Nets and surface area Knowledge-Based Study

Study notes

1. What Is a Net?

   A net is a two-dimensional, flat shape that can be folded along its edges to form a three-dimensional solid. Unfolding a solid allows us to see all of its individual faces laid out flat at the same time.

   Understanding nets helps in analyzing the structure of three-dimensional polyhedra and provides a straightforward way to calculate their total surface area.

   Key Properties of Nets

   • Every face of the three-dimensional solid must appear exactly once in the net.
   • Adjacent edges that join when folded must have identical lengths.
   • Faces must be arranged so that folding along the shared edges forms the solid without overlapping or leaving open gaps.

   Common Solids and Their Nets

   Different geometric solids produce distinct nets based on the number and shape of their faces.

   1. Cube

   A cube consists of 6 identical square faces. There are exactly 11 distinct valid nets for a cube. When folded along its edges, each net forms a cube with 12 edges and 8 vertices.

   2. Rectangular Prism (Cuboid)

   A rectangular prism has 6 rectangular faces arranged in 3 pairs of opposite, congruent rectangles:

   • Two faces of dimensions $l \times w$ (base and top)
   • Two faces of dimensions $l \times h$ (front and back)
   • Two faces of dimensions $w \times h$ (left and right sides)

   3. Triangular Prism

   A triangular prism consists of 5 faces: 2 congruent triangular bases and 3 rectangular lateral faces connecting the corresponding sides of the triangles.

   4. Square-Based Pyramid

   A square-based pyramid consists of 1 square base and 4 isosceles triangular faces that meet at a single apex.

   What Is Surface Area?

   The total surface area of a three-dimensional figure is the total sum of the areas of all its outer faces. Surface area is measured in square units, such as square centimeters ($\text{cm}^2$) or square meters ($\text{m}^2$).

   Using a net is the most direct conceptual method to determine surface area: calculate the area of each separate two-dimensional face in the net, then add those areas together.

   $$\text{Total Surface Area} = \sum \text{Area of each individual face}$$

   Surface Area vs. Lateral Surface Area

   • Total Surface Area (TSA): The sum of all faces, including the bases.
   • Lateral Surface Area (LSA): The sum of the side faces only, excluding the top and bottom bases.

   Formulas for Rectangular Prisms and Cubes

   By analyzing the unfolded net of a rectangular prism with length $l$, width $w$, and height $h$, we find three pairs of matching faces:

   $$\text{Area} = 2(lw) + 2(lh) + 2(wh) = 2(lw + lh + wh)$$

   Special Case: The Cube

   Since all edges of a cube have equal length $s$, each of the 6 faces has an area of $s^2$:

   $$\text{Surface Area of a Cube} = 6s^2$$

   Worked Example

   Find the total surface area of a rectangular prism with length $l = 5\text{ cm}$, width $w = 3\text{ cm}$, and height $h = 4\text{ cm}$.

   1. Calculate the area of the top and bottom faces: $2 \times (5 \times 3) = 2 \times 15 = 30\text{ cm}^2$
   2. Calculate the area of the front and back faces: $2 \times (5 \times 4) = 2 \times 20 = 40\text{ cm}^2$
   3. Calculate the area of the side faces: $2 \times (3 \times 4) = 2 \times 12 = 24\text{ cm}^2$
   4. Sum all areas: $30 + 40 + 24 = 94\text{ cm}^2$

   Thus, the total surface area is 94 $\text{cm}^2$.

   Surface Area of Other Polyhedra

   Triangular Prism

   To find the surface area of a right triangular prism with base triangle dimensions $a, b, c$ (where base is $b$ and triangle height is $h_t$) and prism length $L$:

   • Two triangular bases: $2 \times \left(\frac{1}{2} \times b \times h_t\right) = b \times h_t$
   • Three rectangular faces: $aL + bL + cL = (a + b + c)L$
   • Total Surface Area: $b \times h_t + (a + b + c)L$

   Regular Square Pyramid

   A regular pyramid with a square base of side $s$ and triangular faces with slant height $l_{\text{slant}}$:

   • Base area: $s^2$
   • Four triangular faces: $4 \times \left(\frac{1}{2} \times s \times l_{\text{slant}}\right) = 2s \cdot l_{\text{slant}}$
   • Total Surface Area: $s^2 + 2s \cdot l_{\text{slant}}$

   Summary of Problem-Solving Steps

   1. Sketch or visualize the complete net of the solid.
   2. Identify the shape and dimensions of each individual face.
   3. Compute the area of each face using standard two-dimensional area formulas.
   4. Add all individual areas together to obtain the total surface area.