Tag: defining regular polygons

Questions Related to defining regular polygons

if $\frac { 1 }{ { a } _{ x }+1 } are\quad 8$ vertices of a rectengular octagon where ${ a } _{ k }\epsilon$ R, K =1,2,3,.....,8(where $ i =\sqrt { -1 } )$then area of the regular octagon is

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. $1$

  2. $\sqrt { 2 } $

  3. $\frac { 1 }{ \sqrt { 2 } } $

  4. none

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Correct Option: A

in the given figure,BD is a side a regular hexagon,DC is a side of a regular pentagon and AD is a diameter calculate

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. $\angle ADC$

  2. $\angle BDA$

  3. $\angle ABC$

  4. $\angle AEC$

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Correct Option: A

What is the solid angle subtended by a hemisphere at its center? 

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. $2\pi$ steradian

  2. $\pi$ steradian

  3. $3\pi$ steradian

  4. $4\pi$ steradian

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Correct Option: A
Explanation:
Solid angle $(\Omega)=\dfrac{A}{r^2}$

for a hemisphere, $A=2\pi r^2$

so, $\Omega =\dfrac{2\pi r^2}{r^2}=2\pi$ steradians

Ans is (A).

The area of a regular polygon of $2n$ sides inscribed in a circle is given by?

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. The geometric mean of the areas of the inscribed and circumscribed polygons of $n$ sides.

  2. The arithmetic mean of the areas of the inscribed and circumscribed polygons of $n$ sides.

  3. The harmonic mean of the areas of the inscribed and circumscribed polygons of $n$ sides.

  4. None of the above

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Correct Option: A
Explanation:

Let $a$ be the radius of the circle 


Then,$\displaystyle s _{1}= $ Area of regular polygon of n sides inscribed in the circle $\displaystyle =\frac{1}{2}na^{2}\sin\left ( \frac{2\pi }{n} \right )$

$\displaystyle s _{2}= $  Area of regular polygon of n sides circumscribing in the circle $\displaystyle  = na^{2}\tan \frac{\pi }{n}$

$\displaystyle s _{3}= $ Area of regular polygon of 2n sides inscribed in the circle $\displaystyle  = na^{2}\tan \frac{\pi }{n}$ 

[replacing $n$ by $2n$ is $\displaystyle {(S _{1}}$]

$\displaystyle \therefore $ Geometric mean of $\displaystyle {S _{1}}$ and 

$\displaystyle {S _{2}}$ $\displaystyle = \sqrt{(S _{1}S _{2})}= na^{2}\sin\left ( \frac{\pi }{n}\right ) = S _{3}$

If A B C D E F is a regular hexagon with A B = a and B C = b, then CE equals

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. b-a

  2. -b

  3. b-2a

  4. None of these

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Correct Option: A

If A B C D E F  is a regular hexagon with A B = a and B C = b , then CE equals

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. b-a

  2. -b

  3. b-2a

  4. None of these

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Correct Option: A

If  $\alpha$  is the angle which each side of a regular polygon of  $n$  sides subtends at its centre, then  $1 + \cos \alpha + \cos 2 \alpha + \cos 3 \alpha \ldots + \cos ( n - 1 ) \alpha$  is equal to

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. $n$

  2. $0$

  3. $1$

  4. None of these

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Correct Option: A

Relation between circumradius and number of sides is given by-

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. $Area=\dfrac{r^2n\sin(\dfrac{360}{n})}{3}$

  2. $Area=\dfrac{r^2n\sin(\dfrac{360}{n})}{2}$

  3. $Area=\dfrac{r^2n\cos(\dfrac{360}{n})}{2}$

  4. None of the above

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Correct Option: B

The sum of the radii of inscribed and circumscribed circles of an n sided regular polygon of side $'a'$ is

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. $=\frac{a}{2} \left ( \frac{1}{\sin \pi/2x} + \cot \frac{\pi}{x} \right )$

  2. $=\frac{a}{2} \left ( \frac{1}{\sin \pi/x} + \cot \frac{\pi}{2x} \right )$

  3. $=\frac{a}{2} \left ( \frac{1}{\sin \pi/x} + \cot \frac{\pi}{x} \right )$

  4. None of these

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Correct Option: C
Explanation:

$R\sin \theta  = \frac{a}{2}$
$R = \frac{a}{2\sin \theta }$
$\tan \theta = \frac{a/2}{r}$
$r = \frac{a}{2\tan \theta }                                   \theta = \frac{2\pi}{n} \times\frac{1}{2}$
$R+r = \frac{a}{2} \left ( \frac{1}{\sin \theta}+\frac{\sin \theta}{\cos \theta } \right )             = \frac{\pi}{x}$
    $= \frac{a}{2} \left ( \frac{1}{\sin \pi/x} + \cot \frac{\pi}{x} \right )$

In $\Delta ABC$, there are 35 lines drawn parallel to the base BC such that each line divides the other side into, equal parts. 
If BC =1.8 m find the length of $P _7 Q _7$.

maths understanding 3d and 2d shapes defining regular polygons sum of exterior angles of a polygon regular polygons

  1. 1.8 m

  2. 3.5 m

  3. 0.35 m

  4. 0.18 m

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Correct Option: C