{"id":5936,"date":"2021-12-16T03:06:35","date_gmt":"2021-12-16T03:06:35","guid":{"rendered":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/?page_id=5936"},"modified":"2022-01-03T08:00:43","modified_gmt":"2022-01-03T08:00:43","slug":"number-of-diagonals-of-a-polygon-solved-examples","status":"publish","type":"page","link":"https:\/\/stgwebsite.mindspark.in\/studymaterial\/math-concepts\/number-of-diagonals-of-a-polygon-solved-examples\/","title":{"rendered":"Number of diagonals of a polygon – Solved Examples"},"content":{"rendered":"

[et_pb_section fb_built=”1″ admin_label=”Section” module_class=”mainsec” _builder_version=”4.10.4″ _module_preset=”default” background_color=”#e0f2fd” z_index=”1″ custom_padding=”5px||5px||true|false” locked=”off” collapsed=”off” global_colors_info=”{}”][et_pb_row column_structure=”3_5,2_5″ custom_padding_last_edited=”on|phone” _builder_version=”4.10.8″ _module_preset=”default” background_color=”#FFFFFF” width=”100%” max_width=”1310px” custom_padding=”|51px|40px|51px|false|true” custom_padding_tablet=”” custom_padding_phone=”|40px|30px|40px|false|true” border_radii=”on|10px|10px|10px|10px” global_colors_info=”{}”][et_pb_column type=”3_5″ admin_label=”Column L” _builder_version=”4.9.10″ _module_preset=”default” global_colors_info=”{}”][et_pb_text admin_label=”Acute Angles
\n” _builder_version=”4.13.1″ _module_preset=”default” header_font=”|700|||||||” header_text_align=”left” header_font_size=”50px” header_line_height=”1.18em” custom_padding=”|0px||4px|false|false” header_font_size_tablet=”” header_font_size_phone=”35px” header_font_size_last_edited=”on|phone” global_colors_info=”{}”]<\/p>\n

Number of diagonals of a polygon – Solved Examples<\/h1>\n

[\/et_pb_text][et_pb_text admin_label=”Text” _builder_version=”4.13.1″ _module_preset=”default” text_font_size=”16px” header_2_font=”|600|||||||” header_2_text_color=”#a01414″ header_3_font=”|600|||||||” custom_padding=”15px|15px|54px|4px|false|false” global_colors_info=”{}”]<\/p>\n

Number of diagonals of a polygon\u00a0<\/b><\/h2>\n

A line segment connecting any two vertices (non-adjacent) of a polygon is known as the diagonal of the polygon. A polygon needs to have at least 4 sides to have a diagonal. Triangle having three sides doesn\u2019t have a diagonal because all the three vertices are adjacent to each other.<\/span><\/p>\n

The number of diagonals of a polygon having \u201ca\u201d number of sides is equal to\\frac{a(a-3)}{2}<\/span> where (a > 3).<\/span><\/p>\n

From each vertex (a – 3) diagonals can be drawn. So, from \u201ca \u201cnumber of vertices a total of \u201ca (a – 3)\u201d diagonals can be drawn. But the diagonals drawn from opposite sides should be counted as one. Hence by discarding the duplicates, a total number of \\frac{a(a-3)}{2}<\/span> <\/span>diagonals can be drawn in a polygon having \u201ca\u201d sides.<\/span><\/p>\n

<\/span><\/p>\n

Let us find the number of diagonals of some polygons using the above formula.<\/span><\/p>\n

1. Quadrilateral<\/strong><\/p>\n

Number of sides (a) = 4<\/span><\/p>\n

\\text { Number of diagonals }=\\frac{a(a-3)}{2}=\\frac{4(4-3)}{2}=\\frac{4 \\times 1}{2}=2<\/span><\/span><\/p>\n

<\/span><\/p>\n

2. Pentagon\u00a0<\/strong><\/p>\n

Number of sides (a) = 5<\/span><\/p>\n

\\text { Number of diagonals }=\\frac{a(a-3)}{2}=\\frac{5(5-3)}{2}=\\frac{5 \\times 2}{2}=5<\/span><\/span><\/p>\n

