Polygon – Definition<\/h3>\n
Any closed 2-D shaped figure bounded by three or more sides is a polygon. If all the sides of the polygon are equal, then it is called the regular polygon. Before going to solve frequency polygon problems, check the formulae here.<\/p>\n
Properties of Polygon<\/h3>\n\n- Sum of all the interior angles of a regular polygon: (n-2)180\u00b0<\/li>\n
- Each interior angle of a regular polygon: (n-2)180\u00b0\/n<\/li>\n
- Number of Diagonals: n(n-3)\/2<\/li>\n
- Sum of all exterior angles of a regular polygon: 360\u00b0<\/li>\n<\/ol>\n
Problem 1:<\/strong><\/p>\nThe runs scored by two teams A and B on the first 60 balls in a cricket match are given below.<\/p>\n
\n\n\nS. No<\/th>\n Number of Balls<\/th>\n Team A<\/th>\n Team B<\/th>\n<\/tr>\n<\/thead>\n \n\n1<\/td>\n 1-6<\/td>\n 2<\/td>\n 5<\/td>\n<\/tr>\n \n2<\/td>\n 7-12<\/td>\n 1<\/td>\n 6<\/td>\n<\/tr>\n \n3<\/td>\n 13-18<\/td>\n 8<\/td>\n 2<\/td>\n<\/tr>\n \n4<\/td>\n 19-24<\/td>\n 9<\/td>\n 10<\/td>\n<\/tr>\n \n5<\/td>\n 25-30<\/td>\n 4<\/td>\n 5<\/td>\n<\/tr>\n \n6<\/td>\n 31-36<\/td>\n 5<\/td>\n 6<\/td>\n<\/tr>\n \n7<\/td>\n 37-42<\/td>\n 6<\/td>\n 3<\/td>\n<\/tr>\n \n8<\/td>\n 43-48<\/td>\n 10<\/td>\n 4<\/td>\n<\/tr>\n \n9<\/td>\n 49-54<\/td>\n 6<\/td>\n 8<\/td>\n<\/tr>\n \n10<\/td>\n 55-60<\/td>\n 2<\/td>\n 10<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nRepresent the data of both the teams on the same graphs by frequency polygons. [Hint: First make the class intervals continuous]<\/p>\n
Solution:<\/strong><\/p>\nAs given in the question,<\/p>\n
To make the class intervals continuous, subtract 0.5 from the lowest value and add 0.5 to the highest value<\/p>\n
Therefore, the new intervals are<\/p>\n
\n\n\nS. No<\/th>\n Number of Balls<\/th>\n Team A<\/th>\n Team B<\/th>\n<\/tr>\n<\/thead>\n \n\n1<\/td>\n 0.5-6.5<\/td>\n 2<\/td>\n 5<\/td>\n<\/tr>\n \n2<\/td>\n 6.5-12.5<\/td>\n 1<\/td>\n 6<\/td>\n<\/tr>\n \n3<\/td>\n 12.5-18.5<\/td>\n 8<\/td>\n 2<\/td>\n<\/tr>\n \n4<\/td>\n 19-24.5<\/td>\n 9<\/td>\n 10<\/td>\n<\/tr>\n \n5<\/td>\n 24.5-30.5<\/td>\n 4<\/td>\n 5<\/td>\n<\/tr>\n \n6<\/td>\n 30.5-36.5<\/td>\n 5<\/td>\n 6<\/td>\n<\/tr>\n \n7<\/td>\n 36.5-42.5<\/td>\n 6<\/td>\n 3<\/td>\n<\/tr>\n \n8<\/td>\n 42.5-48.5<\/td>\n 10<\/td>\n 4<\/td>\n<\/tr>\n \n9<\/td>\n 48.5-54.5<\/td>\n 6<\/td>\n 8<\/td>\n<\/tr>\n \n10<\/td>\n 54.5-60.5<\/td>\n 2<\/td>\n 10<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nTo represent the data on the graph, we require the mean value of the interval i.e.,<\/p>\n
1st interval mean value is 0.5 + 6.5 \/ 2 = 3.5<\/p>\n
2nd interval mean value is \u00a06.5+12.5 \/ 2 = 9.5<\/p>\n
3rd interval mean value is \u00a012.5+18.5 \/ 2 = 15.5<\/p>\n
4th interval mean value is 19+24.5 \/ 2 = 21.75<\/p>\n
5th interval mean value is 24.5+30.5 \/ 2 = 27.5<\/p>\n
