Take note Key Concept: Area of a Segment

	–		=
Area of sector	–	Area of triangle	=	Area of segment

Problem 3 Finding the Area of a Segment of a Circle

What is the area of the shaded segment shown below? Round your answer to the nearest tenth.

area of sector  A C B = m A B ⌢ 360 ⋅ π r 2 Use the formula for area of a sector.   = 60 360 ⋅ π ( 18 ) 2 Substitute  60   for  m A B ⌢  and  18   for  r .   = 54 π Simplify. table with 3 rows and 3 columns , row1 column 1 , area of sector eh c b , column 2 equals . fraction m , modified eh b with frown above , over 360 end fraction . dot pi , r squared , column 3 cap use the formula for area of a sector. , row2 column 1 , , column 2 equals , 60 over 360 , dot pi . open 18 close squared , column 3 cap substitute . 60 , for , m , modified eh b with frown above . and , 18 , for , r . , row3 column 1 , , column 2 equals 54 pi , column 3 cap simplify. , end table

Δ A C B cap delta eh c b  is equilateral. The altitude forms a 30 ° − 60 ° − 90 ° 30 degrees , minus , 60 degrees , minus , 90 degrees  triangle.

area of  Δ A C B = 1 2 b h Use the formula for area of a triangle .   = 1 2 ( 18 ) ( 9 3 ) Substitute 18 for  b  and  9 3  for  h .   = 81 3 Simplify . table with 3 rows and 3 columns , row1 column 1 , areaof , cap delta eh c b , column 2 equals , 1 half , b h , column 3 cap usetheformulaforareaofatriangle . . , row2 column 1 , , column 2 equals , 1 half . open 18 close . open , 9 square root of 3 , close , column 3 cap substitute18for . b , and , 9 square root of 3 , for , h . , row3 column 1 , , column 2 equals 81 square root of 3 , column 3 cap simplify , . , end table

area of shaded segment = area of sector  A C B − area of  Δ A C B   = 54 π − 81 3 Substitute .   ≈ 29   .34988788 Use a calculator . table with 3 rows and 3 columns , row1 column 1 , areaofshadedsegment , column 2 equals . areaofsector eh c b minus , column 3 areaof , cap delta eh c b , row2 column 1 , , column 2 equals 54 pi minus 81 square root of 3 , column 3 cap substitute . . , row3 column 1 , , column 2 almost equal to 29 . .34988788 , column 3 cap useacalculator . . , end table

The area of the shaded segment is about 29 . 3   in . 2 . 29 . 3 in , . squared , .

✓ Got It?

3. What is the area of the shaded segment shown below? Round your answer to the nearest tenth.