Test: Polygons - UPSC MCQ

Number of sides in a polygon is 12

Explanation:
Sum of all the exterior angles of any polygon is always 360^o.
As sum of every pair of interior and exterior angle is 180^o,
sum of all the interior and exterior angles of a polygon with n sides, is 180^o×n.

As sum of interior angles of given polygon is 1800^o and sum of exterior angles is 360^o. The Sum of all the angles of given polygon is 2160^o.

As such 180^o×n=2160^o and n=2160^o/ 180^o = 12.

Hence, the number of sides in a polygon is 12

To find the sum of the interior angles of a hexagon, divide it up into triangles. There are four triangles.

 Because the sum of the angles of each triangle is 180 degrees. 

                 4 x 180° = 720°

So, the sum of the interior angles of a hexagon is 720 degrees.

All sides are the same length (congruent) and all interior angles are the same size (congruent).

To find the measure of the interior angles, we know that the sum of all the angles is 720 degrees. And there are six angles.

720° ÷ 6 = 120°

So, the measure of the interior angle of a regular hexagon is 120 degrees.

formula for finding the area of a hexagon is Area = (3√3 s²) / 2 where s is the length of a side of the regular hexagon

where s = 1

Hence area of hexagon will be 

Step 1: Understand the question and draw inferences

To find –x°+ y°

Given – ABCDE is a regular pentagon.

This means, all sides of the pentagon are equal.

Therefore, AB = BC

In triangle ABC, two sides are equal.

Therefore, triangle ABC is an isosceles triangle.

Since angles opposite to equal sides are also equal,

Angle BAC = Angle ACB = x°

Let Angle ABC = i°

Therefore, by Angle Sum Property,

2x° + i° = 180°   .  .  . (1)

Now, considering the right triangle EFG,

Let angle FEG = e°

Therefore, by Angle Sum Property,

e° + y° + 90° = 180°

e° + y° = 90°    .  .  . (2)

Step 2: Finding the required values

We know that for interior angle of an n-sided regular polygon =

So, for n = 5, value of each interior angle =  

Therefore, i° = 108°    . . . (3)

Also, we know that Exterior Angle of an n-sided regular polygon = 360on

So, for n = 5, the value of each exterior angle = 360^o/5 =  72°

Notice that Angle FEG is an exterior angle of the regular pentagon.

Therefore, e° = 72°   .  .  . (4)

Step 3: Calculating the final answer

From (1) and (3), we get:

2x° + 108° = 180°  

Solving this equation, we get x° = 36°

From (2) and (4), we get:

72° + y° = 90°

y° = 18°

Thus, x° + y° = 36° + 18° = 54°

Correct Choice -C

Step 1: Question statement and Inferences

To find the straight line distance between points P and Q, calculate the height of one of the equilateral triangles.

Step 2: Finding required values

Think of the shape as two large, equilateral triangles. If each small equilateral triangle has a perimeter of 3, then it has a side of 1, and each large equilateral triangle has a side of 2.

Create a right triangle inside a large equilateral triangle, and find the third side, shown as h in the drawing.

Step 3: Calculating the final answer

The right triangle has sides 1, 2, and h, where 2 is the hypotenuse. With the Pythagorean Theorem, calculate the length of h:

You have found half the distance. Double this for the full straight-line distance of 2√3

Answer: Option (E)

Step 1: Question statement and Inferences

Since we are not given the figure, let us draw one.

Let the equilateral triangle be ABC.

∠BAC = ∠ABC = ∠BCA = 60⁰

The corners marked in blue are cut from this triangle, thereby resulting in a regular hexagon PQRSTU.

Let each side of this hexagon be x cm

Now, we know that

Exterior angle of a n-sided regular polygon  = 360^o/n

Therefore,

Exterior angle of a regular hexagon 

360^o/6 = 60^o

This means, Angle APQ = 60^o

Consider Triangle APQ:

Every angle in this triangle is 60^o

Therefore, BU = UP = PA = x cm

So, BA = BU + UP + PA = 3x cm

But, we are given that

Each side of triangle ABC measures 9 cm

That is, BA = 9 cm

3x = 9 cm

x = 3 cm

Thus, we have found out that each side of the regular hexagon measures 3 cm.

We need to find the area of this regular hexagon.

Step2: Finding required values

Area of hexagon = (Area of equilateral triangle ABC) -3(Area of equilateral Triangle with side 3 cm)

Area of am equilateral triangle = ; where a is the side of the triangle

so, the area of triangle ABC = 

And, the area of each smaller triangle =
Area of the hexagon = 

Step 3: Calculating the final answer

So, the area of the hexagon is  square cm.

Steps 1 and 2 – Understand the question statement and draw inferences

To find – w₂ : w₁

Given –

(1) Capacity of Parking Lot 1 = 2000 cars

(2) Per the figure, height of both parking lots is the same.  The difference in capacity of the parking lots comes from the difference in the width of the parking lots.

