How to Find Area | Rectangles, Squares, Triangles, & Circles | Math Mr. J

welcome to math with Mr Jay [Music] in this video I'm going to cover how to find the area of rectangles squares triangles and circles now remember area is the amount of surface something covers a shape covers we're going to start with rectangles move to squares then triangles and end with circles let's jump into rectangles starting with number one where we need to find the area of the rectangle for finding the area of a rectangle we can use the formula area equals length times width I used a cursive L for length so it doesn't look like a one not a huge deal though either way as long as you know it's an L when you write out the formula now all we need to do is plug in the length and width and then multiply I'm going to use the longer side for the length so those six meters and then 3 meters for the width so area equals 6 meters times 3 meters and now we can multiply 6 times 3 gives us an area of 18 square meters we measure area in square units so this is square meters before we move on to number two let's take a look at why we do length times width and look at a visual of area we have six meters by three meters so let's draw these out starting with 6 meters so I'm going to Mark each meter here so 1 two three four five and six so we have six meters there and now we can do the width so three meters one two and three you'll notice that we have squares now these are square units and specifically for number one those are square meters that just means that each Square is one meter by one meter and this is a square meter when we find area we find how many square units something covers so for number one how many square meters does this rectangle cover well one two three four five six seven eight nine ten eleven twelve thirteen fourteen fifteen sixteen Seventeen eighteen which is exactly the same thing we got by doing length times width so by drawing all of that out this helps us visualize area and square units but it's much quicker just to do length times width let's move on to number two and for this one we're just going to use the formula so area equals length times width so let's plug in the length we're going to use 19 feet so 19 feet times the width of 5 feet so again I used the longer side for the length now for number two it's a little different than number one because all four side lengths are written out that doesn't mean we need all four side lengths we still just need the length and width we're not multiplying everything that's written out or given so something to keep in mind now we can multiply so 19 times 5 gives us an area of 95 Square feet so there's how to find the area of a rectangle let's move on to finding the area of a square now we're going to take a look at how to find the area of a square let's jump into our examples starting with number one where we need to find the area of that square now four squares we can find the area by squaring a side length so we can use the formula area equals a side length squared and for this Square each side is 8 inches so let's plug in eight inches here for our side length so eight squared 8 squared means eight times eight which gives us an area of 64. Square inches let's move on to number two where we have a square with sides that are 11 meters and you'll notice that we only have one side labeled that's because this is a square and we know all of the sides are going to be the same they're all going to be 11 meters the tick marks on each side of this Square show us that they are all the same we have one tick mark on each side and that indicates that again all of the sides are the same then we have the 90 degree angles marked as well I wanted to include these symbols in case you come across them that way you'll know what they mean as far as area let's use the formula area equals a side length squared and each side is 11 meters so let's plug in 11 meters for the side length so 11 squared that means 11 times 11 that gives us an area of 121 [Music] square meters so there's how to find the area of a square let's move on to finding the area of a triangle now let's take a look at finding the area of a triangle we can either use the formula base times height divided by 2 or 1 half times base times height dividing by 2 and multiplying by one-half are the same thing they're going to give us the same result so either formula will work and give us the correct area think about it if we want to take half of something we can either divide by 2 or multiply by one half we will talk about why taking half of something relates to the area of a triangle in a little bit now for this video I'm going to use the formula base times height divided by 2. but again either formula will work and they both mean the same thing we are taking half a of the base times the height let's jump into number one where we have a triangle with a base of eight feet and a corresponding height of six feet we do not need this 10 feet right here in order to calculate the area we have area equals base times height divided by two let's plug in the base and the height the base is eight feet so we have eight feet times the corresponding height of six feet so eight times six divided by two now we can work through this eight times six gives us 48 and we have 48 divided by 2 here 48 divided by two gives us an area of 24 square feet so our final area for this triangle again 24 square feet now as far as why we take half of the base times height either by dividing by two or multiplying by one half let's take a look right here now we have the same triangle base of eight feet height of six feet but the triangle is doubled so we have two of them and we can fit them together to create a rectangle so we have one triangle two triangles and since they are right triangles they can fit together to create a rectangle now if we aren't working with a right triangle the triangles can still fit together to create a parallelogram so this will still apply we can use the formula base times height to find the area of a parallelogram and if we are working with a rectangle we may also hear length times width so for the area of this parallelogram specifically a rectangle we have base times height let's write it over here we have area equals base times height area equals well the base we have eight feet times the height of six feet eight times six gives us an area of 48 square feet and that's the area of the rectangle now we can see that the triangle is exactly half of the rectangle so we can take half of the area of the rectangle to get the area of the triangle so that's why we take half either by dividing by 2 or multiplying by one-half let's move on to number two where we have a triangle with a base of 9 meters and a corresponding height of 4 meters so let's write our formula here so area equals base times height divided by two now we can plug in