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Geometry - Symmetry in 2D and 3D

Grade 12A Level

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

🔑Concepts

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Line Symmetry (Reflectional Symmetry) occurs when a 2D shape can be folded along a line so that the two halves match exactly. For a regular polygon with nn sides, there are always nn lines of symmetry.

Isosceles triangle with a vertical line of symmetry passing through the top vertex.
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Rotational Symmetry describes how many times a shape fits onto itself during a full 360∘360^\circ turn. This count is called the 'order'. The center of rotation is the fixed point around which the shape turns.

Rectangle showing center of rotation and 180 degree rotation path.
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Planes of Symmetry are the 3D equivalent of lines of symmetry. A plane divides a 3D solid into two mirror-image halves. A sphere has infinite planes of symmetry, while a cylinder has one horizontal plane and infinite vertical planes.

Cylinder cross-section showing a horizontal plane of symmetry.
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Point Symmetry (Order 2 Rotational Symmetry) exists when a shape looks the same when rotated 180∘180^\circ. Every point (x,y)(x, y) on the shape has a matching point (−x,−y)(-x, -y) relative to the center.

📐Formulae

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

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

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

Number of Planes of Symmetry (Rectangular Cuboid, l≠w≠h)=3\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:

For any regular polygon with nn sides, the number of lines of symmetry is equal to nn, and the order of rotational symmetry is also equal to nn. Since an octagon has 8 sides, both values are 8.

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∘72^\circ

Explanation:

Using the formula Angle=360∘n\text{Angle} = \frac{360^\circ}{n}, where nn is the order of symmetry: 360∘5=72∘\frac{360^\circ}{5} = 72^\circ.

Problem 4:

Identify the number of lines of symmetry and the order of rotational symmetry for the rhombus shown, where all sides are equal but interior angles are not 90∘90^\circ.

Rhombus with two diagonal lines of symmetry.

Solution:

  1. Lines of Symmetry: A rhombus has 2 lines of symmetry (the diagonals).
  2. Rotational Symmetry: The shape repeats its appearance twice in a full circle (180∘180^\circ and 360∘360^\circ), so the order is 2.

Explanation:

Unlike a square, a rhombus does not have lines of symmetry through the midpoints of its sides, only through its vertices (diagonals). Because it is not a regular polygon, its order of symmetry is less than the number of sides.

Problem 5:

A right circular cone has a base radius of rr and height hh. Describe the planes of symmetry for this 3D solid.

Side view of a cone showing the vertical axis of symmetry.

Solution:

A right circular cone has infinite planes of symmetry. Every plane that passes through the vertex and the center of the circular base is a plane of symmetry.

Explanation:

Because the base is a circle, any diameter can form the base of a symmetry plane. However, unlike a cylinder, there is no horizontal plane of symmetry because the top (vertex) and bottom (circular base) are different.