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Find the magnitude in radians and degrees of the interior angle of

(A) A regular pentagon

(B) A regular heptagon

(C) A regular octagon

(D) A regular dodecagon and,

(E) A regular polygon of 17 sides

see full answer

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A-

108^{\circ}, \frac{3 π}{5}

. B-

128^{\circ} 34^{'} 17^{''}, \frac{5 π}{7}

. C-

135^{\circ}, \frac{3 π}{4}

.D-

135^{\circ}, \frac{5 π}{6},

E-

{158 . 823}^{\circ}, \frac{15 π}{17}

A-

108^{\circ}, \frac{3 π}{5}

. B-

{128.6}^{\circ}, \frac{5 π}{7}

. C-

135^{\circ}, \frac{3 π}{4}

.D-

135^{\circ}, \frac{5 π}{6},

E-

{158 . 523}^{\circ}, \frac{15 π}{17}

A-

108^{\circ}, \frac{3 π}{5}

. B-

{128.5}^{\circ}, \frac{5 π}{7}

. C-

135^{\circ}, \frac{3 π}{4}

.D-

145^{\circ}, \frac{5 π}{6},

E-

{168 . 823}^{\circ}, \frac{15 π}{17}

A-

108^{\circ}, \frac{3 π}{5}

. B-

{128.5}^{\circ}, \frac{5 π}{7}

. C-

135^{\circ}, \frac{3 π}{4}

.D-

125^{\circ}, \frac{5 π}{6},

E-

{178 . 823}^{\circ}, \frac{15 π}{17}

answer is A.

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Detailed Solution

We have found magnitude in radians and degrees for interior angle
Shape made for any polygon of sides of number = n
Formula for interior angles of polygon with n side,

= (\frac{2 n - 4}{n}) \times 90^{\circ}

(A) Magnitude in radians and degrees for a regular pentagon
Pentagonal has side = 5

= \frac{2 \times 5 - 4}{5} \times 90^{\circ}

= \frac{6}{5} \times 90

= 108^{\circ}

Now, calculate each angle of pentagon

1^{c} = \frac{180}{π}

= \frac{2 \times 5 - 4}{5} \times \frac{π}{2}

= {(\frac{3 π}{5})}^{c}

Hence, the required degree

= 108^{\circ}

and radian

= {(\frac{3 π}{5})}^{c}

(B) Magnitude in radians and degrees for a regular heptagon
Heptagon has side = 7

= \frac{2 \times 7 - 4}{7} \times 90^{\circ}

= \frac{10}{7} \times 90^{\circ} = \frac{900^{\circ}}{7}

= 128^{\circ} 34^{'} 17^{''}

Now, calculate each angle of Heptagon

= \frac{2 \times 7 - 4}{7} \times \frac{π}{2}

= \frac{10}{7} \times \frac{π}{2}

= {(\frac{5 π}{7})}^{c}

Hence, the required degree

= 128^{\circ} 34^{'} 17^{''}

and radian

= {(\frac{5 π}{7})}^{c}

= \frac{2 \times 8 - 4}{8} \times 90^{\circ}

= \frac{12}{8} \times 90^{\circ}

= \frac{12}{8} \times 90^{\circ}

= 135^{\circ}

Now, calculate each angle of Octagon

= \frac{2 \times 8 - 4}{8} \times \frac{π}{2}

= \frac{12}{8} \times \frac{π}{2}

= {(\frac{3 π}{4})}^{C}

Hence, the required degree

= 135^{\circ}

and radian

= {(\frac{3 π}{4})}^{C}

(D) Magnitude in radians and degrees for a regular dodecagon
Dodecagon has side = 12

= \frac{2 \times 12 - 4}{12} \times 90^{\circ}

= \frac{20}{12} \times 90^{\circ}

= 150^{\circ}

Now, calculate each angle of dodecagon

= \frac{2 \times 12 - 4}{12} = \frac{π}{2}

= \frac{20}{12} \times \frac{π}{2}

= {(\frac{5 π}{6})}^{c}

Hence, the required degree

= 150^{\circ}

and radian

= {(\frac{5 π}{6})}^{c}

(E) Magnitude in radians and degrees for a regular polygon of 17 sides
Polygon has side = 17

= \frac{2 \times 17 - 4}{17} \times 90^{\circ}

= \frac{30}{17} \times 90^{\circ}

= {158 . 823}^{\circ}

Now, calculate each angle of polygon

= \frac{2 \times 17 - 4}{17} = \frac{π}{2}

= \frac{30}{17} \times \frac{π}{2}

{= (\frac{15 π}{17})}^{c}

Hence, the required degree

= {158 . 823}^{\circ}

and radian

{= (\frac{15 π}{17})}^{c}

Correct option is 1.

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Find the magnitude in radians and degrees of the interior angle of(A) A regular pentagon(B) A regular heptagon(C) A regular octagon(D) A regular dodecagon and,(E) A regular polygon of 17 sides

Detailed Solution

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Find the magnitude in radians and degrees of the interior angle of

(A) A regular pentagon

(B) A regular heptagon

(C) A regular octagon

(D) A regular dodecagon and,

(E) A regular polygon of 17 sides