geometry problem: circle inscribed in a sector?
a circle having a diameter of 8 cm is inscribed in a sector of a circle whose angle is 80°. find the area of the sector.
- BrianLv 76 years agoFavorite Answer
Since the two straight sides bordering the sector
are tangent to the circle, we know that the right
triangle formed by the centers of the two circles
and the point of tangency of the smaller circle
and either of the sector sides has the radius of
the smaller circle as one leg and the angle opposite
this leg being (1/2)*80 = 40 degrees. The hypotenuse
of this triangle is then the distance between the two
centers, and the radius of the larger circle will then be
this hypotenuse length plus the radius of the smaller
circle. So with the smaller circle having a radius of 4 cm,
the hypotenuse has a length of 4*csc(40), and thus the
radius of the larger circle is 4*csc(40) + 4 = 4*(1 + csc(40)) cm.
The area of the sector is therefore
(1/2)*r^2*(theta) = (1/2)*[4*(1 + csc(40))]^2 * (80 * (pi/180)) =
8*(1 + csc(40))^2 * (4/9) * pi = (32/9)*pi*(1 + csc(40))^2 cm^2,
which is 72.96 cm^2 to 2 decimal places.
- HosamLv 66 years ago
If you draw a sketch you'll find that the inscribed circle has its center on the bisector of the
80 degree angle of the sector. If you draw perpendiculars from the center to the two lines defining
the sector then you can express the distance between the center of the inscribed circle and the
tip of the sector as
r / x = tan(80/2) = tan (40)
Therefroe, x = r / tan(40)
The radius of the sector is x + r = r (1/tan(40) + 1) = 8 ( 1/ tan(40) + 1) = 17.534 cm
Now we can find the area of the sector
Area = 1/2 R^2 (theta), where theta is expressed in radians.
Area = (1/2) (17.534)^2 ( 80 * pi /180 ) = 214.634 cm^2
- Your AnswerLv 66 years ago
area of sector is given by 80 pie R^2 / 360 where R = 4(1 + csc 40) ANSWER
- mohanrao dLv 76 years ago
Area of sector = (x/360)πr^2
where x is central angle of sector
=> A = (80/360)π(8)^2
= 44.68 sq.cm
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- grunfeldLv 76 years ago
A = ( 1 / 2 )r^2 * thetaSource(s): my brain