Area of Uncommon Figures - TEL Education...The picture on the left shows a rectangular prism as a...

Area of Uncommon Figures

Inquire: Larger than Life

Overview Finding the area of some figures requires breaking shapes down into smaller shapes, then finding and summing the areas of the pieces. This method can help us find areas of uncommon shapes like pyramids or some 2-D figures that are difficult to name. An important skill is to recognize that these connected shapes share sides and formulas. By the end of this lesson, students will be able to find the surface area of triangular prisms, triangular pyramids, and square pyramids. They will also be able to calculate the area of trapezoids and irregular figures.

Big Question: How can I find the area or surface area of figures by summing the area of their parts?

Watch: Our Irregular World Life isn’t all triangles and rectangles. Sometimes you need to get a little creative when it comes to finding out how much space an irregular figure takes up. Irregular, or composite, figures are figures that require us to add two or more standard geometric shapes in order to find area. Take a look at the face of a house. Typically, the roof can be modeled with a triangle while the rest of the house can be modeled with a rectangle. Each window can also be modeled with rectangles. If a builder is trying to find out how many bricks he might need to build the face of the house, he might find the area of the triangle roof plus the area of the rectangle base minus the area of the windows. You ever get a good look at a traffic cone? The base is made of a square to keep it stable and a cone to keep its shape. If you look at a traffic cone from the bottom, it is a square with a circle taken out of it. Breaking a shape into its parts makes it easier to understand how to calculate its area by using different formulas. How about a famous composite figure? Let’s take a look at the Washington monument. This composite figure has a rectangular prism for its body with a square pyramid at the top. There is no limit to the creativity of a composite figure. On your mark, get set, go! Runners running a lap around a track will run down a lane, turn a curve, run back down the opposite side, turn another curve, then continue running. If we observe the outline of the track, we can break this field down into two semi-circles and one rectangle.

Quantitative Literacy Module 5 - Area of Uncommon Figures Copyright TEL Library 2018 Page 1

Imagine you are looking for a house to live in, and having a reasonable living space is important to you. You look at a floor plan of a house you are interested in and notice it has three bedrooms, two bathrooms, and a semi-circular patio outside. If you want to know how much space all of the rooms take up, you would need to sum all six areas together. Composite figures are useful to think about when the shape of objects get more complex than a figure or two. What shapes can you identify that are combined together to form composite figures?

Read: Nothing but “Net” To find the surface area of some 3-D figures, we need to think about which 2-D shapes can make up the different parts of a 3-D figure. In this section, we will define nets and apply them to find the surface area of triangular prisms, triangular pyramids, and square pyramids.

Defining Nets In the figure below, we have three ways to view a rectangular prism.

The picture on the left shows a rectangular prism as a 3-D figure. It has a length, width, and height. If we cut the rectangular prism and lay it out flat, we have the middle picture. This is an example of a net. A net is a 2-D model that can be folded into a 3-D figure. A top view of the rectangular prism can reveal what 2-D parts make up a rectangular prism. We have 3 sets of 2 rectangles. That is 6 rectangles altogether. Since surface area is the square measure of the total area of all the sides, we can add all the areas of each 2-D shape to find the surface area of a 3-D figure.

Surface Area of Triangular Prisms Much like how we cut the rectangular prism into a net, we can do the same to a triangular prism to see which 2-D figures we need to find in order to get the surface area.


In the figure above, we can see that triangular prisms are made up of 2 triangular bases and 3 rectangles connected together. This means the surface area of a triangular prism can be written like this:

S = area of both triangles + area of 3 rectangles S = 2 • (area of a triangle) + 3 • (area of a rectangle)

Example 1: A popular chocolate container is an equilateral triangular prism. The triangles have a height of 8 cm and sides that are 10 cm long. The box is 30 cm long. Calculate the surface area of the box to the nearest square centimeter. First, let’s draw a model and list given information and needed formulas. Notice how the side of the triangle is also the width of the rectangle.