Step 3 – Analyse Statement 1

Per statement 1, the total increased capacity of the parking lots is 6000 cars.

• Capacity of Parking Lot 1 = 2000
• Capacity of Parking Lot 2 = 6000 – 2000 = 4000 cars

Area _{Lot 2} = 2 * Area _{Lot 1} … (since capacity of parking lot 1 = 2000 cars and that of lot 2 = 4000 cars)

Since the height of both parking lots is the same, the ratio of area = the ratio of width

Thus, we have the value of w₂ : w₁

Therefore, information provided in Statement (1) is sufficient to arrive at a unique answer.

Step 4 – Analyse Statement 2

Statement 2 provides area of Parking Lot 1 – 10000 m².

So we know that the given area can accommodate 2000 cars.  But we do not know anything about the area of the parking lot 2.  Thus, we do cannot find the relationship between the widths of the two parking lots.

Therefore, statement (2) does not provide sufficient information to arrive at a unique answer.

Step 5 – Analyze both Statements together

This step is not required since statement 1 helps us arrive at a unique answer.

Correct Answer – Choice A

To find – Perimeter of hexagon

Given –

• The two hexagons are identical – the length of the sides are equal.
• Area of rectangle = 256√3 in²

Approach –

• Let ‘a’ be the side of the hexagon and let ‘x’ be the length of the rectangle.
• To find the perimeter of the hexagon, we need to express the sides of rectangle (x) in terms of the side of the hexagon (a). 
• Then given the area we can find the measure of ‘a’
• Hence, we can find the perimeter.

Solution

• Each side of the hexagon is ‘a’ since it is a regular hexagon.
• Consider the middle shaded triangle.
• It is an equilateral triangle since each angle is 60° as explained below. 
  • Interior angle of the hexagon = 120°.
  • The angle made by AB = 180°.
  • Thus, angle of the triangle = 180 – 120 = 60°
  • The purple line is a/2.
  • x/2 = a/2+a+a/2 = 2a
  • x = 4a

Likewise, applying Pythagoras theorem on the shaded side triangle we get the value of y in terms of a

• a²-a²/4 = y²/4
• y = a√3

We can now substitute the values of x and y in terms of a in the equation for area of the rectangle.

• xy = 256√3
• 4a²√3 = 256√3
• a = 8
• Perimeter of hexagon = 8 * 6 = 48 inches

Correct Answer = Choice A

Steps 1 & 2: Understand Question and Draw Inferences

To find the sum of the angles of the polygon, you need to know the number of angles.

Knowing that the polygon is regular, the two visible angles (and all the other hidden angles) are equal. You also know that the visible trapezoid has an angle total of 360°

Step 3: Analyze Statement 1

x°= y°

Knowing x equals y doesn’t tell you anything about the angles of the regular polygon.

Statement (1) is not sufficient.

Step 4: Analyze Statement 2

x°+ y°= 72°

The part of the regular polygon that is visible forms a quadrilateral.

This quadrilateral has 4 angles: x°, y° and two interior angles of the regular polygon.

Let each interior angle of the regular polygon = i°

Since the sum of all angles of a quadrilateral is 360°, we can write:

x° + y° + 2i° = 360°

Inserting the information given in Statement 2 in this equation, we get:

72° + 2i° = 360°

Or, 2i° = 288°

i° = 144°

Thus, we now know that every interior angle of the regular polygon measures 144°

 Let the regular polygon have n sides.

Then 144°*n = (n -2)*180°

⇒ n = 10.

Therefore, sum of angles of the regular polygon = (n-2)180° = 8*180° = 1,440°.

Statement (2) is sufficient.

Step 5: Analyze Both Statements Together (if needed)

You get a unique answer in step 4, so this step is not required

Answer: Option (B)

The sum of interior angles for an ‘n’ sided polygon is
If this value is 900, then the polygon is a heptagon.
The number of diagonals for an ‘n’ sided polygon =

= 14 for a heptagon.

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 = ½ * 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
= ½ * PT * PS * sin (angle TPS) + ½ * PS * PR * sin 36 + ½ * PQ * PR * sin (angle QPR)
= ½ * PT * PS * sin 36 + ½ * PS * PR * sin 36 + ½ * PQ * PR * sin 36
Required ratio = ½ * PS * PR * sin 36 / (½ * PT * PS * sin 36 + ½ * PS * PR * sin 36 + ½ * 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)

Number of diagonals in a regular polygon =ⁿC_2–n
=>ⁿC_2–n

= 135
=> n = 18
Exterior angle = 360/n
=360/18
=20⁰

Volume of the whole pencil = 3.6π cm³
Volume of graphite cylinder =0.4π cm³
Volume of wood part =3.6π−0.4π=3.2π cm³

Cost of the pencil = 
= Rs. 10.56