotenoses, a, b and c, d because they are sides of parallelogram. So therefore, we can say that triangles are congruent. Their areas are the same. Okay, that's actually very important because if you will consider the area of a, b, c, d parallelogram and then you subtract from this this particular triangle which will reduce its area and instead at this triangle c, d, y to increase the area then you will see that we will get a rectangle. Now, this rectangle has exactly the same area as original parallelogram because I reduced the area by the area of this triangle and then added the area which is exactly the same because triangles are congruent. So the area of this rectangle is equal to the area of parallelogram. So this is the property which I'm going to use. Now, what is the area of this particular rectangle? That we know. It's basically multiplication of side by side. So if we are talking about parallelogram then one side of this rectangle is the same as the side of the original parallelogram, right? That was the original parallelogram. So bc is shared as a side by parallelogram a, b, c, d and rectangle x, b, c, y. So one side is this. Now, what is another side of this rectangle, the b? Well, that's actually the height. That's the height of our parallelogram which actually brings us to the formula that the area of the parallelogram is equal to the result of the multiplication of side by the height. Well, what's interesting is what if you start from another side and you will drop perpendiculars instead of this way instead of dropping from this parallel to this parallel you will drop from here to here. So let me draw it with a different color. You drop this perpendicular and this perpendicular. Now, obviously these two triangles a, d, x and b, c, y are exactly the same that are congruent by exactly the same logic and so the area of the same parallelogram would be in this particular case the same as area of this rectangle which means it's the result of multiplication of this side of the parallelogram let's call it whatever m by this height m. So it looks like exactly the same area of the parallelogram we can express either the result of the multiplication of one side multiplied by the height between this side and the parallel line or which is a different formula the result of multiplication of this side which is different from this it's a parallelogram this side by the height which is between these two parallel lines so it looks like no matter how we calculate we take as a base, so to speak and which height we take as another component of the multiplication we will get exactly the same result so whenever you have a parallelogram either you multiply this side by this height or you multiply this side by this height the result should be exactly the same and that's by the way one of the interesting properties of the parallelogram the result of multiplication of this times this is the same as this times this so a times h is equal to b times c or if you wish a divided by b is equal to c divided by h which means that proportion between a and b is reverse to the proportion of corresponding heights the height to the a goes to the denominator and the height to to the b goes to numerator well, all the different variations of how we present this particular property are equivalent so the most important thing is that the error of the parallelogram is equal to multiplication result of the multiplication of the lengths of one of the sides times the lengths of the altitude towards that particular side next is a triangle I will use exactly the same property of equal or rather congruent geometrical figures have equal area, right? so, what I will do here I will draw a parallel line to this and parallel line to this now, what I did I basically converted my triangle a, b, c into a parallelogram a, b, b, c now if this is parallel to this and this is parallel to this then obviously these two triangles are congruent because they have a common side and then this angle is equal to this one because these are parallel and the one which is crossing them and what else and this angle is equal to this one because these two are parallel all right so, we have angle side angle angle side angle, okay so triangles are congruent which means their areas are exactly the same and now what I can say is that I can choose this particular base and this particular height so I have a base of a triangle but now this if I will multiply a by h it would be the area of the parallelogram which is twice as big as the area of one particular triangle so the area of the triangle is this and again as in the case of the parallelogram you can start from any other base you can start from this base multiplied by this height this altitude or you can start from this base and multiplied by this altitude no matter how you do it it's always the result of the multiplication of the lengths of the one of the sides times altitude towards that side not any other altitude towards that particular side divided by two, that's the area of the triangle all right what else do we have we have trapezoid all right it's kind of equivalent more or less and all these proofs so to speak are really trivial and they are basically going towards exactly the same kind of solution now in this particular case since I have two bases parallel to each other then it's not really exactly the same where to start I really have to start from these two bases which are parallel to each other now let's draw a line parallel vx parallel to cg now since this is equal to this bc is equal to xb then this piece is b and therefore this piece is a-b to get the whole length to a right now if this is h the altitude of this trapezoid then what I can say is that the area of this trapezoid is a sum remember measure is additive which means two composed together objects have the area equal to sum of each component so let's calculate the area of triangle abx well as we know this is the multiplication of the base times the altitude altitude is exactly the same so it's a-b times h divided by 2 now the area of parallelogram bcdx is the result of multiplication of the side in this case we choose exactly this particular side times the same altitude so it's plus bh well if you will simplify it it would be a-b times h plus 2bh divided by 2 this is the common denominator multiplied by 2 and divided by 2 if you open these parenthesis it will be ah-bh plus 2bh divided by 2 which is minus bh and plus 2bh is plus bh so it's ah plus bh divided by 2 or a plus b divided by 2h which can be interpreted as this a plus b divided by 2 this is a and this is b so it's half the sum of two bases of the trapezoid so half the sum of the two bases and multiplied by the altitude and by the way I hope you will not forget that half of the sum of two bases is actually the length of the median of the trapezoid so you can always say that the area of the trapezoid is equal to the result of multiplication of the median line of the altitude and by the way this is actually can be this can be proven directly and geometrically not algebraically using the following logic let me draw another picture it will be better so if you have a trapezoid the big one now this is midpoints so this is the median line so this is equal to this this is equal to this now what I will do is I will continue this line here and draw perpendicular here from this point perpendicular to these two parallel lines and from this perpendicular to these lines now it's very easy to prove that these two triangles are congruent as well as these two triangles are congruent it's kind of obvious because we have this midpoint so it's two hypotenoses and you have an angle so same thing here two hypotenoses and the vertical angles so triangles are congruent so their areas are the same now what happens is from our trapezoid we subtract we cut off this particular triangle but we add this one we cut this triangle but we add this one and now, lo and behold we have a rectangle because this is perpendicular to this so it's a rectangle and the rectangle has the area of side which is the median line which we already basically have, this is the median line and times the height and height of this rectangle is exactly the same as the original trapezoid so you see using the different geometrical proof we came to exactly the same formula well as long as we know that the median line of the trapezoid will be equal as we go over 2 then we have exactly the same well that basically includes this lecture about basics of area now, speaking about basics in neither lengths basics or area basics I did not touch circle I didn't go into a direct calculation of what is circumference of a circle or area of a circle that would be a separate topic because it requires a careful approach if you don't mind it's not like straight forward like in the case of triangle or trapezoid or whatever else alright so basically this is a short introduction to basic shapes in geometry and their area all the more difficult stuff would be presented as problems and some theoretical lectures maybe later thank you and don't forget that if you will register to unizor.com you can take exams you can basically build an entire educational process and go from topic to topic which I definitely recommend thank you very much