B = 10 H = 8 L = 10 W = 30 Area of triangle = ½ • BH Area of rectangle = LW Surface area of triangular prism = S = 2 • (area of a triangle) + 3 • (area of a rectangle) Next, we will find the areas of a triangle base and rectangle. Triangle area = ½ • BH Triangle area = ½(10)(8) Triangle area = 40 Rectangle area = LW Rectangle area = (10)(30)


Rectangle area = 300 Finally, we will find the sum of the areas. S = 2 • (area of a triangle) + 3 • (area of a rectangle) S = 2(40) + 3(300) If S = 980, then the surface area of the chocolate container is 980 square centimeters.

Surface Area of Triangular Pyramids A triangular pyramid is made up of 4 triangle faces, as seen in the figure below.

If we sum every face of the 3-D solid, we would have the following formula:

S = area of four triangles S = 4 • (area of triangles)

Let’s try a problem using triangular pyramids. Example 2: Johnny is building an electronic clock in the shape of a triangular pyramid. Each triangle is equilateral, has a base of 8 inches, and a height of 5 inches. Find the surface area in square inches. First, let’s draw a model and list the given information and needed formulas.


H = 5 B = 8 Triangle area = ½ • BH S = 4 • (area of triangles) Then, we will find the area of one of the triangles. Triangle area = ½ • BH Triangle area = ½ (8)(5) Triangle area = 20 Finally, we can calculate the surface area. S = 4 • (area of triangles) S = 4 • (20) If S = 80, then the surface area of the electronic clock is 80 square inches.

Surface Area of Square Pyramids Triangular pyramids aren’t the only pyramids. If we switch the base to a square, we get the appropriately named square pyramid.


Think about the faces of the square pyramid; we have 1 square and 4 triangles. This gets us the following formula:

S = area of the square base + four triangles S = area of a square + 4 • (area of the triangles)

Example 3: Rebecca wants to gift her friend a candy container in the shape of a square pyramid. She wants to wrap it in nice wrapping paper. The side of the base is 30 cm and the height of the triangle face is 24 cm. In square centimeters, how much wrapping paper will she need to cover the gift? First, let’s draw a model and list the given information and needed formulas. Notice that the side of the square is the same as the base of the triangle.

S = 30 B = 30 H = 24 Square area = S2 Triangle area = ½ • BH

S = area of a square + 4 • (area of the triangles) Next, we will find the area of the square and a triangle. Square area = S2

Square area = (30)2 Square area = 900 Triangle area = ½ • BH Triangle area = ½ (30)(24) Triangle area = 360 Finally, we will calculate the surface area. S = area of a square + 4 • (area of the triangles)


S = 900 + 4(360) If S = 2340, then Rebecca will need 2340 square centimeters of wrapping paper.

Reflect Poll: What’s in a Net? Which 3-D solid not discussed in this lesson would you most likely be interested in seeing as a work of art or construction?

● Pentagonal Prism: 2 pentagons connected by 5 rectangles ● Dodecahedron: 12 pentagons in a ball ● Hexagonal Pyramid: a hexagon with triangles connecting to a single point ● Icosahedron: 20 triangles connected together, commonly used as 20-sided dice

Expand: Sum of All Areas

Overview Adding different parts together isn’t exclusively useful for 3-D solids. We can find the area of irregular shapes using the same concept. First, we should find the area of an uncommon 2-D shape: the trapezoid.

Area of Trapezoids A trapezoid is a four-sided figure with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base b, and the length of the bigger base, B. The height, h, of a trapezoid is the distance between the two bases as shown in the figure below.

the trapezoid into two triangles may help us understand the area formula of a trapezoid. In the figure below, the area of the trapezoid is the sum of the areas of the two triangles. Notice that the height is the same for both triangles.

Adding the areas of the triangles and then rewriting the formula can help us get a useful trapezoid area formula.

