Surface Area Of A Triangular Prism Net

🔑 Key Concepts

• A triangular prism net has 2 congruent triangular bases and 3 rectangular lateral faces.
• The 3 rectangular faces can be viewed as one large rectangle.
• The width of that large rectangle is the perimeter of the triangular base.
• The height of that large rectangle is the height or length of the prism.
• The formula \(SA=ab+Ph\) adds the two triangular bases and the lateral area.

✏️ Worked Example

Example from the Video: Combine the Three Rectangles

The triangular faces have side lengths \(9\text{ cm}\), \(12\text{ cm}\), and \(15\text{ cm}\). The width of each rectangle is \(4\text{ cm}\).

Step 1: Find the triangular bases.

One triangle:

\(\frac{9\times12}{2}=54\)

Two triangles:

\(2(54)=108\text{ cm}^2\)

Step 2: Combine the rectangles.

The three rectangle lengths are the side lengths of the triangular base:

\(9+12+15=36\)

The height of the large rectangle is the prism height:

\(h=4\)

Step 3: Find the lateral area.

\(P=9+12+15=36\)

\(Ph=36(4)=144\)

Step 4: Add the bases and lateral area.

\(SA=ab+Ph\)

\(SA=(12)(9)+36(4)\)

\(SA=108+144\)

\(SA=252\)

\(SA=252\text{ cm}^2\)

🧠 Math Vocabulary

• Triangular Prism: a three-dimensional figure with 2 congruent triangular bases and 3 rectangular lateral faces.
• Net: a two-dimensional pattern that can fold into a three-dimensional figure.
• Lateral Faces: the rectangular faces that connect the two triangular bases.
• Perimeter: the distance around the triangular base.
• Lateral Area: the combined area of the rectangular faces.
• Altitude: the perpendicular height of the triangular base.
• Surface Area: the total area of all outside faces of a three-dimensional figure.
• Square Units: units used to measure area, such as \(\text{cm}^2\).

💡 Main Idea

The three rectangular faces of a triangular prism can be combined into one large rectangle. The width of that rectangle is the perimeter of the triangular base, and the height is the prism height. This is why the lateral area is \(Ph\) in the formula \(SA=ab+Ph\).

📚 What You Should Already Know

Students should know how to find the area of triangles and rectangles, calculate perimeter, identify congruent faces, and understand that a net represents the faces of a three-dimensional figure.

🚀 What Comes Next

Students can use this same idea to connect nets, surface area formulas, lateral area, and composite three-dimensional figures.

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