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Using Nets to Find Surface Area

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Using Nets to Find Surface Area

Using nets to calculate the surface area of three-dimensional shapes is a practical and visual method. A net represents a three-dimensional object as a two-dimensional layout that can be folded to reconstruct the object. This method is utilised in various real-world applications, such as packaging design and material estimation.

Surface Area is the combined area of all faces and curved surfaces of a three-dimensional object.

Nets are the unfolded two-dimensional representations of these objects, aiding in the straightforward computation of their surface areas.

Using Nets to Find Surface Area – Explanation

The process of calculating the surface area using nets involves unfolding a three-dimensional figure (like a prism or pyramid) into a flat arrangement of its faces. For this topic, we focus on nets for three shapes: right rectangular prisms (also called cuboids), right rectangular pyramids, and right triangular prisms.

3D Shape Name 3D Shape (Illustration of the 3D version) 2D Shapes Formed (Describe the Net) 2D Net (Illustration Request)
Right Rectangular Prism Illustration request: A 3D right rectangular prism. The net consists of three pairs of rectangles, corresponding to the opposite faces of the prism. Illustration request: A net of a right rectangular prism
Right Rectangular Pyramid Illustration request: A 3D right rectangular pyramid(no labels needed) The net includes one square base and four triangular faces that connect at the apex of the pyramid. Illustration request: A net of a right rectangular pyramid
Right Triangular Prism Illustration request: A 3D right triangular prism. The net features two right triangles for the bases and three rectangles corresponding to the sides of the prism. Illustration request: A net of a right triangular prism - no labels needed.

Using Nets to Find Surface Area – Step-by-Step Process

In this section, we explore how to determine the surface areas of various three-dimensional shapes using their corresponding two-dimensional nets. Mastering these calculations involves foundational concepts such as finding the area of a right triangle, understanding the area of a shape and calculating the area of rectangles. By breaking down complex shapes into simpler 2D nets, we can apply these basic area formulas to find the total surface area effectively.

Surface Area of a Right Rectangular Prism

Steps to Find the Surface Area:

  1. Identify the Dimensions: The dimensions are 6 cm (length), 4 cm (width), and 3 cm (height).
  2. Calculate Area of Each Face:
    • Two rectangles of $6 \times 4$ cm: $2 \times (6 \times 4) = 48$ cm²
    • Two rectangles of $6 \times 3$ cm: $2 \times (6 \times 3) = 36$ cm²
    • Two rectangles of $4 \times 3$ cm: $2 \times (4 \times 3) = 24$ cm²
  3. Add All Areas Together: Total surface area = $48 + 36 + 24 = 108$ cm²
  4. Final Formula: The total surface area $S = 2(lw + lh + wh)$ where $l$, $w$, and $h$ are the dimensions of the prism.

Surface Area of a Right Rectangular Pyramid

Steps to Find the Surface Area:

  1. Identify the Dimensions: This triangular pyramid has a base of 4 cm per side and a slant height of 5 cm for the triangular sides.

  2. Calculate Area of the Base:

    • Base area (square) = $4 \times 4 = 16$ cm²

  3. Calculate Area of Triangular Sides:

    • Triangular side area = $\frac{1}{2} \times 4 \times 5 = 10$ cm²
    • There are four triangular sides: $4 \times 10 = 40$ cm²

  4. Add Base and Side Areas: Total surface area = $16 + 40 = 56$ cm²

Surface Area of a Triangular Prism

Steps to Find the Surface Area:

  1. Identify the Dimensions: For the net of the prism, we have one rectangular base with the dimensions of 7 cm by 10 cm, and two rectangles with dimensions of 6 cm by 10 cm.

  2. Calculate Area of the Triangular Bases:

  3. The triangular base area = $\frac{1}{2} \times 7 \text{ cm} \times 5 \text{ cm} = 17.5 \text{ cm}^2$

  4. Two bases: $2 \times 17.5 \text{ cm}^2 = 35 \text{ cm}^2$

  5. Calculate Area of Rectangular Sides:

  6. Base rectangle: $7 \text{ cm} \times 10 \text{ cm} = 70 \text{ cm}^2$

  7. Side rectangles: $6 \text{ cm} \times 10 \text{ cm} = 60 \text{ cm}^2$ for each, $2 \times 60 \text{ cm}^2 = 120 \text{ cm}^2$ for both

  8. Add All Areas Together: Total surface area = $35 \text{ cm}^2 + 70 \text{ cm}^2 + 120 \text{ cm}^2 = 225 \text{ cm}^2$

Therefore, the surface area of the triangular prism is 225 cm².

Using Nets to Find Surface Area – Practice

Find the surface area of a right rectangular prism with dimensions of $8$ cm by $5$ cm by $3$ cm.
Find the surface area of a right rectangular prism with dimensions of $7$ cm by $4$ cm by $2$ cm.
Find the surface area of a right rectangular pyramid with a base side of 3 cm and slant height of 4 cm for the triangular faces.
Find the surface area of a right rectangular pyramid with a base side of 6 cm and slant height of 5 cm for the triangular sides.
Find the surface area of a triangular prism seen above.
Find the surface area of a triangular prism seen above.

