Depth · one concept from first principles · step 2 of 4
In the figure at the right, a circle with a radius of has its circumference divided into equal parts, with segments drawn between some of the points and a region shaded. What is the area of the shaded part, in ?
Each step round the circle is 30 degrees: A to D is three steps, B to C is three, A round to B is two and C round to D is four.
Angles of 90, 60, 90 and 120 degrees.
7.G.B.5Visualize Spatial RelationshipsA central angle of 90 degrees makes a right triangle whose two legs are radii, so each has legs 12 and 12.
Each triangle is 72 cm².
7.G.B.5Draw A DiagramA sector is the share of the whole circle its angle takes: 60 degrees is a sixth and 120 degrees is a third.
The sectors are 75.36 and 150.72 cm².
7.G.B.4Identify SubproblemsA -degree sector is one sixth of the circle.
Six equal -degree angles at the centre fill one full turn, and they cut the circle into six pieces with no gap and no overlap.
Those six pieces are all the same size, because turning the circle about its centre carries any one of them exactly onto the next.
The two triangles and the two sectors together fill the shaded region exactly.
The shaded area is 370.08 cm².
7.G.B.4Identify SubproblemsA shape with no formula can be cut at the centre into sectors and triangles, which do have one.
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