6th interval mean value is \u00a030.5+36.5 \/ 2 = 33.5<\/p>\n
7th interval mean value is 36.5+42.5 \/ 2 = 39.5<\/p>\n
8th interval mean value is 42.5+48.5 \/ 2 = 45.5<\/p>\n
9th interval mean value is 48.5+54.5 \/ 2 = 51.5<\/p>\n
10th interval mean value is 54.5+60.5 \/ 2 = 57.5<\/p>\n
The final graph is<\/p>\n
![\"\"](\"https:\/\/www.worksheetsbuddy.com\/wp-content\/uploads\/2021\/04\/Cricket-team-Graphs-300x210.png\")
<\/p>\n
Hence, the frequency polygons graph data is represented here.<\/p>\n
Problem 2:<\/strong><\/p>\nRepresent the data on the same graphs by frequency polygons.<\/p>\n
\n\n\nClass Interval (price of pen)<\/th>\n 10-20<\/th>\n 20-30<\/th>\n 30-40<\/th>\n 40-50<\/th>\n 50-60<\/th>\n<\/tr>\n<\/thead>\n \n\nFrequency (Number of pens sold)<\/strong><\/td>\n15<\/td>\n 20<\/td>\n 30<\/td>\n 25<\/td>\n 5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\nSolution:<\/strong><\/p>\nAs given in the above table,<\/p>\n
The frequency has gradually increased and then decreased. Hence the graph is as follows.<\/p>\n
![\"\"](\"https:\/\/www.worksheetsbuddy.com\/wp-content\/uploads\/2021\/04\/Frequency-Polygons-Graph-300x187.png\")
<\/p>\n
Hence, the frequency polygons graph data is represented here.<\/p>\n
Problem 3:<\/strong><\/p>\nDraw a histogram, a frequency polygon and frequency curve of the following data:<\/p>\n
\n\n\nMarks<\/th>\n 0-10<\/th>\n 10-20<\/th>\n 20-30<\/th>\n 30-40<\/th>\n 40-50<\/th>\n 50-60<\/th>\n<\/tr>\n<\/thead>\n \n\nNumber of Students<\/strong><\/td>\n5<\/td>\n 12<\/td>\n 15<\/td>\n 22<\/td>\n 14<\/td>\n 4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\u00a0Solution:<\/strong><\/p>\nBy considering the above-given data![\"\"](\"https:\/\/www.worksheetsbuddy.com\/wp-content\/uploads\/2021\/04\/Frequency-Polygon-Graph-300x128.png\")
<\/p>\n
Hence, the histogram, frequency polygons graph, and curve are represented here.<\/p>\n
Problem 4:<\/strong><\/p>\nEach interior angle of a regular polygon is three times its exterior angle, then the number of sides of the regular polygon is?<\/p>\n
Solution:<\/strong><\/p>\nAs given in the question,<\/p>\n
Each interior angle of a regular polygon is three times its exterior angle<\/p>\n
Suppose the exterior angle = 1<\/p>\n
Then the interior angle = 3<\/p>\n
The sum of interior and exterior angle = 4<\/p>\n
Each side of a regular polygon = 180\u00b0<\/p>\n
Therefore, the angle of one side = 180\/4 = 45\u00b0<\/p>\n
The complete angle of a polygon = 360\u00b0<\/p>\n
No of sides of a polygon = 360\u00ba\/N<\/p>\n
= 360\/45<\/p>\n
= 8<\/p>\n
Therefore, the regular polygon has 8 sides<\/p>\n
Problem 5:<\/strong><\/p>\nWith the help of the frequency distribution for the calculus table, draw the frequency polygon graph?<\/p>\n
