Surface Area Of A Composite Solid

🔑 Key Concepts

• Composite figures are made by combining two or more three-dimensional shapes.
• This problem is an enrichment-style surface area problem because students must reason about exposed and hidden surfaces.
• Only exposed outside surfaces should be included in the surface area.
• Surfaces that are covered, attached, or inside the object should not be counted.
• The rectangular parts and curved half-cylinder part are calculated separately and then added together.
• When using \(3.14\) for \(\pi\), round the final answer according to the directions.

✏️ Worked Example

Composite Figure Surface Area

Given:

Length: \(8\text{ in.}\)

Width/Diameter: \(6\text{ in.}\)

Radius: \(r=3\text{ in.}\)

Use \(3.14\) for \(\pi\).

Step 1: Find the exposed rectangular surfaces.

\(2(6\times6)+3(6\times8)\)

\(2(36)+3(48)\)

\(72+144=216\)

Step 2: Find the curved half-cylinder surface.

The curved part is half of a cylinder surface.

\[ \frac{2\pi r^2+\pi dh}{2} \]

\[ \frac{2\pi(3)^2+\pi(6)(8)}{2} \]

Step 3: Simplify the curved surface area.

\[ \frac{18\pi+48\pi}{2} \]

\[ \frac{66\pi}{2}=33\pi \]

\(33\pi\approx33(3.14)=103.62\)

Step 4: Add the exposed areas.

\(216+103.62=319.62\)

Rounded Surface Area:

\(SA\approx320\text{ in}^2\)

🧠 Math Vocabulary

• composite figure: a figure made by combining two or more simpler shapes or solids.
• surface area: the total area of all exposed outside surfaces of a three-dimensional figure.
• exposed surface: a surface that can be seen or touched from the outside of the figure.
• hidden surface: a surface that is covered, attached, or inside the composite figure and should not be counted.
• cylinder: a three-dimensional figure with circular bases and a curved surface.
• radius: the distance from the center of a circle to its edge.
• diameter: the distance across a circle through its center; \(d=2r\).
• rounding: rewriting a number to a requested place value, such as the nearest square inch.

💡 Main Idea

Surface area of composite figures requires more than applying a single formula. Students must break the object into familiar parts, identify which surfaces are exposed, ignore hidden or covered surfaces, and then add the exposed areas together.

📚 What You Should Already Know

Students should already know how to find the area of rectangles, calculate surface area of rectangular prisms, use cylinder surface area formulas, substitute values into formulas, and round decimal answers.

🚀 What Comes Next

Composite surface area problems are a strong enrichment step after standard prism and cylinder surface area. They prepare students for more advanced geometry, modeling, engineering-style design problems, and real-world situations where not every surface is visible or exposed.

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