CAT Exam Model Paper 4 with solutions for free online practice

Question : 3 of 60

Marks: +1, -0

In the figure given below, ABCDEFGH is a regular octagon of side ‘a’ units inscribed in a circle of radius ‘r’ units. If a

^{2}

= 2 -

√2

, what is the area (in square units) of the shaded region?

Solution:

Applying cosine rule,

\cos 45 ° = { r ^ { 2 } + r ^ { 2 } - a ^ { 2 } }/ { 2 r ^ { 2 } }

⇒\ { 1 }/ { √{ 2 } } = { 2 r ^ { 2 } - a ^ { 2 } } /{ 2 r ^ { 2 } }

⇒r ^ { 2 } = { a ^ { 2 } } /{ 2 - √ { 2 } } = 1

Area of shaded region
=

1/2

[Area of circle - Area of octagon]

= { 1 }/ { 2 } [ \π r ^ { 2 } - 8 ( { 1 } { 2 } × r ^ { 2 } × \sin 45 °) ]

= { 1 }/ { 2 } ( \π- 4 × { 1 }/ { \√ { 2 } } )

= { \π - 2 { 2 } }/ { 2 }

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