Depth · one concept from first principles · step 2 of 4
The figure at the right shows congruent equilateral triangles drawn inside a circle, with some of them shaded. What is the total area of the shaded parts, in ?
The circle is made of six congruent triangles inside the hexagon and six congruent pieces between the hexagon and the circle.
Twelve pieces, in two matching sets of six.
6.G.A.1Visualize Spatial RelationshipsThree of the six triangles are shaded, and three of the six outer pieces are.
Exactly half of each kind.
6.G.A.1Change Focus Count The ComplementHalf the triangles plus half the outer pieces is half of everything in the circle -- the shaded parts slide together into a semicircle.
The shaded area is half the circle.
7.G.B.4Visualize Spatial RelationshipsThe shaded pieces together are half the circle.
The circle is made of six triangles and six outer pieces, with nothing left over and nothing counted twice.
The six triangles are all the same size and so are the six outer pieces, because turning the picture a sixth of a turn carries each onto the next.
Exactly three of each kind are shaded, so the shading takes half of every kind and therefore half of everything.
Work out the circle and halve it.
The shaded parts come to 226.08 cm².
7.G.B.4Draw A DiagramIf the shading takes half of every kind of piece, it takes half the whole.
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