\n\n\nLower Bound<\/th>\n Upper Bound<\/th>\n Frequency<\/th>\n Cumulative Frequency<\/th>\n<\/tr>\n<\/thead>\n \n\n49.5<\/td>\n 59.5<\/td>\n 5<\/td>\n 5<\/td>\n<\/tr>\n \n59.5<\/td>\n 69.5<\/td>\n 10<\/td>\n 15<\/td>\n<\/tr>\n \n69.5<\/td>\n 79.5<\/td>\n 30<\/td>\n 45<\/td>\n<\/tr>\n \n79.5<\/td>\n 89.5<\/td>\n 40<\/td>\n 85<\/td>\n<\/tr>\n \n89.5<\/td>\n 99.5<\/td>\n 15<\/td>\n 100<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\u00a0Solution:<\/strong><\/p>\nAs given in the question,<\/p>\n
We consider the scores and test scores of frequency distribution for the calculus<\/p>\n
The frequency polygon distribution graph is<\/p>\n
![\"\"](\"https:\/\/www.worksheetsbuddy.com\/wp-content\/uploads\/2021\/04\/Frequeny-Polygon-Graph-300x210.png\")
<\/p>\n
Hence, the frequency polygons graph data is represented here.<\/p>\n
Problem 6:<\/strong><\/p>\nWith the help of the frequency distribution, draw the frequency polygon graph?<\/p>\n
\n\n\nAge (in years)<\/th>\n 0-5<\/th>\n 5-10<\/th>\n 10-20<\/th>\n 20-40<\/th>\n 40-50<\/th>\n 50-80<\/th>\n<\/tr>\n<\/thead>\n \n\nNumber of Persons<\/strong><\/td>\n7<\/td>\n 9<\/td>\n 14<\/td>\n 12<\/td>\n 8<\/td>\n 15<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\u00a0Solution:<\/strong><\/p>\nAs we know,<\/p>\n
Adjusted frequency of a class = Minimum class size \/ Class size * Frequency<\/p>\n
The frequency polygon distribution graph is as follows<\/p>\n
![\"\"](\"https:\/\/www.worksheetsbuddy.com\/wp-content\/uploads\/2021\/04\/Frequency-Polygon-Graph-1-300x282.png\")
<\/p>\n
Hence, the frequency polygons graph data is represented here.<\/p>\n
Problem 7:<\/strong><\/p>\nWith the help of the frequency distribution, draw the frequency polygon graph?<\/p>\n
\n\n\nCB<\/th>\n Frequency<\/th>\n<\/tr>\n<\/thead>\n \n\n0.5 – 10.5<\/td>\n 2<\/td>\n<\/tr>\n \n10.5 – 20.5<\/td>\n 3<\/td>\n<\/tr>\n \n20.5 – 30.5<\/td>\n 6<\/td>\n<\/tr>\n \n30.5 – 40.5<\/td>\n 9<\/td>\n<\/tr>\n \n40.5 – 50.5<\/td>\n 4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\u00a0Solution:<\/strong><\/p>\nAs we know,<\/p>\n
The frequency polygon distribution graph is as follows<\/p>\n
![\"\"](\"https:\/\/www.worksheetsbuddy.com\/wp-content\/uploads\/2021\/04\/Frequency-Polygon-300x251.png\")
<\/p>\n
Hence, the frequency polygons graph data is represented here.<\/p>\n
Problem 8:<\/strong><\/p>\nWith the help of the frequency distribution, draw the frequency polygon graph?<\/p>\n
\n\n\nUpper Bound<\/th>\n Lower Bound<\/th>\n Midpoint<\/th>\n Frequency<\/th>\n<\/tr>\n<\/thead>\n \n\n10<\/td>\n 19<\/td>\n 14.5<\/td>\n 4<\/td>\n<\/tr>\n \n20<\/td>\n 29<\/td>\n 24.5<\/td>\n 5<\/td>\n<\/tr>\n \n30<\/td>\n 39<\/td>\n 34.5<\/td>\n 7<\/td>\n<\/tr>\n \n40<\/td>\n 49<\/td>\n 44.5<\/td>\n 5<\/td>\n<\/tr>\n \n50<\/td>\n 59<\/td>\n 54.5<\/td>\n 5<\/td>\n<\/tr>\n \n60<\/td>\n 69<\/td>\n 64.5<\/td>\n 4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\u00a0Solution:<\/strong><\/p>\nAs we know,<\/p>\n
