Finding Surface Area of Figures with Complex Shapes

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

This lesson will define complex shapes and surface areas. We will then use an example to walk us through two ways of finding the surface area of figures with complex shapes as their bases.

Surface Area

Suppose that the stage crew of a play is building and painting props for the play. They need to build a prop that is a cube with a triangular prism attached to the side of it. First, they build the cube with side length 5 feet. They then build the triangular prism such that the height of the prism is 5 feet, and the base is an equilateral triangle with side length 5 feet and height 5.6 feet.


surfarecomp1


Before putting the two parts together to create the prop, they want to paint each one separately, so they need to know how much paint they will need. This involves the surface area of each part. The surface area of a three-dimensional figure is the total area that the figure's surface takes up. The formula for the surface area of a cube with side length a is as follows:

  • 6(area of one face) = 6a2

The formula for the surface area of a triangular prism with height a, and bases that are equilateral triangles with side length a and height h is as follows:

  • 2(area of one base) + 3(area of one square side) = 2(1/2)ah + 3a2 = ah + 3a2

We just need to plug a = 5 and h = 5.6 into these formulas to determine how much paint we need to paint each of these parts.


surfarecomp2


Great! We know we need enough paint to cover 150 square feet for the cube and 103 square feet for the prism.

Surface Area of Complex Shapes

After they've painted both parts of the prop, they connect them to create the desired figure. When they do this, they realize that the prop would look better if they painted it all one color, so they have to repaint the whole thing. This time, the two parts are connected, so we need to find the surface area of this new figure to determine how much paint they will need.


surfarecomp5


The prop has a base that is two shapes put together, a square and an equilateral triangle. When a shape is a combination of shapes like this, we call it a complex shape. Because the prop has a complex shape as its base, we don't have a nice formula for its surface area. When we need to find the surface area of figures with complex shapes as their bases, we have two options.

  1. Find the area of the base by splitting it up into shapes with known area formulas, then adding up the areas of all of the surfaces of the figure.
  2. We can use known formulas of cubes, prisms, spheres, pyramids or other figures to produce a formula for the figure being considered.

Let's take a look at how to do this by finding the surface area of the prop.

Example

To find the surface area of the entire prop, let's first consider the first option of finding the area of the base, then adding up the areas of all of the surfaces of the figure. We can think of the base as a square, with side length 5 feet, with an equilateral triangle, with side length 5 feet and height 5.6 feet, attached to it. Therefore, to find the area of the base, we simply add up the areas of the square and the triangle.


surfarecomp7


The area of a square with side length a is a2, and the area of an equilateral triangle with side length a and height h is (1/2)ah. To find the area of the base, we plug a = 5 and h = 5.6 into these formulas and add them up.

  • a2 + (1/2)ah = 52 + (1/2)(5)(5.6) = 25 + 14 = 39

We have that the area of each of the bases is 39 square feet.

Now we just consider the rest of the faces of the prop, which are 5 squares with side length 5 feet, so each of them has area 52 or 25 square feet. Perfect! All we have to do is add up all of these areas to find our surface area.

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