Thread: Segment of a Circle

What is the approximate area of a segment of a circle with a height 6 m and the length of the chord is 20 m? Round answer to the nearest whole number.

Originally Posted by magentarita

What is the approximate area of a segment of a circle with a height 6 m and the length of the chord is 20 m? Round answer to the nearest whole number.

chord = 2 sqrt [ height x ( 2 x radius - height) ]

Given the values 20 and 6;

20 m = 2 sqrt 6 m x ( 2 x radius - 6 m )
20 m / 2 = 2 sqrt 6 m x ( 2 x radius - 6 m ) / 2
10 m = sqrt [ 6 m x ( 2 x radius - 6 m ) ]
10 m squared = sqrt 6 m x ( 2 x radius - 6 m ) squared
100 sq m = 6 m x ( 2 x radius - 6 m )
100 sq m = 12 m x radius - 36 sq m
100 sq m + 36 sq m = 12 m x radius - 36 sq m + 36 sq m
136 sq m = 12 m x radius
136 sq m / 12 m = 12 m x radius / 12 m
11.333 m = radius

A = r squared x arc cosine ( ( r - h ) / r ] - ( r - h ) x sqrt ( 2 x r x h - h squared )

A = 128.444 sq m x arc cosine ( 5.333 m / 11.333 m ) - 5.333 m x sqrt ( 2 x 68 sq m - 36 sq m )

A= 128.444 sq m x 1.0808 radians - 5.333 m x 10 m

A= 85.4 sq m= 85 sq m

Originally Posted by TheMasterMind

chord = 2 sqrt [ height x ( 2 x radius - height) ]

Given the values 20 and 6;

20 m = 2 sqrt 6 m x ( 2 x radius - 6 m )
20 m / 2 = 2 sqrt 6 m x ( 2 x radius - 6 m ) / 2
10 m = sqrt [ 6 m x ( 2 x radius - 6 m ) ]
10 m squared = sqrt 6 m x ( 2 x radius - 6 m ) squared
100 sq m = 6 m x ( 2 x radius - 6 m )
100 sq m = 12 m x radius - 36 sq m
100 sq m + 36 sq m = 12 m x radius - 36 sq m + 36 sq m
136 sq m = 12 m x radius
136 sq m / 12 m = 12 m x radius / 12 m
11.333 m = radius

A = r squared x arc cosine ( ( r - h ) / r ] - ( r - h ) x sqrt ( 2 x r x h - h squared )

A = 128.444 sq m x arc cosine ( 5.333 m / 11.333 m ) - 5.333 m x sqrt ( 2 x 68 sq m - 36 sq m )

A= 128.444 sq m x 1.0808 radians - 5.333 m x 10 m

A= 85.4 sq m= 85 sq m

Wonderfully done!