Question

Show that the area of a regular 8-gon is equal to the product of its longest diagonal and its shortest diagonal.

Answer 1

the octagon is regular giving it equal sides and angles thus all the triangles have the same area and both rectangles have the same area.

now,

we have 8 triangles with area= 1/2ab giving,
total area of the triangles =8(1/2)ab =4ab

also we have 2 rectangles with area =bc giving
total area of the rectangles =2bc

thus total area of the octagon =4ab+2bc *****

now,

the length of the longest diagonal call it X = a+c+a =2a +c
the length of the shortest diagonal call it Y = 2b

multiplying X and Y gives XY=(2a +c)(2b)=4ab+2bc =area of octagon

end of proof