what's going on besties in this video we're going to be tackling AIT 7 mathematics when it comes to Perimeter circumference area and volume let's get started let's start off by exploring the concepts of perimeter area and volume as a whole to quantify a one-dimensional object we use a one-dimensional measure known as length for instance the length of a given line might be precisely 1 cm a standard unit of measure when it comes to length if we take that same line and we extend it perpendicularly by 1 cm this is going to transform into a two-dimensional shape specifically a square two-dimensional shapes are Quantified by a two-dimensional measure referred to as area since our initial line of 1 cm in length was extended by 1 cm to the right the resulting Square covers an area of 1 square cm a typical unit of measure when we are trying to figure out area now let's consider extending that two-dimensional shape up from its plane of 1 cm this new action creates a three-dimensional shape known as a cube to quantify a three-dimensional object such as this we're going to utilize the three-dimensional measure known as volume when we talk about volume we're quantifying the amount of three-dimensional space inside our object in which it fills so what is the volume of a cube given that our volume of Cube was formed by extending a square cm up into a third dimensional shape by 1 cm the volume of this cube is defined as 1 cubic cm a standard unit of measure when we are measuring volume to make it easier area is measured as square units and volume in cubic units square units results from multiplying two onedimensional units cimer * cimer often expressed in exponent notation as CM squar or CM raised to the second power conversely cubic units arise from the product of three one-dimensional units centimeter * centimeter * centimeter this is abbreviated in its exponent notation as cm cubed or CM raised to the 3 power let's break each one of these three concepts down further and apply them based on the guidelines of the te's so let's start by diving into the fundamental of geometry when it comes to Perimeter it refers to the total distance or length around a shape now you might wonder what does it mean to measure the distance around a shape well distance or length is the concept that exists in one dimension and is quantifiable in units such as centimeters inches or miles this means that perimeter is also a one-dimensional measure Quantified with units of length so when we talk about a shape's perimeter we're not just saying 10 we're saying 10 cm or it's not just 25 it's 25 miles the specifity of units is crucial when we're discussing perimeter but then what do you mean by around the shape what we mean by this is It's the absolute shortest path around a shape this is the distance that you cover if you were to trace a line around the shape's border or its Edge a helpful way to understand perimeter is to imagine yourself walking along the edge of a shape such as this Pentagon visualize yourself beginning at one end of the pentacon and walking all along the edge of the shape until you reach back to your starting point that complete distance that you cover is equal to the perimeter of the shape in our example we know that every side of the Pentagon measures 10 m transversing all five sides of this shape is going to equal a total distance of 50 m another insightful method to grasp the concept of perimeter is to imagine taking the shape like our Pentagon and separating it from its Corners you can unfold that shape into a straight line that length from the start of your line to the end of your line is your perimeter calculating the perimeter of polygons which are shapes that only have straight edges is pretty straightforward you simply sum the lengths of all the sides and the result of the polygon is the perimeter let's practice some examples of how this actually works we're going to start with a right triangle the triangle sides are 3 cm 4 cm and 5 cm to determine the triangle's perimeter we need to add the lengths of each individual side so we calculate 3 + 4 + 5 total gives us 12 CM and that is our right triangle's perimeter in our next example we have a hexagon a polygon with six equal sides this hexagon is described as regular meaning that every single side is the exact same length this actually simplifies our task because only one size length is actually given to us which is 4 cm we can infer that every single side along this hexagon is going to be 4 cm long instead of adding each side individually as we did in our previous example we can actually take a shortcut and use multiplication since all the sides are of equal length that's because multiplication is really just repeated addition to find the total perimeter we would simply multiply the number of sides that would be six by our length of the one side that we know which is 4 cm 4 * 6 is going to give us 24 cm which is the overall perimeter of our hexagon let's do a more complicated example this time we're looking at a polygon with six sides again but this one isn't regular that means that the lengths of each side is going to vary this example is a bit more challenging because the diagram only reveals the lengths of four sides leaving us with two sides that