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Two points are drawn on each side of a square with an area of 81 square units dividing the side into 3 congruent parts. Quarter-circle arcs connect the points on adjacent sides to create the figure shown. What is the length of the boundary of the bolded figure? Express your answer as a decimal to the nearest tenth.

 

tertre  Jul 23, 2017

Best Answer 

 #1
avatar+4174 
+1

The area of the square is 81 square units.

81 un2  =  (side length)2

  9 un   =   side length

 

The two points divide the side length into 3 congruent parts.

So, each piece will be  9/3  =  3  units long.  Now we know...

radius of each quarter circle  =  3 un             and...

        length of straight piece  =  3 un

 

On the shape's perimeter...there are  4  quarter circles, and there are 4  straight pieces.  So...

perimeter  =  4[ circumference of quarter circle ]  +  4[ length of straight piece ]

                 =  4[ (2π * radius)/4 ]  +  4[ length of straight piece ]

                 =  [ 2π * radius ] + 4[ length of straight piece ]

                 =  [ 2π * 3 ] + 4[ 3 ]

                 =  6π + 12

                 ≈  30.8 un

hectictar  Jul 23, 2017
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1+0 Answers

 #1
avatar+4174 
+1
Best Answer

The area of the square is 81 square units.

81 un2  =  (side length)2

  9 un   =   side length

 

The two points divide the side length into 3 congruent parts.

So, each piece will be  9/3  =  3  units long.  Now we know...

radius of each quarter circle  =  3 un             and...

        length of straight piece  =  3 un

 

On the shape's perimeter...there are  4  quarter circles, and there are 4  straight pieces.  So...

perimeter  =  4[ circumference of quarter circle ]  +  4[ length of straight piece ]

                 =  4[ (2π * radius)/4 ]  +  4[ length of straight piece ]

                 =  [ 2π * radius ] + 4[ length of straight piece ]

                 =  [ 2π * 3 ] + 4[ 3 ]

                 =  6π + 12

                 ≈  30.8 un

hectictar  Jul 23, 2017

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