# Counting

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## Counting Rectangles

Q1) Count the number of rectangles in the given figure 1)
15
2) 16
3) 18
4) 20

Solution:

The figure may be labelled as shown. The rectangles composed of two components each are HIJE, EKJ,F, FMNG, GPQH, AEOH, EBFO, OFCG and HOGD i.e. 8 in number.
The rectangles composed of four components each are ABFH, BCGE, CDHF, DAEG and EFGH i.e. 5 in number.
The rectangles composed of six components each are IJFG, KLGH, MNHE and PQEF i.e. 4 in number.
The rectangles composed of eight components each are IJMN, KLPQ and ABCD i.e. 3 in number.
Thus, there are 8 + 5 + 4 + 3 = 20 rectangles in the given figure.
(Here note that the squares are also counted amongst rectangles)

Q2) Count the number of rectangles in the given figure 1)
7
2) 8
3) 9
4) 10

Solution:

The figure may be labelled as shown. The simplest rectangles are ABJI, BCKJ, IJFG and JKEF i.e. 4 in number.
The rectangles composed of two components each are ACKI, BCEF, IKEG and ABFG i.e. 4 in number.
The only rectangle composed of four components is ACEG.
Thus, there are 4 + 4 + 1 = 9 rectangles in the given figure.

### Read More: Questions on Counting number of Squares

Q3) Count the number of rectangles and hexagons in the given figure 1)
30, 5
2) 32, 3
3) 28, 5
4) 30, 3

Solution:

The figure may be labelled as shown. Rectangles:
The simplest rectangles are CVSR, VETS, RSWM and STKW i.e. 4 in number.
The rectangles composed of two components each are CETR, VEKW, RTKM and CVWM i.e. 4 in number.
The rectangles composed of three components each are ACRP, PRMO, EGHT and THIK i.e. 4 in number.
The rectangles composed of four components each are CEKM, AVSP,PSWO, VGHS and SHIW i.e. 5 in number.
The rectangles composed of five components each are AETP, PTKO, CGHR and RHIM i.e. 4 in number.
The rectangles composed of six components each are ACMO and EGIK i.e. 2 in number.
The rectangles composed of eight components each are AGHP, PHIO, AVWO and VGIW i.e. 4 in number.
The rectangles composed of ten components each are AEKO and CGIM i.e. 2 in number.
AGIO is the only rectangle having sixteen components.
Total number of rectangles in the given figure
= 4 + 4 + 4 + 5 + 4 + 2 + 4 + 2 + 1 = 30.

Hexagons:
The hexagons in the given figure are CDEKLM, CEUKMQ, CFHJMQ, BEUKNP and BFHJNP.
So, there are 5 hexagons in the given figure.

### Read More: Shortcuts for Geometrical Patterns

Q4) Count the number of rectangles in the given figure 1)
8
2) 17
3) 18
4) 20

Solution:

The figure may be labelled as shown. The simplest rectangles are ABQP, PQNO, BCDN, NDEM, MEFL, LFJK, FGHR and RHIJ i.e. 8 in number.
The rectangles composed of two components each are ABNO, BCEM, NDFL, MEJK and FGIJ i.e. 5 in number.
The rectangles composed of three components each are ACDO, BCFL, NDJK and LGIK i.e. 4 in number.
There is only one rectangle i.e. BCJK composed of four components.
Total number of rectangles in the figure = 8 + 5 + 4 + 1 = 18.