The frequency polygon distribution graph is as follows<\/p>\n
![\"\"](\"https:\/\/www.worksheetsbuddy.com\/wp-content\/uploads\/2021\/04\/Frequency-Polygon-1-300x180.png\")
<\/p>\n
Hence, the frequency polygons graph data is represented here.<\/p>\n
Problem 9:<\/strong><\/p>\nThe following is the distribution of workers of a factory. On the basis of this information, construct a histogram and convert it into a frequency polygon.<\/p>\n
Problem 1:<\/strong><\/p>\n The runs scored by two teams A and B on the first 60 balls in a cricket match are given below.<\/p>\n Represent the data of both the teams on the same graphs by frequency polygons. [Hint: First make the class intervals continuous]<\/p>\n Solution:<\/strong><\/p>\n As given in the question,<\/p>\n To make the class intervals continuous, subtract 0.5 from the lowest value and add 0.5 to the highest value<\/p>\n Therefore, the new intervals are<\/p>\n To represent the data on the graph, we require the mean value of the interval i.e.,<\/p>\n 1st interval mean value is 0.5 + 6.5 \/ 2 = 3.5<\/p>\n 2nd interval mean value is \u00a06.5+12.5 \/ 2 = 9.5<\/p>\n 3rd interval mean value is \u00a012.5+18.5 \/ 2 = 15.5<\/p>\n 4th interval mean value is 19+24.5 \/ 2 = 21.75<\/p>\n 5th interval mean value is 24.5+30.5 \/ 2 = 27.5<\/p>\n 6th interval mean value is \u00a030.5+36.5 \/ 2 = 33.5<\/p>\n 7th interval mean value is 36.5+42.5 \/ 2 = 39.5<\/p>\n 8th interval mean value is 42.5+48.5 \/ 2 = 45.5<\/p>\n 9th interval mean value is 48.5+54.5 \/ 2 = 51.5<\/p>\n 10th interval mean value is 54.5+60.5 \/ 2 = 57.5<\/p>\n The final graph is<\/p>\n Hence, the frequency polygons graph data is represented here.<\/p>\n Problem 2:<\/strong><\/p>\n Represent the data on the same graphs by frequency polygons.<\/p>\n Solution:<\/strong><\/p>\n As given in the above table,<\/p>\n The frequency has gradually increased and then decreased. Hence the graph is as follows.<\/p>\n Hence, the frequency polygons graph data is represented here.<\/p>\n Problem 3:<\/strong><\/p>\n Draw a histogram, a frequency polygon and frequency curve of the following data:<\/p>\n \u00a0Solution:<\/strong><\/p>\n By considering the above-given data Hence, the histogram, frequency polygons graph, and curve are represented here.<\/p>\n Problem 4:<\/strong><\/p>\n Each interior angle of a regular polygon is three times its exterior angle, then the number of sides of the regular polygon is?