are unknown encountering incomplete information is very common in mathematics and also on the t's the strategy here is to leverage the information that we do have to deduce the information that we're missing here's how we're going to approach this pay attention to the two vertical sides that we have here on our screen 4 in and 6 in now imagine that you could slide these two vertical sides across the opposite side and both of those sides are going to equal the length of that missing side so as we know we have four and we have six so when you add these together we actually get a total of 10 we can apply this similar logic to our horizontal sides as well by shifting these lengths that we have available to us of 10 in and 5 in so if we slide those down we have 10 and we have five we're going to add those two totals together in order to get the horizontal side that we're missing so our missing side is 15 in by utilizing the lengths we knew we managed to deduce the lengths that were unknown to us with the knowledge in Hand of every s's length now we can actually calculate our perimeter so 10 which is our number up here + 4 + 5 + 6 + 15 + 10 gives us a total of 50 in which is the perimeter of our irregular polygon now let's step it up a notch by talking about how we calculate circumference and area of circles it's crucial that you familiarize yourself with the formulas for both of these as they are going to be fundamentally covered when it comes to the t's the formula to calculate circumference of a circle is going to be circumference is equal to pi multiplied the diameter in sh hand we have C is equal to Pi * D where C is our circumference and D is our diameter so you might be asking yourself what the heck is a diameter right so if we were to take a circle and we were to draw a line through the center of that Circle that line from one end of the circle to the other end of the circle that measurement is considered our diameter an easy sentence to remember this equation is Cherry Pi's delicious where C in our Cherry is equal to our circumference Pi is of course equal to Pi and the D and delicious is equal to diameter now how do we calculate a circle's area so the area which is the region enclosed within our circle's boundaries is area equal pi multiplied radius squared so what exactly is a radius so our radius is actually half of our diameter so as we know we drew a line down the center of our Circle in order to get a diameter we're just going to take the diameter and divide it by the number two in order to get our radius which is half the diameter an easy sentence to remember this equation is Apple Pi's R2 where the A and apple is our area the pi is of course our PI the r and our R is our radius and the two is going to be the square of our equation a crucial point to remember is that R squar does not mean 2 * R this is frequently misunderstood among te test takers who first learned to calculate area of a circle when we're examining both formulas closely their similarities become apparent each formula involves multiplying Pi by a specific Circle measurement to determine either circumference or area for circumference the calculation involves pi multiplied by the diameter while for Circle it's Pi multili the radius squared how do we distinguish between a diameter and a radius well as we know the diameter is the line that we draw across the center of a circle that divides a circle into two equal halves that line measurement from one end of the circle to the other end of the circle is our diameter sometimes the t's is going to give you questions that's only going to give you the radius well in order for us to figure out the diameter based on the radius is we're going to take the radius and multiply it by two because 2 * the radius is going to give us the full length of our diameter so when we're talking about area we're going to do something a little bit different when it comes to our radius we're going to actually Square our radius and when we talk about squaring it is not the same thing as 2 * R when we Square we actually multiply the number by itself so if our radius was 2 2 * 2 is going to equal 4 if our radius was 5 5 * 5 is going to equal 25 so that is the big key differences between diameter and radius when we're trying to figure out circumference and area let's take a look at an example of how this will actually be applied on the t's so starting with our first example we're given the exact same Circle and our circle is given us a radius of eight so when it comes to considering the circumference of a circle we're actually going to multiply the radius by two like we discussed before so in this case 2 * 8 is equal to 16 so we're going to plug that into our equation Pi is just a fancy way of saying 3.14 when it comes to your t's so you can automatically manipulate the pi to be 3.14 so we're going to multiply that by 16 and that is going to give us our circumference of this circle as being 50.2 4 M next we're going to figure out the area of the exact same Circle so again we know that the radius is 8 but instead of multiplying it by two like we did in circumference this time we're going to square it so 8 * 8 gives us 64 M squared we're going to multiply that by pi which is 3.14 and that is going to