Find the number of sides of a polygon if the sum of its interior angles is equal to 1800^{0}?
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
What is the measure of each interior angle of a regular hexagon (in degrees)?
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.
The figure below is a regular polygon. Find the measure of angle x (in degrees)?
If ABCDEF is a regular hexagon and AB = 1cm, then what is the area of this regular hexagon ABCDEF?
formula for finding the area of a hexagon is Area = (3√3 s^{2}) / 2 where s is the length of a side of the regular hexagon
where s = 1
Hence area of hexagon will be
The area of polygon ABCD is equal to 8√3 cm^{2}. If triangle ABD and BCD are equilateral triangles, then find the perimeter of the polygon ABCD?
ABCDE is a regular pentagon. Sides AE and ED of the pentagon are extended as shown to form triangle EFG, which is rightangled at F. Vertices A and C are joined to form the triangle ABC. What is the value of sum (x° + y°)?
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 nsided regular polygon =
So, for n = 5, value of each interior angle =
Therefore, i° = 108° . . . (3)
Also, we know that Exterior Angle of an nsided 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
The above shape is made of eight identical equilateral triangles, each having a perimeter of 3. What is the straightline distance between points P and Q?
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 straightline distance of 2√3
Answer: Option (E)
Corners of an equilateral triangle were cut off such that a regular hexagon is formed. If each side of the original triangle measures 9 cm, find the area of the hexagon in square cm.
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 = 60^{0}
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 nsided 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.
Dodger’s stadium, which is rectangular in shape, currently has one parking lot that has a maximum capacity of 2000 cars. In order to accommodate the increased demand due to District Level games, the management plans to increase the capacity of the parking lot by adding one more parking lot – parking lot 2. Assuming the shape of the structure as shown in the figure, what is the ratio of the width of parking lot 2 (w_{2}) to the width of parking lot 1(w_{1}). Also assume that the height of both the triangular parking lots is same as the width of the rectangle and that the allowed spacing between the cars is same in both parking lots.
(1) After the addition of parking lot 2, the total parking capacity of the stadium is 6000 cars.
(2) The area of parking lot 1 is 10000 m^{2}
Steps 1 and 2 – Understand the question statement and draw inferences
To find – w_{2 }: w_{1}
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.
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 w_{2 }: w_{1}
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 m^{2}.
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
Two identical regular hexagons are inscribed in a rectangle whose area is 256√3 square inches. What is the perimeter of each hexagon in inches?
To find – Perimeter of hexagon
Given –
Approach –
Solution
Likewise, applying Pythagoras theorem on the shaded side triangle we get the value of y in terms of a
We can now substitute the values of x and y in terms of a in the equation for area of the rectangle.
Correct Answer = Choice A
A regular polygon is partially obscured by a box. What is the sum of the angles of the polygon?
(1) x°= y°
(2) x°+ y°= 72°
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 = (n2)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)
If the sum of interior angles of a polygon is 900 degrees, what is the number of diagonals it has?
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.
In a regular pentagon PQRST, what is the ratio of the area of triangle PRS to the area of the pentagon PQRST?
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)
A regular polygon P has 135 diagonals. Find the exterior angle of the polygon P.
Number of diagonals in a regular polygon =^{n}C_{2–n}
=>^{n}C_{2–n}
= 135
=> n = 18
Exterior angle = 360/n
=360/18
=20^{0}
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.
Volume of the whole pencil = 3.6π cm^{3}
Volume of graphite cylinder =0.4π cm^{3}
Volume of wood part =3.6π−0.4π=3.2π cm^{3}
Cost of the pencil =
= Rs. 10.56
Use Code STAYHOME200 and get INR 200 additional OFF

Use Coupon Code 








