Question

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. shaded area = area of the $\boldsymbol{\text{dropdown1}}$ $-$ area of the $\boldsymbol{\text{dropdown2}}$ $+$ area of the $\boldsymbol{\text{dropdown3}}$ $-$ area of the $\boldsymbol{\text{dropdown4}}$

Explanation:

Step1: Analyze the outermost shape

The outermost shape is the regular hexagon, so the first term is the area of the regular hexagon.

Step2: Analyze the next inner shape

Inside the hexagon is the regular pentagon, so we subtract the area of the regular pentagon.

Step3: Analyze the next inner shape

Inside the pentagon is the square, so we add the area of the square.

Step4: Analyze the innermost shape

Inside the square is the equilateral triangle, so we subtract the area of the equilateral triangle.

Answer:

Shaded area = area of the regular hexagon − area of the regular pentagon + area of the square − area of the equilateral triangle