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Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn how you can break up the shapes in a combined figure to easily find the perimeter, area, and volume of the whole figure.

Combined Figures

We learn about triangles, squares, and circles in math. We know how to find the perimeter, the area, and the volume of each of these shapes on their own. But what happens when you have a combined figure, a shape that is the combination of several simpler shapes? Many shapes that we come across are combined figures. Look at your house, and you can see that you have a rectangular box plus a triangular box on top. How would you find the perimeter, the area, and the volume of such a figure? This is the question that we will be answering in this video lesson.

We will look at this combined figure that is the combination of a rectangle and a triangle:

Combined figure

It kind of looks like a house. All the lengths are in feet, and we'll look at how to find the perimeter of this shape, the area of it, and also the volume of the shape when our shape is in three dimensions. The method that you will learn in this video lesson can be applied to any combined figure.

Finding the Perimeter

We begin with finding the perimeter. Our shape looks like we drew a house on a piece of paper. To find the perimeter of our shape, we can highlight just the sides that make up the outside of our shape. Think of the outside as the fence that keeps our shape contained. So, we need to know the distance of this fence.

We need to know the length of each side. If we don't know the length of some of the sides, we need to figure them out. To find the length of some sides, we can look at the other sides to see if we can calculate them. For example, for our shape, we have four sides that are labeled with their lengths, but the fifth side isn't. However, because the fifth side is the other side of our triangle, we can calculate it, since this side is the same length as the side that is opposite it. In our case, 6 feet.

Given side lengths

Now we have all the lengths for our sides. So, all we have to do now is add up all these lengths to find our perimeter. Our perimeter is 5 + 5 + 6 + 8 + 6 = 30 feet. Notice how I've basically gone around the shape. This way I make sure that I am adding up all my sides. I recommend you do the same. Start with one side and then methodically work your way around the shape until you get back to the side you started with.

Finding the Area

Next, we need to find the area of our combined figure. For this part, we want to separate our shape into its separate shapes. We already know that we have a rectangle with a triangle on top. So, what we can do is to draw a line to separate the two shapes. Now we can clearly see our two shapes:

Separating the two shapes makes it easier to find the area.

We do this to make finding our area easier. We can easily find the area of a rectangle as well as of a triangle. So, if we separate our combined figure into these easy shapes, then all we have to do is to find the area of each of our simple shapes and then add the areas up.

We first calculate the area of our rectangle. We know the area of a rectangle is its length multiplied by its width, so we have 6 times 8. This equals 48, so the area of our rectangle is 48 square feet. Next, we calculate the area of our triangle. The formula for the area of a triangle is half the base times the height: (1/2) * bh. Our base is the longer side of our triangle.

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We can see that this equals 8 from the other lengths that are given to us. Our height is given to us, and it's 4. If we are missing any one of these numbers, we might need to use our other math skills to figure them out. Now that we have all our needed measurements, we can calculate the area of our triangle as (1/2) * 8 * 4. This equals 16 square feet. To find the total area of our shape, we add our two separate areas. We get 48 + 16, which equals 64. So, the area of our combined figure is 64 square feet.

Finding the Volume

To find our volume, we use the skills we learned to find the area of our combined shape and apply it here as well. We do the same as we did for our area and separate our shape into easy, simple shapes. So far, we've dealt with our shape in two-dimensions. Let's add a third dimension to our shape by turning it into a cake that is shaped like a house. We now have a third dimension: the height of our cake.

We separate our shape into its two simple shapes. We draw a line to cut our cake into its rectangular part and its triangular part. We can then easily find the volume of each part separately and then add them together to find the total volume of our combined figure. We see that our cake has a height of 0.5 feet:

Diagram to help you find the volume of the combined figure

To find the volume of these pieces, we multiply the height of each of our pieces by the area of our flat, two-dimensional shape. Our triangle has a flat area of 48 square feet. Multiplying this by the height of 0.5 feet gives us 24 cubic feet. Our triangle has a flat area of 16 square feet. Multiplying this by 0.5 feet gives us 8 cubic feet. Adding these two volumes together gives us 32 cubic feet for our combined three-dimensional figure.

Lesson Summary

We've covered how to find the perimeter, area, and volume now. So, let's review what we've learned. A combined figure is a shape that is the combination of several simpler shapes. To find the perimeter, we add up all the outside sides of our shape. To find the area, we divide our shape into its simple shapes, calculate the area of these shapes separately, and then add these areas up to get our total. To find the volume, we also divide our shape into its simple shapes, calculate the volume of each separately, and then add up the pieces to get the whole volume.

Learning Outcomes

After reviewing this lesson, you should have the ability to:

Define combined figure

Explain how to find the perimeter, area, and volume of a combined figure

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