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Understanding Elementary Shapes - Polygons

Grade 6CBSE

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

🔑Concepts

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A polygon is a simple closed figure made up entirely of line segments. Visually, imagine drawing several straight lines on a piece of paper such that they all connect end-to-end and the starting point meets the ending point, creating an enclosed space.

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Polygons are classified by the number of sides they have: a Triangle has 33 sides, a Quadrilateral has 44 sides, a Pentagon has 55 sides, a Hexagon has 66 sides, and an Octagon has 88 sides. Visually, as you add more sides, the shape begins to appear more like a circle.

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A 'Regular Polygon' is a shape where all sides have the same length and all interior angles have the same measure. For example, a Square is a regular quadrilateral. In contrast, an 'Irregular Polygon' has sides or angles that are not all equal, such as a rectangle that is longer than it is tall.

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The 'Sides' are the line segments that form the polygon, and a 'Vertex' (plural: vertices) is the point where two sides meet. A 'Diagonal' is a line segment connecting two non-adjacent vertices. Visually, in a square, the lines forming the border are sides, the corners are vertices, and a line drawn from one corner to the opposite corner through the center is a diagonal.

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Triangles (33-sided polygons) are classified by their sides as Scalene (all sides different), Isosceles (at least two sides equal), or Equilateral (all three sides equal). They can also be classified by their angles as Acute (all angles less than 90∘90^{\circ}), Right (one angle exactly 90∘90^{\circ}), or Obtuse (one angle greater than 90∘90^{\circ}).

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Quadrilaterals (44-sided polygons) include specific shapes like the Parallelogram (opposite sides are parallel and equal), the Rhombus (a parallelogram with all four sides equal), the Rectangle (a parallelogram with four right angles), and the Trapezium (a quadrilateral with at least one pair of parallel sides).

📐Formulae

Sum of interior angles of a triangle = 180∘180^{\circ}

Sum of interior angles of a quadrilateral = 360∘360^{\circ}

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

Perimeter of a regular polygon = n×sn \times s, where nn is the number of sides and ss is the length of one side

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

💡Examples

Problem 1:

Calculate the sum of the interior angles of a pentagon.

Solution:

Step 1: Identify the number of sides in a pentagon. A pentagon has n=5n = 5 sides. Step 2: Use the formula for the sum of interior angles: (n−2)×180∘(n - 2) \times 180^{\circ}. Step 3: Substitute n=5n = 5 into the formula: (5−2)×180∘(5 - 2) \times 180^{\circ}. Step 4: Calculate the value: 3×180∘=540∘3 \times 180^{\circ} = 540^{\circ}.

Explanation:

To find the total degrees inside any polygon, we subtract 22 from the number of sides and multiply the result by 180∘180^{\circ} because every polygon can be divided into (n−2)(n-2) triangles.

Problem 2:

A regular hexagon has a side length of 77 cm. Find its perimeter.

Solution:

Step 1: Identify the shape and number of sides. A hexagon has n=6n = 6 sides. Step 2: Since it is a 'regular' hexagon, all 66 sides are equal. Step 3: Use the perimeter formula: P=n×sP = n \times s. Step 4: Substitute the values: P=6×7 cm=42 cmP = 6 \times 7\text{ cm} = 42\text{ cm}.

Explanation:

The perimeter is the total distance around the edge of a shape. For regular polygons, we simply multiply the side length by the total number of sides.