give us our total area of 200.96 M squared now that we have a better understanding of one-dimensional values of perimeter and two-dimensional values of area when it comes to a circle let's explore other shapes when it comes to polygons and area so to grasp this concept of area we imagine that that 1 M line and length is going to be dragged across perpendicularly to create another another 1 M line when we do this we create a two-dimensional shape in this case it's a square so when we talk about area we're trying to figure out the measurement within this shaded region when it comes to our different polygons so we're going to start by trying to figure out the measurement for three shapes squares rectangles and triangles to determine the area of a square or a rectangle we simply multiply the dimensions of its side typically we're only measuring two sides which would be our length times our width so if we have in our first example we have a square that has two sides of 2 cm each so we're going to multiply length * width 2 * 2 gives us 4 cm squar so the area of this square is equal to 4 cm squar just like with our Square we're going to use the same formula when it comes to our rectangle so in this case we have a length of 4 cm and we have a width of 2 cm so we're just going to multiply length * width WID 4 * 2 is going to give us 8 cm sared which is going to give us the overall area when we're trying to consider our squares and our rectangles now how do we figure out the area of a triangle so we're going to begin with a rectangle once more and this time we have a dimension of 3 cm by 4 cm using our previously discussed formula the area of a rectangle is calculated as 3 * 4 is equal to 12 s CM now let's imagine that we're slicing this rectangle in half diagonally from one corner to the opposite corner this action creates two triangles each occupying half of the area of our original rectangle given that the rectangle's area is 12 square cm the area of each triangle would be half of the overall area that we saw in our rectangle meaning that each triangle is 6 square cm this Revelation leads us to the formula for the area of a triangle being essentially half of the area of a rectangle so instead of area equals length time width the formula of triangle modifies to area equals 12 of length time width however there's a slight adjustment that we need to note for our triangle dimensions we actually refer to them as base time height instead of length time width here's the reason the labels length time width are suitable for a right triangle as a right triangle forms precisely 1/2 of our rectangle however these terms don't fit as well with other triangle types like we see with acute and obtuse triangles where it's less clear which side should be labeled what in the case of these triangles we adopt a different approach we select one of the three sides as our base the choice of the base is really up to you and often the base is predetermined in mathematical problems anyways and once we select That Base we envision placing that triangle on the ground with its base being horizontal now we have to figure out what is going to be our height we do this by finding our triangle's Peak also known as the Apex of our triangle once we find that we're going to highlight that and draw a little line down until we hit the base that is going to be the height of our triangle in certain cases such as with a cute triangles the line is going to fall within our triangle boundaries but for obtuse triangles this height line extends outside of our triangle's perimeter regardless of its position the formula to calculate the area of a triangle is going to remain the same consistently area equals 12 of our base multiplied by our height let's take a look at a couple different examples when it comes to triangles we're going to start off with our acute triangle so as we know our acute triangle formula is going to be2 times our base times our height because as we know any triangle is basically 1/2 of a rectangle so we have our base is equal to 5 m and our height is equal to 8 m we're just going to go ahead and plug in play we've got 5 * 8 is going to be equal to 40 40 m squar of course because we're dealing with area and then we are just going to divide that overall number by two so the area of our acute triangle is going to be 20 M squared now let's take a look at our obtuse triangle so again using the same formula we we have a base of four and we have a height of s so we're going to plug those numbers into our equation 4 * 7 is equal to 28 in squared and we're just going to take that number and divide it by two so the area of our OB triangle is equal to 14 in squared now let's talk about parallelograms and trapezoids because they are most commonly tested on your te's starting with a parallelogram it kind of resembles a rectangle but it lacks those four right angles typically that you would see when it comes to our rectangles to determine a rectangle's area we typically use the formula area equals length time width we apply a similar approach when we're looking at parallelograms so let's imagine taking a segment of our parallelogram and sliding it across the other side of our shape fitting it in like the perfect puzzle piece so if we take