Understanding the concept of nets and their application in real-world and mathematical problems is crucial for CIE IGCSE students. This section focuses on how to draw nets for basic 3D shapes such as cubes, cuboids, prisms, and pyramids, and how to use these nets to calculate volumes and surface areas. This skill is not only foundational for geometric reasoning but also enhances spatial visualisation.

What are Nets?

A net is a two-dimensional shape that can be folded to form a three-dimensional object. Each face of the 3D object is represented on the net. Nets are particularly useful for understanding the structure of 3D shapes and for calculating their surface areas.

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Drawing Nets

Cubes

• Definition: A cube is a three-dimensional shape with six square faces of equal size.
• Net Construction: A cube’s net consists of six squares arranged in a cross shape. This shape includes four squares in a row with one square on each side of the second square from one end.

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Cuboids

• Definition: A cuboid is a three-dimensional rectangle box, characterised by length, width, and height.
• Net Construction: A cuboid’s net consists of six rectangles. The arrangement depends on the dimensions of the cuboid but typically involves three pairs of identical rectangles.

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Prisms

• Definition: A prism is a solid figure with two parallel faces called bases, which are congruent polygons. The other faces, known as lateral faces, are rectangles.
• Net Construction: The net of a prism includes the shapes of the bases and a series of rectangles that connect the corresponding sides of the two bases.

Pyramids

• Definition: A pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex.
• Net Construction: The net of a pyramid consists of the base shape and as many triangles as the base has sides, all meeting at a common point that represents the apex.

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Calculating Volumes and Surface Areas from Nets

Volume Calculation

The volume of a 3D shape is the amount of space it occupies, measured in cubic units. The formulas vary by shape:

• Cube: $V = a^3$, where $a$ is the length of a side.
• Cuboid: $V = l \times w \times h$, where $l$, $w$, and $h$ are the length, width, and height.
• Prism: $V = B \times h$, where $B$ is the area of the base, and $h$ is the height.
• Pyramid: $V = \frac{1}{3}Bh$, where $B$ is the area of the base, and $h$ is the vertical height from the base to the apex.

Surface Area Calculation

The surface area is the total area of all the faces of a 3D shape, measured in square units.

• Cube: $SA = 6a^2$.
• Cuboid: $SA = 2(lw + lh + wh)$.
• Prism: $SA = 2B + Ph$, where $P$ is the perimeter of the base.
• Pyramid: $SA = B + \frac{1}{2}Pl$, where $P$ is the perimeter of the base, and $l$ is the slant height.

Worked Examples

Example 1: Calculating the Surface Area of a Cuboid

A cuboid has dimensions 5 cm by 3 cm by 2 cm. Calculate its surface area using its net.

Solution:

• Identify the dimensions:

Length $(l)$ = 5 cm, Width $(w)$ = 3 cm, Height $(h)$ = 2 cm

• Use the surface area formula for a cuboid:

• Calculate:

$SA = 2(5 \times 3 + 5 \times 2 + 3 \times 2)$

$= 2(15 + 10 + 6)$

$= 2 \times 31$

$= 62 \text{ cm}^2$

Example 2: Drawing a Net for a Prism and Calculating its Volume

Draw the net of a triangular prism with a base area of $20 \text{ cm}^2$ and a height of 10 cm. Calculate its volume.

Solution:

• The net includes two triangles for the bases and three rectangles that connect corresponding sides of the triangles.
• For the volume, use the formula $V = B \times h$.
• Calculate: $V = 20 \times 10 =200 \text{ cm}^3$.

Practice Questions

Problem 1:

Draw the net of a pyramid with a square base of side 4 cm and a slant height of 6 cm. Calculate its surface area.

Solution:

Given: Square base side = 4 cm, Slant height = 6 cm.

Surface Area $(SA)$ calculation:

$SA = B + \frac{1}{2}P \times l$

$= 4^2 + \frac{1}{2} \times 4 \times 4 \times 6$

$= 64 \text{ cm}^2$

Problem 2:

A cube has a side length of 3 cm. Draw its net and calculate both its surface area and volume.

Solution:

Given: Side length = 3 cm.

Surface Area $(SA)$ calculation:

$SA = 6a^2$

$= 6 \times 3^2$

$= 54 \text{ cm}^2$

Volume $(V)$ calculation:

$V = a^3 = 3^3 = 27 \text{ cm}^3$