Finding the Area of a Rectangle with a Semi-circle
This video shows how to find the area of a rectangle with a semi-circle. To do this, you will need to find the area of the rectangle, then find the area of the half circle, and add them together. To find the area of the rectangle, you need to know the length and width and multiply them. The area of a semi-circle is 1/2 πr2.
Area of a Rectangle with a Half Circle
A tall window is in the shape of a rectangle with a half circle attached (that’s that picture right there). If the window is 3 feet wide, and its total height is 5 and a half feet, what is its area to the nearest square inch?
We’re going to start by pulling out our important information and putting it on our picture. It says the window is 3 feet wide. That means that this length (from here to here) is 3 feet, and the total height of the window is 5 and a half feet, so the total height (that means from the very top all the way down to the very bottom) is 5 and 1/2 feet.
It says, “What is its area to the nearest square inch?” Well, that’s important since all of our dimensions were given in feet. We’re going to have to do some conversions before we can do area, but before we even get to that, we need to figure out these different parts of our picture.
Our picture is made up of this rectangle and this half of a circle, so to find the area of the whole window, we need to find the area of the rectangle, then find the area of the half circle, and add those areas together. In order to find the area of our rectangle, we need to know what our length and our width is.
We know our width (that was given to us) it’s 3 feet, but what we don’t know is what the length is, since all we know is the total length, 5 and 1/2 feet. We can find it, since 3 feet is all the way across our rectangle, but it’s also all the way across our semicircle, and the segment that goes all the way across your semicircle through the center of the circle is the diameter.
That means that 3 feet is the diameter of our semicircle. If 3 feet is the diameter, then 1 and 1/2 feet is the radius, so that means that this distance here is 1 and 1/2 feet, but also this length is 1 and 1/2 feet.
If our total height is 5 and 1/2 feet, and it’s 1 and 1/2 feet from the top of the rectangle to the top of the semicircle, then we can subtract 5 and 1/2 minus 1 and 1/2, and we can find the length of our rectangle, so this distance (here to here) is 4 feet.
Now that we know both the length and the width of our rectangle, we can find the area of the rectangle. Again, we need to find it to the nearest square inch, so we’re going to start by converting. 1 foot is 12 inches, so 4 feet is 48 inches, (12 times 4, 48) and 3 feet is 36 inches (3 times 12 is 36).
Those are the dimensions we’ll use to find the area of the rectangle. Area of a rectangle is length times width, and our length and width are 3 feet and 4 feet, or 36 inches and 48 inches. The area is 48 inches times 36 inches, so the area is (48 inches times 36 inches is) 1,728 inches squared.
That’s the area of the rectangle part of the window, so now we need to find the area of the semicircle. The area of a circle is Pi times radius squared, but this is not a complete circle, it’s only half of a circle.
Therefore, it has half of the area of a circle, so the area of this semicircle is 1/2 times Pi times the radius squared. Since our diameter is 3 feet, we talked about the fact that our radius is 1and 1/2 feet, but again, we need our area to the nearest square inch, so we need to convert 1 and 1/2 feet into inches. 1 and 1/2 feet is, 1 foot is 12 inches, 1/2 a foot is 6 inches, so 12 plus 6 is 18 inches, so our radius is 18 inches. The area is 1/2 times Pi, times 18 inches squared (for the radius).
Now I’m going to simplify 18 inches squared. The area is 1/2 times Pi, times 18 inches squared is 324 inches squared, and then we’ll take half of 324. Half of 324 is 162 Pi inches squared. We need to go ahead and multiply times Pi, since they asked us to find our answer to the nearest square inch, so we’re not going to leave Pi in our answer.
That means that the area of the semicircle is about 509 inches squared. Now that we know the areas of both parts of our window, we can add them together to find the total area of the window. The area is 1,728 inches squared plus 509 inches squared, which is 2,237 inches squared, and that’s the area of the window.