Surface area measures the total amount of outside space covering a three-dimensional shape. One helpful way to understand surface area is to imagine unfolding the shape into a net. A net shows all of the flat pieces that make up the outside of the solid. To find surface area, we find the area of each piece and then add those areas together.

In this topic, we will use nets to find the surface area of shapes such as rectangular prisms, triangular prisms, and cylinders. For prisms, the net includes two matching bases and several side faces. For a cylinder, the net includes two circles and one rectangle that wraps around the side.

We will also look for patterns that make surface area easier to calculate. For any prism, the surface area can be found using:

SA = 2(area of base) + (perimeter of base)(height)

This formula makes sense because a prism has two matching bases, and the side faces together form a rectangle-like strip whose length is the perimeter of the base and whose height is the height of the prism. As future teachers, it is important to help students see that surface area formulas come from covering the outside of a shape, not from memorizing a rule.

 

Student Learning Goals

By the end of this topic, students should be able to:

  • Define surface area as the total area covering the outside of a 3D shape.

  • Use nets to represent 3D shapes.

  • Find the surface area of a rectangular prism.

  • Find the surface area of a triangular prism.

  • Find the surface area of a cylinder.

  • Explain how surface area is different from volume.

  • Use the formula SA = 2B + Ph for prisms.

  • Connect surface area formulas to the areas of individual faces.

Key Vocabulary

  • Surface Area - The total area covering the outside of a 3D shape.

  • Net - A flat pattern that can be folded into a 3D shape.

  • Face - A flat surface of a 3D shape.

  • Base - The repeated face or faces used to name a prism or cylinder.

  • Lateral Faces - The side faces that connect the bases.

  • Lateral Area - The total area of the side surfaces, not including the bases.

  • Perimeter of the Base - The distance around the base shape.

  • Height of a Prism - The distance between the two bases.

  • Cylinder - A 3D shape with two circular bases and one curved side surface.

Surface Area Strategy

  • Rectangular Prism - 6 rectangles - Find the area of each rectangular face and add them.

  • Triangular Prism - 2 triangles and 3 rectangles - Add the areas of the two triangular bases and the three rectangular side faces.

  • Cylinder - 2 circles and 1 rectangle - Add the areas of the circular bases and the rectangular side surface.

  • General Prism - 2 bases and side rectangles - Use SA = 2B + Ph.

Formula Connection

For a prism:

SA = 2(area of base) + (perimeter of base)(height)

This can be written more compactly as:

SA = 2B + Ph

where:

  • B - Area of one base

  • P - Perimeter of the base

  • h - Height of the prism

The formula works because the prism has two matching bases, giving 2B, and the side faces wrap around the base. When the side faces are laid flat, their combined area is the perimeter of the base times the height of the prism, or Ph.

Teacher Connection

Surface area is a strong place to emphasize meaning before formulas. Students often want to jump straight to a formula, but nets help them see what is actually being measured. The question is:

How much material would it take to cover the outside of this shape?

For a box, that might mean wrapping paper. For a cylinder, it might mean the label around a can plus the top and bottom. For a triangular prism, it might mean covering every face of a tent-shaped object.

Helpful teacher questions include:

  • What faces make up this shape?

  • What would the net look like?

  • Have we included every outside piece?

  • Are we covering the shape or filling it?

  • Which pieces are the bases?

  • What is the perimeter of the base?

  • How does the side strip connect to PhPhPh?

  • Why do we multiply the base area by 2?

Quick Reflection Question

Why does the formula SA = 2B + Ph make sense for prisms? How does a net help explain each part of the formula?