down here one part of our parallelogram which is shaped like a triangle and then add our triangle here down to the end of our parallelogram this rearrangement is going to transform our parallelogram into a rectangle without altering its Dimensions therefore we can still utilize the formula area equals base time height to calculate the area of our parallelogram following those same principles that we see with our rectangle so in this example we have a base of five and a height of two we're just going to go ahead and plug those numbers into our equation and that is going to give us the overall area of our parallelogram is 5 * 2 is equal to 10t squared now finishing off this area section with a trapezoid you might have noticed that the formula for calculating this trapezoid Bears a remarkable resemblance when it comes to triangles the area of a triangle can be expressed as base time height / 2 however for a trapezoid the formula adapts to include both bases base 1 plus base 2times our height is all divided by two let's explore why that is a trapezoid can effectively be segmented in into two triangles by drawing a diagonal line from the bottom of one end to the top of the other end depicted by the red line that I just drew for you this division splits the trapezoid into two triangles the bases of these triangles align with the lengths of the trapezoids top and bottom edges and their assured height is identical to the trapezoids vertical height in order to find the area of a trapezoid we could calculate the area of each triangle separately and then sum them up together but to save ourselves from that frustration this step can be streamlined by simply adding both bases together base one and base two multiplying that outcome by its height and then dividing it by two so taking a look at our example that we have here we know that we have a base of eight a base of four and our height overall is six so we're just going to add our two bases together multiply it by its height and divide it by 2 so 8 + 4 multipli by 6 gives us 72 and then we're just going to divide that by two to give us the overall area of our trapezoid as being area is equal to 36 in squared so let's take a look at one more example of a complex shape so here we have a heart and we're trying to figure out what the overall area of that heart is so Step One is we want to identify the shapes and formulas that we're going to use in order to find the area of our heart so we have a couple different shapes we have have a square which is this area that's right here and then we have a circle formula which are these areas right here step two is we're going to start with our easier shape we're going to start with our square and figure out the area for that so we know that the square area is length times width so we have a length of four and we also have a width of four so we're going to multiply four * four and that's going to give us the overall area of our Square as being 16 cm squ so step three is we want to calculate the next shapes formula so as we know we have two semicircles along the edges of our Square so we need to figure out what is going to be the area of our semicircles and all we do is we take our equation when it comes to area of a circle and then we divide it by two in order to get the overall area of a semicircle so starting with our equation we have apple Pi's R2 which is a is equal to pi r R 2 that's how we're going to figure out the area so in this particular case we know we have a diameter of four and we know that in order for us to determine the radius is we have to divide that diameter by two so we're going to divide by two and that is going to give us two as our radius so now we just need to plug it in so a is equal to PK * 2^ 2 we know that 2^2 is equal to 4 4 * 3.14 which is the equation of a pi is equal to 12.56 now we need to divide this number by two because we don't have a full circle we only need half the circle so as we divide by two that is going to give us overall area of our semicircle as being 6.28 cm squared and finally in step four we can add all of our areas together we have our two semicircles so each one was 6 . 28 this is semicircle 1 semicircle 2 and then our last shape number three which we know was our square is equal to 16 if we add 6.28 + 6.28 + 16 is going to give us the overall area of our irregular shape as being 28.57 cm squared now let's talk about volume as we discussed before volume quantifies the amount of three-dimensional space an object fills we create three-dimensional objects by extending a two-dimensional shape up and height from its plane to create a three-dimensional object starting with square and rectangle projecting these shapes into the third dimension yields a three-dimensional figure known as a square prism or cube and a rectangular prism similarly if we extend a triangle into its Third Dimension it's going to produce a triangular prism to determine the volume you can simply calculate the area of an initial two-dimensional shape and then multiply that by the height in which you projected it so if you can understand the area when it comes to squares and rectangles we know that volume's going to be really easy because the only additional thing that we are measuring here is the