<\/span><\/p>\n

3. Hexagon\u00a0<\/strong><\/p>\n

Number of sides (a) = 6<\/span><\/p>\n

\\text { Number of diagonals }=\\frac{a(a-3)}{2}=\\frac{6(6-3)}{2}=\\frac{6 \\times 3}{2}=9<\/span><\/span><\/p>\n

<\/span><\/p>\n

4. Heptagon\u00a0<\/strong><\/p>\n

Number of sides (a) = 7<\/span><\/p>\n

\\text { Number of diagonals }=\\frac{a(a-3)}{2}=\\frac{7(7-3)}{2}=\\frac{7 \\times 4}{2}=14<\/span><\/span><\/p>\n

<\/span><\/p>\n

\"\"<\/span><\/p>\n

<\/span><\/p>\n

Given Below is the table showing the number of diagonals for different types of polygon.<\/span><\/p>\n

<\/span><\/p>\n

\"\"<\/span><\/p>\n

<\/span><\/p>\n

Solved Examples<\/b><\/h2>\n

<\/b><\/p>\n

1. How many diagonals does a nonagon have?<\/b><\/p>\n

Nonagon is a polygon having 9 sides<\/span><\/p>\n

Number of sides (a) = 9<\/span><\/p>\n

Number of diagonals =\\frac{a(a-3)}{2}=\\frac{9(9-3)}{2}=\\frac{9 \\times 6}{2}=27<\/span><\/span><\/p>\n

Hence there are 27 diagonals in a nonagon.<\/span><\/p>\n

<\/span><\/p>\n

2. A polygon has 54 diagonals. Find the number of sides of this polygon.<\/b><\/p>\n

Number of diagonals\u00a0 =\\frac{a(a-3)}{2}<\/span><\/span><\/span><\/p>\n

\\Rightarrow \\quad 54=\\frac{a(a-3)}{2}<\/span><\/span><\/p>\n

\\Rightarrow a(a-3)=54 \\times 2=108<\/span><\/span><\/p>\n

\\Rightarrow \\quad a=12<\/span><\/span><\/p>\n

Hence the polygon having 54 diagonals have 12 sides.<\/span><\/p>\n

<\/span><\/p>\n

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Explore Other Topics<\/h1>\n

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Geometry<\/a><\/div>\n
Trigonometry<\/a><\/div>\n
Operations<\/a><\/div>\n
Numbers<\/a><\/div>\n<\/div>\n

[\/et_pb_text][et_pb_text admin_label=”Related Concepts
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Related Concepts<\/h1>\n

[\/et_pb_text][et_pb_text _builder_version=”4.13.1″ _module_preset=”default” text_line_height=”2.2em” link_font_size=”16px” custom_margin=”||0px||false|false” custom_padding=”10px|15px|10px|28px|true|false” locked=”off” global_colors_info=”{}”]<\/p>\n

\n
Convex Polygon<\/a><\/div>\n
Area of Octagon<\/a><\/div>\n
Area of regular Hexagon<\/a><\/a><\/div>\n<\/div>\n

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Frequently Asked Questions\u00a0<\/span><\/span><\/h1>\n
    <\/ol>\n

    Q1.\u00a0<\/strong>Define the diagonal of a polygon?<\/b>
    <\/strong><\/h3>\n

    Ans:\u00a0<\/strong><\/span>A line segment connecting any two vertices (non-adjacent) of a polygon is known as the diagonal of the polygon.<\/span><\/p>\n

    Q2.\u00a0<\/strong>What is the minimum number of sides required in a polygon to have a diagonal?<\/b><\/h3>\n

    Ans:\u00a0<\/strong>A polygon needs to have at least 4 sides to have a diagonal. Triangle having three sides doesn\u2019t have a diagonal because all the three vertices are adjacent to each other.<\/p>\n

    Q3.\u00a0<\/span>How to calculate the exact number of diagonals in a polygon?<\/b><\/h3>\n

    Ans:<\/strong> The number of diagonals of a polygon having \u201ca\u201d number of sides is equal to \\frac{a(a-3)}{2}<\/span> where (a > 3).<\/p>\n

    [\/et_pb_text][\/et_pb_column][\/et_pb_row][\/et_pb_section]<\/p>\n","protected":false},"excerpt":{"rendered":"

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