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8.2 Inscribing shapes in circles

3 Shen wants to estimate the area of a hexagon inscribed in a circle of radius 6 cm.
He takes these steps.

Step 1 Draw a circle of radius 6 cm.

Step 2 Construct an inscribed hexagon. 5.2 cm

Step 3 Draw a circle inside the hexagon so that
it touches all the sides of the hexagon. 6 cm

Step 4 Measure the radius of the smaller circle.

Step 5 Area of large circle = Π × 6 = 113.04 cm
Step 5 2 2
Area of small circle =
Area of small circle = Π × 5.2 = 84.91 cm 2
2
The area of the hexagon must be bigger than 84.91 cm but
2
smaller than 113.04 cm .
2
.
+
.
Halfway between 84.91 and 113.04 is 84 91 113 04 = 98 975
.
2
I estimate the area of the hexagon to be 99 cm .
2

Use Shen’s method to make the constructions and work out an estimate for the area of:
a a hexagon inscribed in a circle of radius 7 cm
b an octagon inscribed in a circle of radius 6 cm
c an octagon inscribed in a circle of radius 7 cm.

4 Anders inscribes an octagon in a circle of radius 4.5 cm.
Harsha inscribes an octagon in a circle of radius 9 cm.

That means that the area of
I estimate the area of my inscribed
my inscribed octagon must be
octagon to be about 60 cm .
2
about 120 cm , as my radius is
2
double your radius.
a Draw an accurate diagram and make appropriate calculations to show that Anders has made a
correct estimate.
b Without drawing a diagram, how can you tell that Harsha’s statement is false?
c Draw an accurate diagram and make appropriate calculations to show that Harsha is wrong.

80 8 Constructions and Pythagoras’ theorem