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6. Find the area of the shaded sector. Round to the nearest tenth.

6. Find the area of the shaded sector. Round to the nearest tenth.-example-1
User Yeny
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Explanation:

the whole 120° sector of the circle is the sum of the white isoceles triangle (120° top angle, 2 equal sides of 4) and the shaded segment.

so, to get the area of the shaded segment, we need to calculate the area of the sector and subtract the area of the triangle.

as the area of the full circle (360°) is

pi × r²,

the area of a sector with angle theta is

theta/360 × pi×r²

simply the theta/360 part of the whole circle.

so, in our case that means

120/360 × pi×4² = 1/3 × 16pi = 16pi/3

the area of the white triangle is a bit trickier.

in general it is

baseline × height / 2

for a right-angled triangle that means

leg1 × leg2 / 2

now, if we draw the height in the main triangle, this splits the main triangle into 2 equal right-angled triangles. theta gets split in half as well (120/2 = 60°).

and the area of one of them is then

(half of main baseline) × height / 2

and we get 2 of them, so the area the main triangle is

(half of main baseline) × height

how long are the height and half of the main baseline ?

we know from trigonometry that such a right-angled triangle with theta/2 as angle at the center of the circle makes

half of main baseline = sin(theta/2)×r

height = cos(theta/2)×r

remember, in any circle larger than r = 1 we need to multiply the trigonometric functions sine and cosine by the radius to get the actual lengths.

so, the area of the main triangle is

sin(theta/2)×r × cos(theta/2)×r =

= sin(theta/2)×cos(theta/2)×r² =

= sin(60)×cos(60)×4²

and therefore, the area of the shaded segment is

16pi/3 - sin(60)×cos(60)×4² =

= 16pi/3 - sin(60)×1/2 × 16 =

= 16pi/3 - sin(60)×8 = 9.826957589... ≈ 9.8 units²

User Guillermo Calvo
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