Calculate a set of Cartesian coordinates with respect to the origin for firstly, the carbon atoms only of benzene given that the C-C bond distance is 1.40 A and the benzene ring is a regular hexagon, below Figure. Secondly, the carbon atoms only of the cyclopentadiene anion given that the C-C bond distance is 1.40 A and the five membered ring is a regular pentagon. To tackle this question find the coordinates of C^{1} and then apply the appropriate symmetry operations to find the coordinates of the remaining carbon atoms, here focus on Molecular Structure Analysis of Cyclic Molecules.

**Solution**

The C-C distance in benzene is 1.40 A with the angle between neighbouring C-atoms of 360°/6 = 60°. So neighbouring C-atoms form an isosceles triangle with the origin, so half the angle and half the bond distance gives us a right angle triangle with the hypotenuse of (1.40 A/2)/sin(60^{0}/2) = 1.40 A. In below Figure the coordinates of C^{1} are (1.40, 0, 0) after dropping the units for clarity.

C^{2} is symmetry related to C^{1} as a rotation around the z-axis of 300°

C^{3} is a reflection in the yz plane of C^{2} and only changes the x coordinate of C^{2} to —x

C^{4} is a reflection in the yz plane of C^{1} and only changes the x coordinate of C^{1} to —x

C^{5} is an inversion through the origin of C^{2} will change all three coordinates to their negative values

C^{6} is a reflection in the zx plane of C^{2} and only changes the y coordinate of C^{2} to —y

The C-C distance in the cyclopentadiene anion is 1.40 A with the angle between neighbouring C-atoms of 360°/5 = 72°. So neighbouring C-atoms form an isosceles triangle with the origin, so half the angle and half the bond distance gives us a right angle triangle with the hypotenuse of (1.40 A/2)/sin(72o/2) = 1.19 A. So in Fig. 5.19 C^{1} has the coordinates (1.19, 0, 0) after dropping the units for clarity.

The take home message is that without the use of matrix algebra of symmetry operations these structural problems in Chemistry would involve some complicated non-trivial trig calculations.