An equilateral triangle is shown inside a square inside a regular pentagon inside a regular hexagon. write an expression for the area of the shaded regions.

We know that
the area of the shaded regions is equal to
area of the shaded region 1+area of the shaded region 2

step 1
find the area of the shaded region 1
area of the shaded region 1=Area regular Hexagon-Area regular Pentagon

step 2
find the area of the shaded region 2
area of the shaded region 2=Area of a square-Area of the equilateral triangle

therefore
the area of the shaded regions is equal to
[Area regular Hexagon-Area regular Pentagon]+[Area of a square-Area of the equilateral triangle]

The expression would be written as

Area of hexagon - Area of the pentagon + Area of the square - area of the rectangle.

We can see that the Hexagon has three shapes inside it

PentagonSquare Rectangle

We have to find the areas of each of these shapes and perform basic arithmetic to find the area of the shaded regions.

Read more on the area of a shape here:

link

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