Question #260238

The angle of a sector is 30deg and the radius is 15cm.




Find the area of the sector.




10. A circle having an area of 224sq.m is inscribed in an




octagon. Find the area of the octagon.




11. A circle is circumscribed about a hexagon. Determine




the area of the hexagon if the area outside the hexagon




but inside the circle is 15sq.cm.





1
Expert's answer
2021-11-16T19:15:32-0500

9) Area of a sector =θ360πr2=30360×227×152=58.93cm2\frac{\theta }{360}\pi r^2=\frac{30}{360}\times \frac{22}{7}\times 15^2=58.93cm^2

10) Area of the circle =πr2=224=\pi r^2=224

r=224/π=8.4444mr=\sqrt{224/\pi}=8.4444m

From the centre of the octagon, it can be divided into 8 equal triangles.

Angle at the centre for one triangle =360/8=450=360/8=45^0

Height of one triangle =radius of the circle =8.4444m

Base angles of the triangle =18004502=67.50=\frac{180^0-45^0}{2}=67.5^0

Hypotenuse of the triangle =8.444sin67.50=9.14m=\frac{8.444}{sin67.5^0}=9.14m

Area of the octagon =area of one triangle ×8=1/2×9.14×sin450×8=236.29m2\times 8=1/2\times 9.14 \times sin 45^0\times 8=236.29m^2


11) A hexagon can be divided into 6 equal triangles

Angle at the centre =3600/6=600=360^0/6=60^0

The triangles are equilateral, meaning all sides are equal = x cm

The area of the hexagon if the area outside the hexagon but inside the circle=(600/3600×π×x21/2×x2×sin600)6=15= (60^0/360^0\times \pi \times x^2-1/2 \times x^2 \times sin60^0)6=15

0.5236x20.433x2=15/6=2.50.5236x^2-0.433x^2=15/6=2.5

0.0906x2=2.50.0906x^2=2.5

x2=27.5938x^2=27.5938

x=5.253cmx=5.253cm

Area of the hexagon = area of one triangle ×6=1/2×5.253×5.253×sin600×6=71.69cm2\times 6=1/2\times 5.253\times 5.253\times sin60^0\times 6=71.69cm^2


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