<\/p>\n Solution:<\/strong><\/p>\n As given in the question,<\/p>\n Each interior angle of a regular polygon is three times its exterior angle<\/p>\n Suppose the exterior angle = 1<\/p>\n Then the interior angle = 3<\/p>\n The sum of interior and exterior angle = 4<\/p>\n Each side of a regular polygon = 180\u00b0<\/p>\n Therefore, the angle of one side = 180\/4 = 45\u00b0<\/p>\n The complete angle of a polygon = 360\u00b0<\/p>\n No of sides of a polygon = 360\u00ba\/N<\/p>\n = 360\/45<\/p>\n = 8<\/p>\n Therefore, the regular polygon has 8 sides<\/p>\n Problem 5:<\/strong><\/p>\n With the help of the frequency distribution for the calculus table, draw the frequency polygon graph?<\/p>\n \u00a0Solution:<\/strong><\/p>\n As given in the question,<\/p>\n We consider the scores and test scores of frequency distribution for the calculus<\/p>\n The frequency polygon distribution graph is<\/p>\n Hence, the frequency polygons graph data is represented here.<\/p>\n Problem 6:<\/strong><\/p>\n With the help of the frequency distribution, draw the frequency polygon graph?<\/p>\n \u00a0Solution:<\/strong><\/p>\n As we know,<\/p>\n Adjusted frequency of a class = Minimum class size \/ Class size * Frequency<\/p>\n The frequency polygon distribution graph is as follows<\/p>\n Hence, the frequency polygons graph data is represented here.<\/p>\n Problem 7:<\/strong><\/p>\n With the help of the frequency distribution, draw the frequency polygon graph?<\/p>\n \u00a0Solution:<\/strong><\/p>\n As we know,<\/p>\n The frequency polygon distribution graph is as follows<\/p>\n Hence, the frequency polygons graph data is represented here.<\/p>\n Problem 8:<\/strong><\/p>\n With the help of the frequency distribution, draw the frequency polygon graph?<\/p>\n \u00a0Solution:<\/strong><\/p>\n As we know,<\/p>\n The frequency polygon distribution graph is as follows<\/p>\n Hence, the frequency polygons graph data is represented here.<\/p>\n Problem 9:<\/strong><\/p>\n The following is the distribution of workers of a factory. On the basis of this information, construct a histogram and convert it into a frequency polygon.<\/p>\n\n\n
\n \nS. No<\/th>\n Number of Balls<\/th>\n Team A<\/th>\n Team B<\/th>\n<\/tr>\n<\/thead>\n \n 1<\/td>\n 1-6<\/td>\n 2<\/td>\n 5<\/td>\n<\/tr>\n \n 2<\/td>\n 7-12<\/td>\n 1<\/td>\n 6<\/td>\n<\/tr>\n \n 3<\/td>\n 13-18<\/td>\n 8<\/td>\n 2<\/td>\n<\/tr>\n \n 4<\/td>\n 19-24<\/td>\n 9<\/td>\n 10<\/td>\n<\/tr>\n \n 5<\/td>\n 25-30<\/td>\n 4<\/td>\n 5<\/td>\n<\/tr>\n \n 6<\/td>\n 31-36<\/td>\n 5<\/td>\n 6<\/td>\n<\/tr>\n \n 7<\/td>\n 37-42<\/td>\n 6<\/td>\n 3<\/td>\n<\/tr>\n \n 8<\/td>\n 43-48<\/td>\n 10<\/td>\n 4<\/td>\n<\/tr>\n \n 9<\/td>\n 49-54<\/td>\n 6<\/td>\n 8<\/td>\n<\/tr>\n \n 10<\/td>\n 55-60<\/td>\n 2<\/td>\n 10<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n \n\n
