consider the composite figure shown on the screen how can we find its area in this video we're going to focus on finding the area of composite figures that contain semicircles now if you have a rectangle with length L and W width W the area of the rectangle is simply length time width now for a circle the area of a circle is pi r 2 for a semicircle the area is going to be half of that so it's 12 pi r 2 likewise for a quarter Circle the area what just happened here is going to be 1/4 of a full circle so it's 1/4 r^ 2 so using these formulas we can calculate the area of this particular composite figure so for the rectangle it's simply left time width 6 * 10 and that's going to be 60 square ft now let's focus on the semicircle portion of the figure what we need to determine is the radius of the semicircle the radius is going to be the distance from the center to any end point on a semicircle so this value will be R now if this entire distance is 6t R is going to be half of that so R is 3T and the area of the semicircle is going to be 12 pi r 2 so that's half of piun * 3^ 2 3^ 2 is 9 divided by or half of nine that's 4.5 Pi so you have 3 squ which is 9 if you divide 9 by 2 you get 4.5 so now the total area is going to be the area of the rectangle which is 60t plus the area of the semicircle which is 4.5 Pi now that is the exact answer but if you want to get a rounded answer you can plug this in your calculator and you can replace Pi with 3.14159 so replacing Pi with that figure we we get an approximate area of 74137 square ft so that's going to be the total area of this particular figure now let's move on to our next example if you want to try this problem feel free to pause the video and give it a try so we could immediately determine the area of the rectangle it has a length of 16 and a width of five so 5 * 16 we can multiply it as 5 * 10 + 6 10 + 6 is 16 5 * 10 is 50 5 * 6 is 30 50 + 30 is 80 so the area of the rectangle is 80 square cm now in order to determine the area of the semicircle we need to determine the radius how can we find the value of the radius now we know this part is 16 and this part is six so the diameter of the semicircle is going to be the difference of 16 and 6 16 - 6 is 10 so this part is 10 cm that's the diameter the radius is going to be half of that so the radius is 5 cm the radius is always half of the diameter so now that we know let me just put a this is the center of the semicircle this part is five this is five 5 5 5 and six adds up to 16 so now that we know the radius of the semicircle is 5 cm we can calculate the area of the semicircle which it's 12 < R 2 so we're going to plug in five for the radius 5^ 2 is 25 and half of 25 is 12.5 so the area of the semicircle is 12.5 piun square cm so now we can find a total area of the entire figure so it's going to be the area of the rectangle which is 80 square cm plus the area of the semicircle which is 12.5 Pi square cm we can leave our answer like this which is an exact answer or we can get a rounded decimal answer so replace in pi with 3.14159 we're going to get 11.3 square cm so that's it for the second example that's how we can calculate the area of this particular composite figure for those of you who want to quickly access my math and science video playlists feel free to check out the website video dtor you'll find playlists on algebra geometry trig pre-cal calculus General chemistry organic chemistry physics statistics and other topics as well and you can also access access my final exam review videos on this website in addition to my test prep videos and there's some other links that you can explore here as well so feel free to take a look at that when you get a chance now for our next example what we have here is a quarter circle that has been cut out of a square knowing this information go ahead and calculate the area of this figure so if we were to extend this part we could see this quarter Circle and this right here would represent the radius of that quarter circle now if this is 18 and this is N9 that means this part has to be N9 and you could see it from the other side as well if this is 18 and this is 9 this has to be 9 as well so what that tells us is that the radius is 9 in so how can we get the answer for this problem in this case what we need to do is we need to use subtraction instead of addition the total area is going to be the difference of the area of the square that is this square right here and once we have the area of the square we need to subtract the area of the quarter Circle that will give us the remaining area of the figure so it's going to be the area of the square minus the area of the quarter Circle the area of the square if we call this s and this s is simply s s you can also use the area of a rectangle formula length times wi and you'll get the same result now the area of the quarter circle is going to be not 1/2 but 1/4 of the area of a full circle so it's 1/4 PK r^ 2 so this is going to be s^2 which is 18 2 - 1/4 piun and the radius of the quarter circle is not now 18 s 18 * 18 is 324 and 9^ SAR is 81 so the exact answer is going to be 324 minus 81 over4 Pi with the units Ines squared but now let's get our red answer so if you plug in 324 minus 81 Pi over4 in your calculator you should get 2604 if you round it to the nearest 10th Place Square Ines so that's going to be the area of the Shaded region or the composite figure now let's move on to our last example of this video and we have a composite figure that has a similar shape to a cupcake feel free to try this problem go ahead and calculate the area of this composite figure so we can see that the diameter of the semicircle is 16 units that means that the radius is going going to be eight now what we have here is a trapezoid we can calculate the area of the trapezoid or we can break it up into two triangles and a rectangle which I'm going to see it that way now let's focus on the triangles both of these are right triangles and notice that this is 10 which means this portion is 10 so I'm going to redraw the semicircle so this part is 10 which means this part is 10 so I'm going to put a 10 right here now the total of this segment is 16 and this is basically a symmetrical figure or at least we could assume that is symmetrical otherwise we can't solve it therefore if this is 10 and the entire thing is 16 the remainder must be six if we split six in half that means this part is three and the other part is three so this part is three this is three if this part is eight that means this SE this section here is five and this is five but the radius is still 8 it's 5 + 3 so I'm just going to put R is equal to 8 now focusing on this triangle this is three this is five do you know what the missing side is this is going to be a 345 right triangle if you know your special right triangles they are 3 45 5 12 13 7 24 25 and there's also the 8 15 17 now if you didn't know your special right triangles you can solve for this missing variable using the Pythagorean theorem so if this if we call C the hypotenuse and 3 is a and the missing side is B we could use the formula a 2 + b^ 2 is equal to c^ 2 a is 3 we want to calculate b c is 5 3^ S which is 3 * 3 that's 9 5^ 2 which is 5 * 5 that's 25 subtracting both sides by 9 we get that if we bring this down B ^2 is 25 - 9 which is 16 taking the square root of both sides the square root of b^2 is B the sare otk of 16 is four so that's how we can show our work for that but if you know your special right triangles if you have three and five you know the Miss side is going to be four so this part here is four and this is 4 as well so now we have everything that we need to get the area of this composite figure so focusing on the rectangle we have a length of 10 a width of four so that gives us an area of 40 square units now for this triangle the area of a triangle is 12 base time height the base of the triangle is 3 units long the height of the triangle is 4 units it really doesn't matter the order in which you write it you'll still get the same answer 3 * 4 is 12 half of 12 is six so each of these two triangles will have an area of six square units now let's focus on the semicircle the area for that is going to be 12 pi r 2 and R is 8 8^ SAR or 8 * 8 is 64 half of 64 is 32 so the area of the semicircle is 32 Pi so now we can calculate the total area by adding up the individual areas so it's going to be 40 + 6 + 6 + 32 Pi 6 + 6 is 12 + 40 That's 52 so we get 52 + 32 Pi square units since we weren't given a unit for these values in this problem which is going to say square units now let's get the decimal equivalent of this answer so 52 + 32 piun is approximately 152 5 square units so that's basically it for this video now you know how to calculate the area of composite figures that contain semicircles