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Understanding Quadrilaterals - Polygons and their Classification

Grade 8CBSE

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

🔑Concepts

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A polygon is a simple closed curve made up entirely of line segments. Visually, a polygon like a triangle or a pentagon consists of straight sides that close a space without intersecting each other.

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Polygons are classified by the number of sides or vertices they possess. For example, a 33-sided polygon is a triangle, a 44-sided one is a quadrilateral, a 55-sided one is a pentagon, and an nn-sided one is generally called an nn-gon.

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In a convex polygon, no portion of its diagonals lies in the exterior. All its interior angles are less than 180∘180^{\circ}, which means all its vertices point 'outwards' from the center of the shape.

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A concave polygon has at least one interior angle greater than 180∘180^{\circ} (a reflex angle). Visually, it appears as if one or more vertices are 'caved in', and it is possible to draw a diagonal that passes through the exterior of the shape.

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A regular polygon is both equilateral (all sides are of equal length) and equiangular (all interior angles are of equal measure). For instance, a square is a regular quadrilateral, whereas a rectangle is irregular because its sides are not all equal.

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A diagonal is a line segment connecting two non-consecutive vertices of a polygon. You can visualize diagonals as internal lines that 'cut across' the shape, such as the two crossing lines inside a rectangle.

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The Angle Sum Property states that the sum of the interior angles of a quadrilateral is 360∘360^{\circ}. For any nn-sided polygon, the sum of interior angles can be visualized by dividing it into (n−2)(n - 2) triangles, leading to a total sum of (n−2)×180∘(n - 2) \times 180^{\circ}.

📐Formulae

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

Sum of the exterior angles of any polygon = 360∘360^{\circ}

Number of diagonals in a polygon with nn sides = n(n−3)2\frac{n(n - 3)}{2}

Measure of each interior angle of a regular nn-sided polygon = (n−2)×180∘n\frac{(n - 2) \times 180^{\circ}}{n}

Measure of each exterior angle of a regular nn-sided polygon = 360∘n\frac{360^{\circ}}{n}

💡Examples

Problem 1:

Find the number of sides of a regular polygon if each exterior angle has a measure of 45∘45^{\circ}.

Solution:

  1. We know that the sum of all exterior angles of any polygon is 360∘360^{\circ}.
  2. For a regular polygon, all exterior angles are equal.
  3. Let the number of sides be nn.
  4. The formula for the number of sides is n=360∘Measure of each exterior anglen = \frac{360^{\circ}}{\text{Measure of each exterior angle}}.
  5. Substitute the given value: n=360∘45∘=8n = \frac{360^{\circ}}{45^{\circ}} = 8.

Explanation:

Since the polygon is regular, we divide the total sum of exterior angles (360∘360^{\circ}) by the measure of a single exterior angle to find the number of vertices, which corresponds to the number of sides.

Problem 2:

Find the value of the unknown angle xx in a quadrilateral if the other three angles are 110∘,80∘,110^{\circ}, 80^{\circ}, and 70∘70^{\circ}.

Solution:

  1. The sum of the interior angles of a quadrilateral is 360∘360^{\circ}.
  2. Set up the equation: x+110∘+80∘+70∘=360∘x + 110^{\circ} + 80^{\circ} + 70^{\circ} = 360^{\circ}.
  3. Combine the known angles: x+260∘=360∘x + 260^{\circ} = 360^{\circ}.
  4. Subtract 260∘260^{\circ} from both sides: x=360∘−260∘=100∘x = 360^{\circ} - 260^{\circ} = 100^{\circ}.

Explanation:

This solution uses the angle sum property of quadrilaterals, which dictates that the four interior angles must always total 360∘360^{\circ} regardless of the shape's specific dimensions.