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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 of only line segments. Polygons are classified based on the number of sides (or vertices) they have: Triangle (3), Quadrilateral (4), Pentagon (5), Hexagon (6), and so on.

Classification of polygons showing a triangle, quadrilateral, and pentagon.
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A diagonal is a line segment connecting two non-consecutive vertices of a polygon. For a polygon with nn sides, the number of diagonals is given by n(n−3)2\frac{n(n - 3)}{2}.

A pentagon showing diagonals drawn from one vertex to non-consecutive vertices.
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Polygons are Convex if all diagonals lie in the interior, or Concave if at least one diagonal (or part of it) lies in the exterior. In a convex polygon, every interior angle is less than 180∘180^{\circ}.

Comparison of a convex quadrilateral and a concave polygon where a diagonal lies outside.
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A Regular Polygon is both equiangular (all angles equal) and equilateral (all sides equal). Examples include equilateral triangles and squares. Irregular polygons do not have all sides and angles equal.

📐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.

Problem 3:

Find the value of xx in the given pentagon where four interior angles are 100∘,120∘,90∘,100^{\circ}, 120^{\circ}, 90^{\circ}, and 110∘110^{\circ}.

A pentagon with four interior angles labeled and one marked as x.

Solution:

The sum of interior angles of a pentagon (n=5n=5) is: Sum =(5−2)×180∘=3×180∘=540∘= (5 - 2) \times 180^{\circ} = 3 \times 180^{\circ} = 540^{\circ} Summing the given angles: 100∘+120∘+90∘+110∘+x=540∘100^{\circ} + 120^{\circ} + 90^{\circ} + 110^{\circ} + x = 540^{\circ} 420∘+x=540∘420^{\circ} + x = 540^{\circ} x=540∘−420∘x = 540^{\circ} - 420^{\circ} x=120∘x = 120^{\circ}

Explanation:

First, we calculate the total sum of interior angles for a 5-sided polygon using the formula (n−2)×180∘(n-2) \times 180^{\circ}. Then, we subtract the sum of the four known angles from the total to find the missing angle.

Problem 4:

Calculate the measure of each interior angle of a regular hexagon.

A regular hexagon with equal interior angles labeled 'a'.

Solution:

A regular hexagon has n=6n = 6 sides. Sum of interior angles =(6−2)×180∘=4×180∘=720∘= (6 - 2) \times 180^{\circ} = 4 \times 180^{\circ} = 720^{\circ} Since it is a regular polygon, all interior angles are equal. Each interior angle =720∘6=120∘= \frac{720^{\circ}}{6} = 120^{\circ}

Explanation:

For any regular polygon, each interior angle is the total sum of angles divided by the number of sides. For a hexagon, this is 720∘/6720^{\circ} / 6.