Geometry - Symmetry in 2D and 3D

Grade 12IGCSE

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

🔑Concepts

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Line Symmetry (2D): A shape has line symmetry if a mirror line can be drawn through it such that one half is the reflection of the other.

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Rotational Symmetry (2D): The number of times a shape fits into itself during a full 360-degree turn is called the 'order' of rotational symmetry.

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Planes of Symmetry (3D): A flat surface that divides a 3D object into two identical halves that are reflections of each other.

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Axes of Symmetry (3D): A line about which a 3D solid can be rotated by a certain angle to look exactly as it did in its original position.

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n

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Center of Symmetry: A point such that for every point on the figure, there is another point directly opposite and at the same distance from the center.

📐Formulae

\text{Angle of Rotational Symmetry} = \frac{360^\circ}{n}

\text{Order of Symmetry for Regular } n\text{-gon} = n

\text{Number of Planes of Symmetry (Cube)} = 9

\text{Number of Planes of Symmetry (Rectangular Cuboid, } l \neq w \neq h) = 3

💡Examples

Problem 1:

Determine the number of lines of symmetry and the order of rotational symmetry for a regular octagon.

Solution:

Lines of symmetry: 8; Order of rotational symmetry: 8.

Explanation:

n

Problem 2:

Identify the number of planes of symmetry in a square-based pyramid where all triangular faces are isosceles.

Solution:

4 planes of symmetry.

Explanation:

Two planes pass through the vertices of the square base (diagonals) and the apex. Two planes pass through the midpoints of the opposite sides of the square base and the apex.

Problem 3:

A shape has a rotational symmetry of order 5. Calculate the smallest angle through which the shape must be rotated to coincide with its original position.

Solution:

72^\circ

Explanation:

\text{Angle} = \frac{360^\circ}{n}