**CAT Questions on Polygons With Solutions:**

Geometry Polygons questions and answers for CAT. Polygons problems with solutions for CAT exam. Download excellent CAT preparation app to directly solve and practice questions.

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Question 1:Â If the sum of interior angles of a polygon is 900 degrees, what is the number of diagonals it has?

a) 9

b) 14

c) 20

d) None of these

Question 2:Â In a regular pentagon PQRST, what is the ratio of the area of triangle PRS to the area of the pentagon PQRST?

a) 1 / (sin 72 + 4)

b) 1 / (cos 72 + 4)

c) 1 / (4 cos 72 + 1)

d) 1 / (4 sin 72 + 1)

Question 3:Â A regular polygon P has 135 diagonals. Find the exterior angle of the polygon P.

a) $$18^{\circ}$$

b) $$20^{\circ}$$

c) $$25^{\circ}$$

d) $$30^{\circ}$$

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Question 4:Â A cylindrical pencil consists of a layer of wood surrounding a solid cylinder of graphite. The radius of the pencil is 6 mm, the radius of the graphite cylinder is 2 mm and the length of the pencil is 10 cm. Find the cost of the material used in a pencil, if the cost of wood is Rs. 0.70/cm3 and that of graphite is Rs. 2.80/cm3.

a) Rs. 7.85

b) Rs. 8.76

c) Rs. 9.36

d) Rs. 10.56

Question 5:Â If the shortest diagonal in a regular hexagon is 10 cm, what is the length of the longest diagonal (in cm)? Round your answer to the nearest integer. Given $$\sqrt3 = 1.732$$.

a) 12

b) 13

c) 14

d) 15

**Solutions forÂ CAT Questions on Polygons With Solutions:**

**Solutions:**

**1) Answer (B)**

The sum of interior angles for an â€˜nâ€™ sided polygon is (2n-4) \times 90 degrees.

If this value is 900, then the polygon is a heptagon.

The number of diagonals for an â€˜nâ€™ sided polygon = $$ \frac{n(n-3)}{2}$$

= 14 for a heptagon.

Factors of a number – Formulas for CAT

Formulas on Number system and factorialsÂ

**2) Answer (C)**

Each angle of a regular pentagon is 108 degrees. In triangle PTS, angle T = 108 degrees, and PT = TS. So, angle TPS = angle TSP = 36 degrees.

Similarly, angle QPR = 36 degrees. So, angle SPR = 108 – (36 + 36) = 36 degrees

Also, PS = PR

Area of triangle PSR = Â½ * PS * PR * sin 36

Area of the pentagon = Sum of areas of the three triangles = Area of triangle PTS + Area of triangle PSR + Area of triangle PQR

= Â½ * PT * PS * sin (angle TPS) + Â½ * PS * PR * sin 36 + Â½ * PQ * PR * sin (angle QPR)

= Â½ * PT * PS * sin 36 + Â½ * PS * PR * sin 36 + Â½ * PQ * PR * sin 36

Required ratio = Â½ * PS * PR * sin 36 / (Â½ * PT * PS * sin 36 + Â½ * PS * PR * sin 36 + Â½ * PQ * PR * sin 36)

= PR / (PT + PR + PQ)

PT = PQ = SR

SR = 2 * PR cos (angle PSR) = 2 * PR * cos 72

Ratio = PR / (2 * PR cos 72 + PR + 2 * PR cos 72) = 1 / (4 cos 72 + 1)

**3) Answer (B)**

Number of diagonals in a regular polygon = $$^nC_2 – n$$

=> $$^nC_2 – n$$ = 135

=> n = 18

Exterior angle = $$\frac{360}{n}$$ = $$\frac{360}{18}$$ = $$20^{\circ}$$

**4) Answer (D)**

Volume of the whole pencil = $$3.6\pi$$ cm^3

Volume of graphite cylinder = $$0.4\pi$$ cm^3

Volume of wood part = $$3.6\pi-0.4\pi=3.2\pi$$ cm^3

Cost of the pencil

= $$3.2\times\frac{22}{7}\times0.7+0.4\times\frac{22}{7}\times2.8$$

= Rs. 10.56

**5) Answer (A)**

Measure of internal angle in a regular hexagon = 120 degrees

Diagonal AD bisects angle EDC. So, angle EDA = 60 degrees

In triangle AED, angle AED = 90 degrees, angle ADE = 60 degrees

AE is the shortest diagonal of the hexagon and AD is the longest diagonal.

AE = 10 cm

Sin 60 = AE/AD => AD = AE/sin 60 = $$10/(\sqrt3/2)$$ = $$20/\sqrt3$$ cm = 20/1.732 = 11.54 cm = 12 cm (approx.)

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