the base is nine meters times the height of 4 meters divided by two nine times four gives us 36 so we end up with 36 divided by two which gives us an area of 18 square meters now just to show again why we take half I'm going to double this triangle so let's make a copy of this triangle to create a parallelogram so we have the copy of the triangle right here it's not the best but we have a parallelogram right here so we have the base of 9 times the height of four that gives us an area of 36 square meters for that parallelogram and then we take half to find the area of the triangle let's move on to number three where we have a triangle with a base of 5 yards and a corresponding height of 11 yards now this one looks a little confusing as far as how it's drawn out but we are finding the area of this triangle right here the dashed parts off to the side are to show the base being extended out so we have the base right here and then this part extends out and we have the height going up from that extension so the dashed parts are to show the height so don't let it confuse you now we can calculate the area of this triangle so we have area equals base times height divided by 2. let's plug in the base and the height so the base is 5 yards times the height of 11 yards divided by 2. let's work through this now 5 times 11. is 55. divided by 2. that gives us an area of 27 and a half or 27 and 5 10. and this is square yards so final area 27 and a half square yards let's make a copy of this triangle as well to show the parallelogram that we took half of so this is the copy of the triangle and our parallelogram is right here this parallelogram has a base of 5 yards and a height of 11 yards 5 times 11 gives us 55. so that parallelogram has an area of 55 square yards we took half of that in order to find the area of the triangle so there's how to find the area of a triangle let's move on to finding the area of a circle now let's take a look at finding the area of a circle now the formula we use for the area of a circle is area equals pi r squared so pi times the radius squared so we do need to be familiar with both diameter and radius when we work with circles diameter is the distance all the way across a circle so it's a line segment that connects two points on the circle and it has to go through the center and then the radius is half the diameter so it goes from the center to any point on the circle and then lastly before we get into our example is pi which is the ratio of a circle circumference to its diameter Pi is a constant so it's the same for all circles now Pi is irrational it goes on forever so we are going to use use an approximate or rounded version of Pi 3.14 with that being said let's jump into our example where we have a circle with a radius of 9 centimeters now the first thing that I'm going to do is write out the formula for the area of a circle so area equals pi r squared then I'm going to plug in the radius so area equals pi times the radius which is 9 centimeters squared after that let's do 9 squared which means 9 times 9 not 9 times 2. that's a common mistake so remember 9 squared when you square a number that means multiply that number by itself so 9 squared means 9 times 9 which is 81. so we end up with pi times 81. now we can rewrite this as area equals 81 pi one way to write the area of a circle is to write it in terms of Pi meaning we can write our number in front of the pi symbol and put our unit of measure which is centimeters and when working with area it's going to be square units so we have area equals 81 Pi square centimeters so again one way to write our answer is in terms of Pi so we have our number and then our PI symbol followed by the unit of measure squared now another way we can do this is to use an approximate version of Pi like I mentioned earlier we're going to use 3.14 so pi is approximately 3.14 so that's squiggly equal sign there that means an approximate value unlike an equal sign which means an exact value so something to keep in mind when you see that so let's go through our process here and we're going to start by writing out our formula so area equals pi r squared next let's plug everything in so we have area is approximately and I'm using the approximately symbol since we are using an approximate version of Pi so 3.14 that we're using for pi times our radius of 9 centimeters squared so 9 squared is going to give us well we have 3.14 and 9 squared equals 81. so we end up with 3.14 times 81 and that's going to give us an approximate area of 254 and 34 hundredths Square centimeters and again that's an approximate area because we used an approximate version of Pi so that gave us an approximate area there in decimal form in comparison to this area over here which we wrote in terms of Pi so there's how we calculate the area of a circle given the radius let's move on to finding the area given the diameter for this example we have a circle with a given diameter of 12 feet now the first thing that I'm going to do is write out the formula for the area of a circle so area equals pi r squared next we can plug in our radius so area equals pi times now we are given the diameter so we need to divide that diameter by 2 to get the radius because remember the radius is half of the diameter 12 divided by 2 gives us a radius of 6 feet let's plug that in and that's going to give us 6 squared now 6 squared that means 6 times 6 not 6 times 2. when we square a number we multiply that number by itself so let's do 6 squared which gives us 6 times 6 which equals 36 so we end up with pi times 36. now remember just like in our first example one way to write our area is to write it in terms of Pi and again that means we can write our number in front of the pi symbol and then put our unit of measure so we can put 36 pi and this is square feet for our unit of measure so that's one way we can write out our area of that Circle another option we can use an approximate rounded version of Pi like I mentioned earlier we're going to use 3.14 so we can use this to get an approximate area of this circle in decimal form so again we're going to use pi is approximately 3.14 let's write out our formula which is area equals pi r squared and then we can plug in so area is approximately I'm going to use the approximate symbol there because again we're using an approximate version of Pi so we are going to get an approximate area let's plug in 3.14 and then multiply by our radius which is 6. squared and now we can calculate here let's do 6 squared so 3.14 times 6 squared which equals 36. and now let's multiply so 3.14 times 36 gives us an approximate area of 113 and four hundredths and this is square feet so that is an approximate area of that Circle so there you have it there's how to find the area of rectangles squares triangles and circles I hope that helped thanks so much for watching until next time peace