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Quadrilaterals - Tiling the Plane Using Any 4-gon

Grade 9CBSE

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

🔑Concepts

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Tessellation or tiling of a plane is the process of covering a surface with a repeating pattern of shapes with no overlaps and no gaps. A fundamental property of geometry is that any quadrilateral (convex or concave) can tile the plane because the sum of the interior angles of a quadrilateral is 360∘360^{\circ}.

A series of identical quadrilaterals joined at their vertices and edges to cover a surface without gaps.
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To create a tiling with an arbitrary quadrilateral, you rotate the shape 180∘180^{\circ} around the midpoint of each side. This ensures that at every vertex where the shapes meet, one of each of the four interior angles (∠A,∠B,∠C,∠D∠A, ∠B, ∠C, ∠D) is present.

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The tiling property relies on the Angle Sum Property of a Quadrilateral. Since ∠A+∠B+∠C+∠D=360∘\angle A + \angle B + \angle C + \angle D = 360^{\circ}, arranging these four angles around a single point creates a full circle, allowing the plane to be perfectly filled.

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Even non-convex (concave) quadrilaterals can tile the plane. The logic remains the same: four such shapes can be oriented so that their four distinct angles meet at a common vertex point to form 360∘360^{\circ}.

📐Formulae

Sum of interior angles=(n−2)×180∘\text{Sum of interior angles} = (n-2) \times 180^{\circ}

For a quadrilateral: ∠A+∠B+∠C+∠D=(4−2)×180∘=360∘\text{For a quadrilateral: } \angle A + \angle B + \angle C + \angle D = (4-2) \times 180^{\circ} = 360^{\circ}

Condition for point-tessellation: ∑angles at a vertex=360∘\text{Condition for point-tessellation: } \sum \text{angles at a vertex} = 360^{\circ}

💡Examples

Problem 1:

Show that four identical quadrilaterals with angles 70∘70^{\circ}, 110∘110^{\circ}, 80∘80^{\circ}, and 100∘100^{\circ} can meet at a single point to form a tiling. Draw the vertex point and the meeting angles.

Four angles meeting at a central point summing to 360 degrees.

Solution:

  1. Verify the sum: 70∘+110∘+80∘+100∘=360∘70^{\circ} + 110^{\circ} + 80^{\circ} + 100^{\circ} = 360^{\circ}.
  2. Since the sum is exactly 360∘360^{\circ}, the four different angles can be arranged around a single vertex point such that no gaps exist.
  3. By repeating this arrangement and rotating the quadrilateral 180∘180^{\circ} about side midpoints, the entire plane is covered.

Explanation:

Because the interior angles sum to a full circle, placing one of each angle at a junction point creates a perfect fit.

Problem 2:

Given a concave quadrilateral with interior angles 40∘40^{\circ}, 30∘30^{\circ}, 50∘50^{\circ}, and a reflex angle of 240∘240^{\circ}, explain how it tiles the plane despite its shape.

A concave quadrilateral showing a reflex angle which can be filled by other quadrilaterals during tiling.

Solution:

  1. Sum of interior angles: 40∘+30∘+50∘+240∘=360∘40^{\circ} + 30^{\circ} + 50^{\circ} + 240^{\circ} = 360^{\circ}.
  2. Even though it is concave, the angle sum remains 360∘360^{\circ}.
  3. In a tiling, the reflex angle (240∘)(240^{\circ}) is 'filled' by the other three angles (40∘+30∘+50∘+...40^{\circ} + 30^{\circ} + 50^{\circ} + ...) from adjacent quadrilaterals at various vertices.

Explanation:

Tessellation works for all quadrilaterals because the topological property of having four angles that sum to 360∘360^{\circ} is universal for 4-gons.