Using Nets to Find Surface Area – Summary

Key Learnings from this Text:

  • Nets simplify the calculation of surface area by presenting 3D objects as connected 2D shapes.
  • Skills in using nets are applicable in real-world scenarios such as packaging design and material estimation.
  • The process includes unfolding a 3D shape into a net, calculating the area of each face, and summing these areas to find the total surface area.

Using Nets to Find Surface Area – Frequently Asked Questions

What is a net in geometry?
Why is calculating the surface area important?
Can nets be used for any 3D shape?
How do you find the area of a triangle from its net?
What are common mistakes when drawing nets?
Is there a tool or software that helps create nets accurately?
How does the surface area calculation differ between different types of prisms?
Can the surface area of curved objects be calculated using nets?
What is the difference between surface area and volume?
Why is it important for students to learn about nets and surface area in school?

Using Nets to Find Surface Area exercise

Would you like to apply the knowledge you’ve learnt? You can review and practice it with the tasks for the learning text Using Nets to Find Surface Area.

  • Understand what a net is and how to use them to find the surface area of a 3D shape.
    Hints

    A net is like a flattened-out outline of a 3D shape.

    If you have a rectangular prism, and you unfold it so it lays flat, the net would be the shape you'd see. It's all the faces spread out in 2D. Think of it like opening up a cardboard box so it's flat on the floor - that flat shape is the net of the box.

    To help you answer the question, refer to the images seen in the problem.

    Surface area is the total area of all of the surfaces of a three-dimensional object.

    Imagine a cardboard box: the surface area is the amount of space covering the outside of the box, including all of its sides. If you were to unfold the box and lay it flat, the surface area would be the total area of all of the cardboard pieces you see.

    Solution

    The measurement that calculates the sum of the area of each of the faces of the prism is called the surface area.

    A 3D shape can be unfolded to reveal a 2D flattening or what is known as a net. Finding the area of each face can help find the surface area.

  • Identify the process to finding the surface area of a net.
    Hints

    To find the surface area, add the area of each face of the net.

    There are 2 correct answers in the problem that would find the surface area.

    Solution

    To find the surface area, add the area of each face of the net. The two answers that would work to find the surface area are:

    • $12 + 12 + 27 + 27 + 36 + 36$
    • $(12 \times 2) + (27 \times 2) + (36 \times 2)$
    Both of these expressions have a sum of $150\:\text{m}^2$.

  • Identify the area of the net to find the total surface area.
    Hints

    To find the area of a rectangle, multiply the length by the width.

    Look at the face that has a measurement of 3 m and 10 m. To find the area of this face, we will use the formula $A=lw$.

    Substitute in the values for $l$ and $w$.

    $A=10 \times 3$

    Find the product to find the area,

    $A=30\:cm^2$.

    Solution

    The area of each face was found and then added together to find the total surface area.

    $60 + 60 + 18 + 18 + 30 + 30 = 216\:cm^2$

  • Demonstrate your understanding of the process to find the surface area using a net.
    Hints

    A 3D rectangular prism can be flattened out into a 2D shape, which is called a net. The net can help us find the surface area.

    To find the surface area of a net, you must find the area of each face.

    The formula used to find the area of each face of a rectangular prism is $A=lw$. The $l$ stands for the length, and the $w$ stands for the width.

    Solution

    1) Find the net for the rectangular prism.

    2) Substitute the $l$ and $w$ in to the formula $A=lw$ and find the area.

    3) The areas for each face:

    • $A=6\:cm^2$
    • $A=6\:cm^2$
    • $A=12\:cm^2$
    • $A=12\:cm^2$
    • $A=18\:cm^2$
    • $A=18\:cm^2$
    4) Add the areas together. $6 + 6 + 12 + 12 + 18 + 18$

    5) The surface area is $62\:cm^2$.

  • Find the surface area of a rectangular prism as a net.
    Hints

    To find the surface area, add up the area of each face of the rectangular prism.

    A rectangular prism has a total of 6 faces. There are two sets of each face with identical area measurements.

    Addition can be used to find the total surface area. Can you find the sum?

    $15 + 15 + 15 + 15 + 9 + 9 = ?$

    Solution

    The surface area of the rectangular prism is $78\:m^2$.

    To find this, each area can be added together like this:

    $15 + 15 + 15 + 15 + 9 + 9 = 78$.

    Don't forget to add in the units, which are $m^2$.

  • Using nets to find surface area.
    Hints

    The first step is to find the area of each face. It is helpful to find the net of this rectangular prism first to lay out all of the faces.

    When a rectangular prism is turned into a net, there are six 2D rectangles.

    Solution

    The areas for the faces of the rectangular prism are:

    • $15\:cm^2$
    • $21\:cm^2$
    • $35\:cm^2$
    Since there are two of each we add each face two times to find the surface area.

    $15(2) + 21(2) + 35(2)$

    The sum of the faces of the net is equal to the surface area.

    $SA=142\:cm^2$

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