Question

A stop sign is a regular octagon. The standard stop sign has a 16.2 inch radius. The area of a standard stop sign is _______________
square inches. Please round your answer to the nearest whole number

Answer 1

The area of standard stop sign regular octagon is 742.3 squared inches, if the standard stop sign has a 16.2 inch radius.

Step-by-step explanation:

The given is,

               Radius of standard stop sign is 16.2 inch radius

Step:1

              The regular octagon is equal to the 16 right angled triangle,

                    Angle of right angle triangle  = $\frac{360}{16}$ = 22.5°

              Ref attachment,

              From the OAB right angle triangle,  

              Trigonometric ratio,

                                 sin ∅ = $\frac{Opp}{Hyp}$        

              Where, ∅ = 22.5°

                           Hyp = Radius = 16.2 inches

              Ratio becomes,

                           sin 22.5°= $\frac{b}{16.2}$

                                       b = (0.374607)(16.2)

                                       b = 6.19947 inches

             Trigonometric ratio,

                                 cos ∅ = $\frac{Adj}{Hyp}$        

              Where, ∅ = 22.5°

                           Hyp = Radius = 16.2 inches

                            Adj = h

              Ratio becomes,

                           cos 22.5°= $\frac{h}{16.2}$

                                       h = (0.9238795)(16.2)

                                       h = 14.967 inches

Step:2

           From the triangle OAC,

                             Area, $A = \frac{1}{2} (Height )( Base)$

          Where, Height = 14.967 inches

                       Base = b + x = 6.19947 + 6.19947

                                             = 12.3989 inches

           From the values equation becomes,

                                       A = $\frac{1}{2} (14.967)(12.39894)$

                                       A = 92.7875 Squared inches

Step:3

           The octagon is equal to sum of 8 triangles

            Area of octagon = 8 × Area of triangle

                                        = 8 × 92.7875

                                        = 742.3 squared inches

Result:

        The area of standard stop sign regular octagon is 742.3 squared inches, if the standard stop sign has a 16.2 inch radius.