Area of Complex Figures

Area of Complex Figures

Finding the area of shapes isn't always as simple as it is with squares and rectangles. In this lesson, you'll learn how to find the area of complex figures. This includes finding the area of unusual shapes and finding the area of parts of shapes.

Area in the Real World

Let's say you need to paint some rooms in your house. How much paint will you need? You need to know the surface area of the walls. If you live in a really boring house, all your walls are neat rectangles and squares. But, maybe there are angled ceilings. Or windows - you don't need paint to cover the windows, do you?

In fact, determining the area of shapes in the real world can seem more complicated than your standard 'area of a rectangle' geometry problem. So, let's look at how you can solve area questions with some unusual shapes. It might just save you an extra trip to the store for more paint.

Review of Basic Shapes

Before we get into complex figures, let's quickly review the area formulas we'll need. First up is the rectangle. A rectangle is a four-sided shape where all the angles are 90 degrees. With rectangles, the sides opposite of each other are equal. Remember that - it's important. The area of a rectangle is width times height. In other words, multiply the long side by the short side, and you're done!

Next is the square. A square is a type of rectangle where all the sides are equal in length. Therefore, you only need to know one side's length to know all of the sides. That's going to come in handy. Also, the area of a square is s^2, where s is the length of one of the sides. So again, if you know one side of a square, you have all the power.

Next is the triangle. The area of a triangle is 1/2 * b * h, or 1/2 * base * height. So, you need to know more about triangles to find their areas. But, there are a ton of useful triangle facts that are helpful. For example, in a triangle with one right angle, you can use the Pythagorean theorem, a^2 + b^2 = c^2, to determine the length of a missing side.

Finally, there are circles. If you've ever stared at an extra-large pizza and wondered how much space it takes up, then you've thought about the area of a circle. The formula is pi * r^2, where r is the radius of a circle. That's 3.14 pi, by the way, not pizza pie. Okay, with those formulas in mind, let's get to the task at hand.

Complex Shapes

Let's look at a wall you need to paint.

This room can be divided into rectangles.

Okay, you know the measurements, but how do you account for that space in the corner? Well, what if you draw a line above the space in the corner? A complex shape just became two rectangles, and you're a rectangle ninja. But wait, do you know the measurements for these new shapes? You do!

The small rectangle has a width of 3 feet. And, if the height above it is 4 feet, and the whole wall is 8 feet, then the height of the small rectangle is 4 feet. That rectangle must be 3 times 4, or 12 square feet. As for the big rectangle, again, its height is 8 feet. The width is 10 feet of the whole wall minus the 3 feet of the small rectangle, so it's 7 feet. And, 8 times 7 is 56 square feet. Add 56 and 12, and you get 68 square feet.

Let's look at another. Here's one from that room with a tiny corner:

This room can be divided into a rectangle and a right triangle.

You avoid it because sometimes there are spiders in that corner, but you don't want to have to tell people you didn't paint the whole wall because of spiders. So, what can we do? If we draw a line perpendicular to the floor, we have a rectangle and a right triangle. Awesome. That rectangle is just 8 by 10 feet, or 80 square feet.

And that triangle? You know it's 6 feet high, because your 6`1`` friend always hits his head when he offers to kill the spiders for you. But, how long is it? Well, the whole wall is 17 feet. And, the rectangle is 8 feet wide. So, it's 17 minus 8, or 9 feet. What's the area of a triangle? 1/2 * b * h. So, it's 1/2 * 9 * 6, or 27 square feet. This whole wall, spider territory and all, is 80 plus 27, or 107 square feet.

Parts of Shapes

We've been avoiding walls with doors and windows, but fear not. They're not so bad. Here's a pretty simple wall with one door:

To find the area of the wall space, subtract the area of the door from the area of the room as a whole.

You know the wall is 8 feet high by 12 feet wide. It's a rectangle. So, the area is width times height, or 8 times 12, which is 96 square feet. But, what about the door? Well, the door is also a rectangle. It's 7 feet by 3 feet, so its area is 21 square feet. To determine the amount of wall space, subtract the door's area from the wall. 96 minus 21 is 75. So, this wall is 75 square feet.

Next up is the wall below, which has two windows and a dog door.

To find the area of the wall space, subtract the total area of the windows plus the area of the dog door from the total area of the room.

Little Bagel the beagle would be very upset if you painted his door shut. Let's start with the overall wall. It's an 11 by 8 rectangle. So, it's 88 square feet. The windows are both 3 by 3 squares, making them 9 square feet each, or 18 total square feet. Now, you had to widen Bagel's door when he put on a little weight. It's a 1 by 1.5 rectangle. That's 1.5 square feet. So, there are 19.5 square feet to subtract from the 88, leaving 68.5 square feet.

All this talk of painting is making me hungry. Let's go back to that pizza I mentioned earlier. What if you wanted to know how much empty surface space a box has outside of the pizza - you know, space that could be more pizza but sadly isn't? Okay, that pizza in a box is really just a circle inside a square, like this:

To find the amount of empty box space, subtract the area of the pizza from the area of the box as a whole.

We know this is a 12-inch pizza, which is its diameter. That makes its radius 6.

Therefore, the area is pi * r^2, or 36 * pi. That's about 113 square inches of hot, gooey goodness. Now, how big is the box? It's a square with a 13-inch side. So, the area is 13^2, or 169 square inches. Therefore, the pizza-less space is 169 minus 113, or 56 square inches of cardboard sadness.

Lesson Summary

In summary, the trick with complex shapes is to break them up into rectangles, triangles and other shapes you know. Use the properties of those simpler shapes to find any missing lengths, solve for the smaller areas and then add them together. With parts of shapes, the same rules apply, though sometimes you just need to subtract one or more of the simpler shapes. Now, enough talking about pizza - I think it's time to go find some.