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Shape and Space - Nets of 3D Shapes

Grade 6IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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A net is a two-dimensional (2D) pattern that can be folded to form a three-dimensional (3D) solid. Imagine taking a cardboard box and carefully cutting along some of its edges to lay it completely flat on a table; the resulting shape is the net of that box.

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A cube has 11 possible nets, each consisting of 6 identical squares. A common visual for a cube net is the 'cross' shape, which consists of 4 squares in a vertical column and 2 squares extending horizontally from the sides of the second square down, looking like a lowercase 't'.

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Prisms always have two congruent (identical) bases and several rectangular side faces. For a triangular prism, the net looks like a row of three rectangles placed side-by-side, with two identical triangles attached to the top and bottom edges of one of those rectangles.

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Pyramids consist of one base (like a square or triangle) and several triangular faces that meet at a single point called the apex. The net of a square-based pyramid looks like a central square with a triangle attached to each of its four sides, appearing like a four-pointed star.

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When folding a net into a 3D shape, specific edges and vertices will meet. For example, in a rectangular prism net, the edges that are next to each other in the 2D layout (but not touching) will 'zip' together to form a single edge of the 3D shape.

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The total Surface Area (SASA) of a 3D shape is equivalent to the sum of the areas of all the individual shapes in its net. By calculating the area of each polygon in the net and adding them together, you find the total exterior space of the solid.

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A cylinder unfolds into a unique net consisting of one large rectangle and two circles. The length of the rectangle is equal to the circumference of the circular base (C=2πrC = 2\pi r), and the height of the rectangle is the height of the cylinder.

📐Formulae

Surface Area of a Cube: SA=6s2SA = 6s^2 where ss is the side length.

Surface Area of a Rectangular Prism: SA=2(lw+lh+wh)SA = 2(lw + lh + wh) where ll is length, ww is width, and hh is height.

Area of a Triangle (for pyramid faces): A=12×base×heightA = \frac{1}{2} \times \text{base} \times \text{height}

Area of a Circle (for cylinder bases): A=πr2A = \pi r^2

Circumference of a Circle (for cylinder net width): C=2πrC = 2\pi r

💡Examples

Problem 1:

A net of a cube consists of 6 squares, each with a side length of 4 cm4\text{ cm}. Calculate the total surface area of the cube formed by this net.

Solution:

  1. Find the area of one square face: A=s2=42=16 cm2A = s^2 = 4^2 = 16\text{ cm}^2. \ 2. Since there are 6 identical faces in the net, multiply by 6: SA=6×16=96 cm2SA = 6 \times 16 = 96\text{ cm}^2.

Explanation:

To find the surface area from a net, calculate the area of one individual part and multiply it by the total number of identical parts.

Problem 2:

Identify the surface area of a square-based pyramid from its net. The central square has a side of 6 cm6\text{ cm}, and the four triangles attached to it each have a height (slant height) of 8 cm8\text{ cm}.

Solution:

  1. Area of the square base: Abase=6×6=36 cm2A_{base} = 6 \times 6 = 36\text{ cm}^2. \ 2. Area of one triangular face: Atriangle=12×6×8=24 cm2A_{triangle} = \frac{1}{2} \times 6 \times 8 = 24\text{ cm}^2. \ 3. Total area of 4 triangles: 4×24=96 cm24 \times 24 = 96\text{ cm}^2. \ 4. Total Surface Area: 36+96=132 cm236 + 96 = 132\text{ cm}^2.

Explanation:

The surface area is found by summing the area of the base and the areas of all the triangular lateral faces shown in the net.