Area of a trapezoid = 1/2 bh + 1/2 Bh


Area of a trapezoid = 1/2 h (b + B) Example 1: Vinny has a garden that is shaped like a trapezoid. The trapezoid has a height of 3.4 yards and the bases are 8.2 and 5.6 yards. How many square yards will be available to plant? First, let’s draw a picture and list the known information. b = 5.6 B = 8.2 h = 3.4 A = 1/2 h (b + B) Now, we can plug in values and calculate the area. A = 1/2 h (b + B) A = 1/2 (3.4)(5.6 + 8.2) If A = 19.54, then Vinny’s garden is 19.54 square yards big.

Area of Irregular Figures An irregular figure or composite figure is a figure that is not a standard geometric shape. Its area cannot be calculated using any of the standard area formulas; however, some irregular figures are made up of two or more standard geometric shapes. To find the area of one of these irregular figures, we can split it into pieces whose formulas we know, and then add the areas of the figures. Example 2: Find the area of the shaded region

The given figure is irregular, but we can break it into two rectangles. The area of the shaded region will be the sum of the areas of both rectangles.


The blue rectangle has a width of 12 and a length of 4. The red rectangle has a width of 2, but its length is not labeled. The right side of the figure is the length of the red rectangle plus the length of the blue rectangle. Since the right side of the blue rectangle is 4 units long, the length of the red rectangle must be 6 units.

The area of the figure is 60 square units. Is there another way to split this figure into two rectangles? Try it, and make sure you get the same area. Example 3: Find the area of the shaded region.


We can break this figure into a rectangle and a trapezoid. Notice that the width of the rectangle is the same as the smaller base of the trapezoid. The bottom side of the rectangle is 5 units long, so the height of the trapezoid must be 3 units long.

The area of the shaded region is the sum of the areas between the rectangle (R) and the trapezoid (T). Afigure = AR + AT Afigure = LW + ½ h (b + B) Afigure = (4)(5) + ½ (3)(4 + 7) If Afigure = 36.5, then the area of the figure is 36.5 square units. Can you think of another way to find this area?

Lesson Toolbox

Additional Resources and Readings Additional practice problems for composite figures

● Link to resource: https://www.mathgames.com/skill/6.106-area-of-complex-figures

An interactive site that shows the nets of common 3-D figures and how those figures look from certain angles

● Link to resource: http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.SURF&lesson=html/object_interactives/surfaceArea/use_it.html

An extension interactive that shows the nets of a few figures including dodecahedron and icosahedron

● Link to resource: https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Geometric-Solids/

An interactive site that challenges you to look at a net and determine if it will fold into a cube

● Link to resource: https://www.nctm.org/Classroom-Resources/Illuminations/Interactives/Cube-Nets/

Lesson Glossary composite figure: (irregular figure) figures made up of two or more standard geometric shapes net: a 2-D model that can be folded into a 3-D figure surface area: the square measure of the total area of all the sides


Check Your Knowledge 1. Find the surface area of a triangular prism box if the height of the triangle is 10 inches, the base

of the triangle is 5 inches, and the side of the box is 25 inches. a. 125 b. 250 c. 425 d. 525

2. The height of a trapezoid is 12 feet and the bases are 9 and 15 feet. What is the area? a. 54 b. 72 c. 144 d. 288

3. For 6-9, find the area of the composite figure.

a. 16 b. 18 c. 182 d. 384

Answer Key: 1. C 2. C 3. A

Citations

Lesson Content: Authored and curated by Kashuan Hopkins for The TEL Library. CC BY NC SA 4.0

Adapted Content: Title: 7.2 – Right Prisms And Cylinders. Siyavula. License: CC BY 3.0. Link to resource: https://www.everythingmaths.co.za/read/maths/grade-11/measurement/07-measurement-02 Title: 7.3 – Right Pyramids, Right Cones And Spheres. Siyavula. License: CC BY 3.0. Link to resource: https://www.everythingmaths.co.za/read/maths/grade-11/measurement/07-measurement-03 Title: 9.4 Prealgebra – Use Properties of Rectangles, Triangles, and Trapezoids. Openstax. License: CC BY 4.0. Link to resource: https://cnx.org/contents/[email protected]:HpMawx93/Use-Properties-of-Rectangles-T


