*Kevin Newton*Show bio

Kevin has edited encyclopedias, taught history, and has an MA in Islamic law/finance. He has since founded his own financial advice firm, Newton Analytical.

Lesson Transcript

Instructor:
*Kevin Newton*
Show bio

Kevin has edited encyclopedias, taught history, and has an MA in Islamic law/finance. He has since founded his own financial advice firm, Newton Analytical.

Identifying the area of 2D and 3D shapes can be more difficult as their complexity increases, but is certainly possible using different mathematical tools. Practice finding the area of complex shapes through examples of the mathematical steps.
Updated: 11/20/2021

Let's say that you run a painting company; very often you have to find the area of a wall or house in order to know how much paint to use. However, your customers aren't exactly forthcoming when it comes to providing you with the information you need to know. Much more likely, they simply tell you the dimensions of the house or wall and leave it to you to figure out. Now that means you will have one of two approaches; as you might expect, there aren't neat formulas for finding the area of a wall with different angles or cut outs for windows and doors and there certainly isn't one for finding the surface area of a house with much the same. You could always buy more paint than you need but that would cut into your profits. What you need is a method of finding the area of irregular 2D and 3D shapes. Luckily, that's exactly what this lesson will provide.

First of all, you have to know how to find the **area of a 2D shape** no matter how irregular it is. However, there is a secret. You can always just split the irregular 2D shape into much smaller regular 2D shapes and calculate the area of each of them. In fact, for most of these, you'll only need your most basic area formulas.

Remember that for quadrilaterals, area is b x h, while for triangles, it is b x h/2. Finally for circles, A = r^2 x 3.14. From there, it's merely a mater of adding the areas together.

The same concept can be applied to dealing with cutouts within the shape. Find the area of the total larger piece, then subtract the areas of the cutouts. Using both of these methods, you'll be able to find the area of any irregularly shaped 2D surface.

Let's use this by taking an example that we'll make more detailed in order to make the concept stick. Let's say you've been hired to paint the front of a house; the house is 30 feet tall from the top of the roof to the ground, but only 20 feet tall on the walls. Also, it is 40 feet wide. In other words, it has a triangular roof. Luckily, we can quickly find the area by splitting this shape in to a rectangle and a triangle.

First, let's identify the two areas. The rectangle is 40 feet by 20 feet, which is from the start of the roof to the ground, then the width of the house. That's 800 square feet, since that's what you get when you multiply the base times the height. But what about that triangle? It has a height of 10 feet, since that's the difference between the two heights of the house. It also has a base of 40, since that's the width of the house. If we use our formula for triangles, that gives us an area of 200 square feet. Added with the area of the rectangle, that makes 1000 square feet worth of paint required.

But what about the windows and the door? Easy! All you have to do is calculate the area of them and subtract it from the total. Let's say that the house has big picture windows, each the size of the door to keep our math easy. Therefore, each cut out is 3 feet wide by 8.33 feet tall. That means about 25 square feet per cutout and since you have 4 cutouts, that means 100 square feet total. Subtract that by the 1000 square feet, and you get the real amount of paint needed: 900 square feet.

However as a painter, you often paint more than just one wall; often you will have to do the whole house. In that case, how do you calculate the **surface area**, or the area of each surface on the object of these different shapes? Simply put, you calculate the area of each side of the surface, then add it to the rest. Easy right? If it helps, imagine that you've taken a photograph of each side of the shape and are calculating the area of all those 2D shapes and adding them together. Just don't forget the top and bottom.

So back to our house. You're doing such a good job on the front, the homeowner wants you to paint the rest of the house. The back side is the same as the front, which you remember had an area of 900 square feet that needed painting. The two sides on the other hand, both have an area of 20 feet high times 40 feet long or 800 square feet. Again, there are 100 square feet of windows on each side that don't need paint, so that makes it 700 square feet per side. Since there are two of them, that makes 1400 square feet, added to the 900 square feet for the front, and another 900 square feet for the back, you get 1400 plus 1800 or 3200 square feet of paint.

In this lesson, we learned how to calculate the areas of irregularly shaped 2D and 3D shapes. Remember that the easiest thing to do is just to break the shapes into more regular shapes, then add up the areas. Remember to subtract for any cutouts; also, when finding the surface area of a 3D shape, be sure to remember all sides of it, including the top and the bottom.

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