Answer to Question #260238 in Geometry for Aliposa

Question

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.

Answer 1

9) Area of a sector =$\frac{\theta }{360}\pi r^2=\frac{30}{360}\times \frac{22}{7}\times 15^2=58.93cm^2$

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

$r=\sqrt{224/\pi}=8.4444m$

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

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

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

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

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

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

11) A hexagon can be divided into 6 equal triangles

Angle at the centre $=360^0/6=60^0$

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$= (60^0/360^0\times \pi \times x^2-1/2 \times x^2 \times sin60^0)6=15$

$0.5236x^2-0.433x^2=15/6=2.5$

$0.0906x^2=2.5$

$x^2=27.5938$

$x=5.253cm$

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