height so in this case we have length * width time height will give us the volume of squares and rectangles in our example we have four * 3 * 10 is going to give us the overall volume of 120 cm cubed next let's calculate the volume of a triangular prism just like with the area of the triangle we multiply half of our base times our height but again we've added an additional unit of measure which is our length in this situation so if you can remember the area of a triangle it's going to be really easy for you to remember the area when it comes to a triangular prism because the only thing that we are adding is that additional unit of measurement which is our length so in this case we have our base that is 10 and our height that is eight length that is 50 so we are going to multiply 10 by 8 by 50 and we're going to divide that by two just like we would do with area but this time our volume using our new equation is equal to 2,000 in cubed another easy three-dimensional shape to remember the formula for is going to be our cylinder so again the base of our cylinder is just a circle so how do we find the area of a circle well apple pies are two right so area is equal to Pi R 2 just like we see here at the beginning of our formula but because we made this into a three-dimensional shape we're adding or I should say multiplying one additional unit of measurement and that is our height so we're going to take the formula that we have for an area of a circle and we're going to multiply that by its height so in our example here we have a radius of two so we just plug that into our equation and we have a height of 10 10 m so 2^ 2 is equal to 4 M 4 * 10 is = to 40 * pi which is again 3.14 is going to give us the overall volume of our cylinder as being 1256 cm cubed so not all volumes are going to extend into a perfect polygon or circle in terms of height the tease is going to test you on three additional shapes cone rectangular pyramid and sphere that are not going to follow all of those same rules when calculating the volume of a cone and comparing it to a cylinder of identical radius and height an interesting relationship actually emerges the cone's volume is precisely one3 that of the cylinder to visualize this imagine placing a cone inside a cylinder where both the radius and the height are the same this setup reveals that the cone occupies 1/3 of the cylinder's volume this phenomenon is why when determining the volume of a cone we apply that same formula that we have for a cylinder but we are going to divide the result by three this adjustment to the formula accounts for the cone's proportional volume relative to that of a cylinder so taking a look at our example we have volume is equal to < * R 2 * height / 3 so we know that our radius is five and our height is 12 so we're just going to plug in our numbers so this is going to give us < * 5^ 2 * 12 is / 3 5^ 2 is equal to 25 and we're going to multiply 25 by 12 * pi and that is going to give us 300 Pi all we're going to do now is we're going to divide that total number by three and that is either going to give us 100 Pi or 314 in cubed remember the t's might test you on both of these examples as with cones the same concept is also going to apply when it comes to rectangular pyramid shapes the base of our shape is just a rectangle and the volume of a rectangle pyramid is precisely 13 of a rectangular prism when we're trying to determine the volume we apply that same formula for a re angular prism and we're going to divide the total outcome by three so taking a look at our equation we have volume is equal to length * width time height ided 3 we have a length of eight a width of seven and a height of 11 so we're just going to plug in our numbers 8 * 7 is equal to 56 * 11 is equal to 6166 and then we're just going to divide that overall outcome by three which is going to give us a total volume of 25.3 3 in cubed now finally we have the volume of a sphere it's difficult to explain the volume when it comes to a sphere without getting into a whole bunch of calculus and how this formula is determined to save ourselves some time and frustration I use a sentence to remember this equation four spheres we find the space inside with 4/3 piun * R cubed applied so looking at our formula we have volume is equal to 4/3 * piun * R cubed remember anytime that we're dealing with volume when it comes to Circle we're going to cube the radius any times we're dealing with area when it comes to circles we're going to square the radius so again we know that we have a radius of 10 in so we're just going to go ahead and start plugging in so we have 4/3 * piun * 10 cubed we know that 10 cubed is 1,000 we are going to times that by 4/3 and that is going to give us either two answers depending on how the t's list it it's either going to be 13333 Pi or 4,186 six and that's it besties I hope that this information was helpful in understanding everything you need to know when it comes to te's math perimeter area and volume as always if you have any questions make sure that you leave them down below I love answering your questions head to Nur Chun store.com there's a ton of additional resources to help you pass and Ace those ait's exams and as always I will catch you in the next video bye