\n \nS. No<\/th>\n Number of Balls<\/th>\n Team A<\/th>\n Team B<\/th>\n<\/tr>\n<\/thead>\n \n 1<\/td>\n 0.5-6.5<\/td>\n 2<\/td>\n 5<\/td>\n<\/tr>\n \n 2<\/td>\n 6.5-12.5<\/td>\n 1<\/td>\n 6<\/td>\n<\/tr>\n \n 3<\/td>\n 12.5-18.5<\/td>\n 8<\/td>\n 2<\/td>\n<\/tr>\n \n 4<\/td>\n 19-24.5<\/td>\n 9<\/td>\n 10<\/td>\n<\/tr>\n \n 5<\/td>\n 24.5-30.5<\/td>\n 4<\/td>\n 5<\/td>\n<\/tr>\n \n 6<\/td>\n 30.5-36.5<\/td>\n 5<\/td>\n 6<\/td>\n<\/tr>\n \n 7<\/td>\n 36.5-42.5<\/td>\n 6<\/td>\n 3<\/td>\n<\/tr>\n \n 8<\/td>\n 42.5-48.5<\/td>\n 10<\/td>\n 4<\/td>\n<\/tr>\n \n 9<\/td>\n 48.5-54.5<\/td>\n 6<\/td>\n 8<\/td>\n<\/tr>\n \n 10<\/td>\n 54.5-60.5<\/td>\n 2<\/td>\n 10<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n <\/p>\n\n\n
\n \nClass Interval (price of pen)<\/th>\n 10-20<\/th>\n 20-30<\/th>\n 30-40<\/th>\n 40-50<\/th>\n 50-60<\/th>\n<\/tr>\n<\/thead>\n \n Frequency (Number of pens sold)<\/strong><\/td>\n 15<\/td>\n 20<\/td>\n 30<\/td>\n 25<\/td>\n 5<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n <\/p>\n\n\n
\n \nMarks<\/th>\n 0-10<\/th>\n 10-20<\/th>\n 20-30<\/th>\n 30-40<\/th>\n 40-50<\/th>\n 50-60<\/th>\n<\/tr>\n<\/thead>\n \n Number of Students<\/strong><\/td>\n 5<\/td>\n 12<\/td>\n 15<\/td>\n 22<\/td>\n 14<\/td>\n 4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n <\/p>\n\n\n
\n \nLower Bound<\/th>\n Upper Bound<\/th>\n Frequency<\/th>\n Cumulative Frequency<\/th>\n<\/tr>\n<\/thead>\n \n 49.5<\/td>\n 59.5<\/td>\n 5<\/td>\n 5<\/td>\n<\/tr>\n \n 59.5<\/td>\n 69.5<\/td>\n 10<\/td>\n 15<\/td>\n<\/tr>\n \n 69.5<\/td>\n 79.5<\/td>\n 30<\/td>\n 45<\/td>\n<\/tr>\n \n 79.5<\/td>\n 89.5<\/td>\n 40<\/td>\n 85<\/td>\n<\/tr>\n \n 89.5<\/td>\n 99.5<\/td>\n 15<\/td>\n 100<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n <\/p>\n\n\n
\n \nAge (in years)<\/th>\n 0-5<\/th>\n 5-10<\/th>\n 10-20<\/th>\n 20-40<\/th>\n 40-50<\/th>\n 50-80<\/th>\n<\/tr>\n<\/thead>\n \n Number of Persons<\/strong><\/td>\n 7<\/td>\n 9<\/td>\n 14<\/td>\n 12<\/td>\n 8<\/td>\n 15<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n <\/p>\n\n\n
\n \nCB<\/th>\n Frequency<\/th>\n<\/tr>\n<\/thead>\n \n 0.5 – 10.5<\/td>\n 2<\/td>\n<\/tr>\n \n 10.5 – 20.5<\/td>\n 3<\/td>\n<\/tr>\n \n 20.5 – 30.5<\/td>\n 6<\/td>\n<\/tr>\n \n 30.5 – 40.5<\/td>\n 9<\/td>\n<\/tr>\n \n 40.5 – 50.5<\/td>\n 4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n <\/p>\n\n\n
\n \nUpper Bound<\/th>\n Lower Bound<\/th>\n Midpoint<\/th>\n Frequency<\/th>\n<\/tr>\n<\/thead>\n \n 10<\/td>\n 19<\/td>\n 14.5<\/td>\n 4<\/td>\n<\/tr>\n \n 20<\/td>\n 29<\/td>\n 24.5<\/td>\n 5<\/td>\n<\/tr>\n \n 30<\/td>\n 39<\/td>\n 34.5<\/td>\n 7<\/td>\n<\/tr>\n \n 40<\/td>\n 49<\/td>\n 44.5<\/td>\n 5<\/td>\n<\/tr>\n \n 50<\/td>\n 59<\/td>\n 54.5<\/td>\n 5<\/td>\n<\/tr>\n \n 60<\/td>\n 69<\/td>\n 64.5<\/td>\n 4<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n <\/p>\n
![\"\"](\"https:\/\/www.worksheetsbuddy.com\/wp-content\/uploads\/2021\/04\/Frequency-Polygon-2-300x233.png\")
<\/p>\n![\"\"](\"https:\/\/www.worksheetsbuddy.com\/wp-content\/uploads\/2021\/04\/Frequency-Polygon-3-300x240.png\")
<\/p>\n