Title: 9.5 Prealgebra – Solve Geometry Applications: Circles and Irregular Figures. Openstax. License: CC BY 4.0. Link to resource: https://cnx.org/contents/[email protected]:bKHa6Ab4/Solve-Geometry-Applications-Ci

Attributions “cube unfolded” By Siyavula is licensed under CC BY 3.0. Link to resource: https://www.everythingmaths.co.za/read/maths/grade-11/measurement/07-measurement-02 “triangular prism unfolded” By Siyavula is licensed under CC BY 3.0. Link to resource: https://www.everythingmaths.co.za/read/maths/grade-11/measurement/07-measurement-02 “Triangular prism” By Siyavula altered by TEL Library is licensed under CC BY 3.0. Link to resource: https://www.everythingmaths.co.za/read/maths/grade-11/measurement/07-measurement-02 “Surface area of pyramids” By Siyavula altered by TEL Library is licensed under CC BY 3.0. Link to resource: https://www.everythingmaths.co.za/read/maths/grade-11/measurement/07-measurement-03 “Surface area. Triangular pyramid” By Siyavula altered by TEL Library is licensed under CC BY 3.0. Link to resource: https://www.everythingmaths.co.za/read/maths/grade-11/measurement/07-measurement-03 “Volume. Square pyramid” By Siyavula altered by TEL Library is licensed under CC BY 3.0. Link to resource: https://www.everythingmaths.co.za/read/maths/grade-11/measurement/07-measurement-03 “Volume. Square pyramid #2” By Siyavula altered by TEL Library is licensed under CC BY 3.0. Link to resource: https://www.everythingmaths.co.za/read/maths/grade-11/measurement/07-measurement-03 “Volume. Square pyramid #3” By Siyavula altered by TEL Library is licensed under CC BY 3.0. Link to resource: https://www.everythingmaths.co.za/read/maths/grade-11/measurement/07-measurement-03 “Figure 13” By OpenStax is licensed under CC BY 4.0. Link to resource: https://cnx.org/contents/[email protected]:HpMawx93@20/Use-Properties-of-Rectangles-Triangles-and-Trapezoids “Figure 15” By OpenStax is licensed under CC BY 4.0. Link to resource: https://cnx.org/contents/[email protected]:HpMawx93@20/Use-Properties-of-Rectangles-Triangles-and-Trapezoids “trapezoid . Step 1” By OpenStax is licensed under CC BY 4.0. Link to resource: https://cnx.org/contents/[email protected]:HpMawx93@20/Use-Properties-of-Rectangles-Triangles-and-Trapezoids “Find the Area of Irregular Figures” By OpenStax is licensed under CC BY 4.0. Link to resource: https://cnx.org/contents/[email protected]:bKHa6Ab4@18/Solve-Geometry-Applications-Circles-and-Irregular-Figures “Find the area of the shaded region. Solution” By OpenStax is licensed under CC BY 4.0. Link to resource: https://cnx.org/contents/[email protected]:bKHa6Ab4@18/Solve-Geometry-Applications-Circles-and-Irregular-Figures


“Find the area of the shaded region. Solution part 2” By OpenStax is licensed under CC BY 4.0. Link to resource: https://cnx.org/contents/[email protected]:bKHa6Ab4@18/Solve-Geometry-Applications-Circles-and-Irregular-Figures “60 square units. Solution” By OpenStax is licensed under CC BY 4.0. Link to resource: https://cnx.org/contents/[email protected]:bKHa6Ab4@18/Solve-Geometry-Applications-Circles-and-Irregular-Figures “Find the area of the shaded region.” By OpenStax is licensed under CC BY 4.0. Link to resource: https://cnx.org/contents/[email protected]:bKHa6Ab4@18/Solve-Geometry-Applications-Circles-and-Irregular-Figures “Find the area of the shaded region. 2” By OpenStax is licensed under CC BY 4.0. Link to resource: https://cnx.org/contents/[email protected]:bKHa6Ab4@18/Solve-Geometry-Applications-Circles-and-Irregular-Figures


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