FIGURE PARTITION
The problems on figure partition are based on counting number of figure generated due to partition lines.
(i) If a square is subdivided into n parts on each side, then the total number of squares
(ii) Total number of rectangles (including squares) in a rectangular of size
Ex 1. What is the number of straight lines in the following figure ?
(A) 11 (B) 14 (C) 16 (D) 17
Sol. The figure is labelled as shown. Clearly, there are 3 horizontal lines namely AE, LF and KG. There are 5 vertical lines :
AK, BJ, Cl, DH and EG. There are 6 slanting lines : LC, KE, IF, LI, AG and CF. Thus, there are 3 + 5 + 6 = 14 straight lines in the figure.
Ex.2 How many squares does the figure have ?
(A) 6 (B) 7 (C) 9 (D) 10
Sol. The figure may be labeled as shown :
The squares composed to two components each, are ABKJ, BCLK, CDEL, LEFG, KLGH, and JKHI. Thus there are
6 such squares. Only one square, KCEG is composed of four components. Two squares namely, ACGI and BDFH are
composed of eight components each. Thus, there are 2 such squares. : There are 6 + 1 + 2 = 9 squares in the given figure.
Ex.3 How many parallelograms are there in the figure below ?
(A) 14 (B) 15 (C) 16 (D) 18
Sol. We can label the figure as shown.
The simplest parallelograms are ABFE, BCGF, CDHG, EFJI, FGKJ, and GHKL. These are 6 in number.
The parallelograms composed of two components each, are ACGE, BDHF, EGKI, FHLJ, BCKJ, ABJI, and CDLK.
Thus, there are 7 such parallelograms. The parallelograms composed of three components are ADHE and EHLI i.e. 2
in number. There is only one parallelograms composed of six components, namely ADLI. Thus there are 6 + 7 + 2 + 2 + 1 = 18. parallelograms in the figure. Hence,
Ex.4 What is the number of rectangles in the following figure ?
(A) 6 (B) 7 (C) 8 (D) 9
Sol. The figure is labelled as shown :
Simplest rectangles are AEHG, EFIH, FBKJ, JKCL and GILD. i.e there are 5 such rectangles. The
rectangles composed of two components each are AFIG and FBCL. Thus, there are 2 such rectangles. Only one
rectangles, namely AFLD is composed of 3 components and only one rectangle, namely ABCD is composed
of 5 components. Thus, there are 5 + 2 + 1 + 1=9 rectangles in the figure.
Ex.5 Determine the number is pentagons in the following figure :
(A)5 (B) 6 (C) 8 (D) 10
Sol. The figure may be labelled as follows :
In this case, six pentagons have been formed by the combination of three triangles and two rhombuses
–ADFHJ, CFHJL, EHJLB, GJLBD, ILBDF and KBDFH. Four other pentagons are formed by the combination
of three triangles and BLIFM.
ASSIGNMENT
1. Count the number of squares in the following figure ?
(A) 16 (B) 17 (C) 30 (D) 35
2. Count the number of straight lines and triangles in the following figure ?
(A) 10 straight lines and 34 triangles
(B) 9 straight lines and 34 triangles
(C) 9 straight lines and 36 triangles
(D) 10 straight lines and 36 triangles
3. Count the number of triangles and squares in the following figure ?
(A) 28 triangles, 10 squares
(B) 28 triangles, 8 squares
(C) 32 triangles, 10 squares
(D) 32 triangles, 8 squares
4. How many squares are there in the following figure ?
(A) 13 (B) 14 (C) 16 (D) 15
5. How many squares does the figure have ?
(A) 17 (B) 12 (C) 13 (D) 15
6. How many squares does the following figure have ?
(A) 17 (B) 18 (C) 13 (D) 16
7. How many triangles are there the following figure ?
(A) 19 (B) 21 (C) 27 (D) 48
8. How many rectangles are there in the given figure ?
(A) 6 (B) 7 (C) 8 (D) 9
9. How many triangles does the following figure contain ?
(A) 11 (B) 10 (C) 6 (D) 12
10. How many triangles and squares are there in the following figure ?
(A) 28 triangles, 5 squares
(B) 24 triangles, 4 squares
(C) 28 triangles, 4 squares
(D) 24 triangles, 5 squares
11. How many triangles are there in the figure below ?
(A) 5 (B)6 (C) 8 (D) 10
12. Count the number of squares in the following figure -
(A) 15 (B) 21 (C) 24 (D) 26
13. Count the number of triangles in the following figure.
(A) 8 (B) 10 (C) 12 (D) 14
14. Find the number of convex pentagons -
(A) 2 (B) 3 (C) 4 (D) 6
15 How many triangles are there in the following figure -
(A) 11 (B) 14 (C) 16 (D) 7
16. How many triangles are there in the following figure ?
(A) 6 (B) 7 (C) 8 (D) 9
17. How many quadrilateral there in the following figure ?
(A) 11 (B) 8 (C) 2 (D) 4
18. How many squares does the figure have ?
(A) 10 (B) 11 (C) 12 (D) 14
Directions : (19 to 23) How many triangles are
there in the following figures ?
19. 
(A) 14 or more (B) 12 to 13
(C) 10 to 11 (D) 7 to 9
20. 
(A) 11 or more (B) 10
(C) 9 (D) 8 or less
21. 
(A) 19 (B) 21 (C) 25 (D) 23
22. 
(A) 10 (B) 16 (C) 14 (D) 12
23. 
(A) 21 or more (B) 13 to 20
(C) 6 to 9 (D) 7 to 9
24. In the following figure, the number of triangles are -
(A) 8 (B) 12 (C) 15 (D) 16
ANSWER KEY
1) C 2) C 3) C 4) D 5) A 6) B 7) C 8) D 9) D 10) A 11) D 12) C 13) C
14) D 15) B 16) C 17) A 18) B 19) A 20) A 21) C 22) B 23) B 24) D