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Geometry - Identifying 3D shapes and their nets

Grade 5Cambridge (IGCSE)

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

🔑Concepts

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A net is a 2D pattern that can be folded to make a 3D shape. A cube has a net consisting of six equal squares. There are 11 different net configurations for a cube, but the most common is the 'cross' shape.

A 2D net of a cube showing six squares arranged in a cross shape.
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Prisms have a constant cross-section and two identical ends. A triangular prism has two triangular bases and three rectangular lateral faces. Its net typically looks like three rectangles joined in a row with triangles attached to the sides of one rectangle.

A 2D net of a triangular prism showing three rectangles and two triangles.
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A cylinder consists of two parallel circular bases and a curved surface. When flattened, the curved surface becomes a rectangle, and the net shows two circles attached to the top and bottom of that rectangle.

A net of a cylinder showing a rectangle and two circles.
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Pyramids have a base (which can be any polygon) and triangular faces that meet at a single point called the apex. A square-based pyramid has a square base and four triangular faces.

Net of a square-based pyramid with a central square and four triangles attached to its sides.

📐Formulae

Euler's Formula for Polyhedra: F+V−E=2F + V - E = 2 (where FF = Faces, VV = Vertices, EE = Edges)

Cube Properties: F=6,V=8,E=12F=6, V=8, E=12

Square-based Pyramid Properties: F=5,V=5,E=8F=5, V=5, E=8

Triangular Prism Properties: F=5,V=6,E=9F=5, V=6, E=9

💡Examples

Problem 1:

Identify the 3D shape that consists of 2 circular faces and 1 curved surface. What would its net look like?

Solution:

Cylinder.

Explanation:

A cylinder has two identical circular bases at the top and bottom. Its net consists of two circles and one rectangle (which wraps around to form the curved surface).

Problem 2:

A 3D shape has 4 triangular faces and 4 vertices. Name the shape and describe its net.

Solution:

Triangular-based pyramid (or Tetrahedron).

Explanation:

Since all 4 faces are triangles, it is a tetrahedron. Its net consists of four triangles joined together; commonly seen as one central triangle with three other triangles attached to its sides.

Problem 3:

If a shape is a hexagonal prism, how many faces, vertices, and edges does it have?

Solution:

Faces: 8, Vertices: 12, Edges: 18.

Explanation:

A hexagonal prism has 2 hexagonal bases and 6 rectangular side faces (2+6=82 + 6 = 8). It has 6 vertices on the top base and 6 on the bottom (6+6=126 + 6 = 12). It has 6 edges on the top, 6 on the bottom, and 6 vertical edges connecting them (6+6+6=186 + 6 + 6 = 18).

Problem 4:

Which 3D shape has a net made of one square and four triangles?

Solution:

Square-based pyramid.

Explanation:

The square acts as the base of the pyramid, and the four triangles fold up to meet at a single point (vertex) above the center of the square.

Problem 5:

Identify the 3D shape formed by the following net and calculate its total number of edges EE.

Net showing five rectangles in a row and two pentagons attached to one rectangle.

Solution:

E=15E = 15

Explanation:

The net consists of two pentagonal faces and five rectangular faces. This identifies the shape as a pentagonal prism. For any prism with an nn-sided base, the number of edges is 3n3n. Here, n=5n = 5, so E=3×5=15E = 3 \times 5 = 15.

Problem 6:

A 3D shape is a cone. Describe the components of its net and identify the shape of the lateral surface.

A net of a cone showing a large sector and a small circle.

Solution:

The net of a cone consists of a circle (the base) and a sector of a larger circle (the lateral surface).

Explanation:

When a cone is 'unrolled', the curved surface forms a sector (a pie slice shape). The distance along the arc of this sector must be equal to the circumference of the circular base, C=2πrC = 2 \pi r.