Unknown Area Problems on the Coordinate Plane 07:06 minutes

Video Transcript

Transcript Unknown Area Problems on the Coordinate Plane

Welcome to Grid City. Mr. Tad Dunlop, a real estate tycoon, is working on a project to build two new skyscrapers. But just as they are about to lay the foundation, the construction site manager reports that they’ve miscalculated the amount of cement needed. So it looks like Mr. Dunlop and the site manager are going to have to go back to the architect's plans for the project. This way they can figure out the area of composite shapes on a coordinate plane. Let's apply a coordinate plane to the architect's plans. The shapes represent the foundations of the skyscrapers. Each of the boxes on the grid represents an area of 100 square meters, which will be the scale for our grid. For squares and rectangles, we can find the area by counting the number of squares inside the shape. But for all other shapes, we're going to have to find another way to calculate area. Let's take a look at the plan for building 1. To figure out how much concrete is needed, we could just count the number of square units inside. But this could take a long time, and we can't count the partial squares accurately. So to calculate the area, we can’t just count the squares inside the shape. But what we can do, is break it up into smaller shapes that are easier to work with. When we combine smaller shapes to make one larger shape, we call the larger shape a 'composite shape'. If we divide here and here, we can create a rectangle, a triangle, and a half-circle. Okay, now let’s calculate the area of these individual shapes. To find the area of the rectangle, we first need to know the side lengths. In order to find out how long the sides are, we first need to look at the coordinates that make up the rectangle. Notice how the y-coordinates are the same here and here, so to find the length of this side, we simply need to find the difference in the x-coordinates. 10 minus 2 is 8, so this side length is 8. We also need the height of the rectangle. Finding this is easy! We just have to subtract the y-coordinates. Hey! Would you look at that! Both of the sides are the same length, which makes this rectangle a square. So to find the area, we just take our 8 units and square it. That gives us 64 square units. Now for the area of the triangle. We already know the base is 8 units long because it's the same length as the square's side. The height can be found by finding the difference between the y-coordinates of this vertex and the base. Since the area of a triangle is one half the base times the height, we can just substitute our known values and calculate the area. That gives us a total of 20 square units. For our final piece, let’s calculate the area of the semicircle. The area of a circle is pi, 'r' squared. So a semicircle, or half a circle, is one-half times that. The radius is equal to 4 units. So we square that, then divide by half to get 8π. Using 3.14 for pi, we get about 25.12 square units. Now, to calculate the area of the whole composite shape, just add these three areas together. So, the composite area of the foundation of building 1 is approximately 109.12 square units. There is one more building in this development. We can see that it has four sides, but it’s not a square or a rectangle. Can you see any simple shapes in this composite shape? Look at that! That looks like a trapezoid, and a triangle. The formula for the area of a trapezoid is one-half times the sum of the two bases times the height. That formula looks weird, right? But let's look at the pieces of the formula. Look at that! One-half times the quantity base 1 plus base 2. That's the formula for the average length of the two bases! In order to find out how long the bases are, we first need to look at the coordinates that make up the trapezoid. Notice how the y-coordinates are equal. To find the length of base 1, we simply need to find the difference in the x-coordinates. To find the length of base 2, we just have to repeat the same process. We also need the height of the trapezoid. We can get the height of the trapezoid by subtracting the y-coordiates of our two bases. 10 - 6 is 4, so our height is 4 units. Now to do the math. We substitute our base lengths and height into the equation. As always, parentheses first, then multiplication, then multiplication one more time. The area of the trapezoid is 40 square units. Let's look at the triangle next. The base of the traingle is the same as base two of the trapezoid. The height is established by subtracting the y-values, 4 from 6. We can substitute our base and height values. So the area of the triangle is 8 square units. Now we just need to add the area of the trapezoid with the area of the triangle to find the total composite area. Remember, since each of our units is 100 square meters, we need to multiply both our final composite shapes by 100 to find the total area of the foundation of the buildings. By combining all the shapes we know the total area. That results in 15,712 square meters to cover in cement. Now Mr. Tad Dunlop and his site manager have enough information to order the cement and it looks like it's just arriving to the building site. Hey that's a nice statue, but I didn't see it in the building plans. We might have to take another look at those numbers.