Jared and Morgan are making two gazebos in the shape of regular polygons, Jared's gazebo is a heptagon and has an apothem of 13.6 feet and an area of 623.5 ft and Morgan's gazebo is an octagon has an
apothem of 12.5 feet and an area of 517.8 ft". Which gazebo has the longest side length?

The gazebo with the longest side length is the heptagon.

A heptagon is a polygon with 7 sides. It has 7 edges and vertices.

The length of the side of a heptagon = (area of the heptagon x 2) / (number of sides x length of the apothem)

= (623.5 x 2) / (7 x 13.6)

= 1247 / 95.2

= 13.1 feet

A octagon is a polygon with 8 sides. It has 8 edges and vertices. The sum of internal angles in 135 degrees.

The length of the side of a octagon = (area of the octagon x 2) / (number of sides x length of the apothem)

= (517.8 x 2) / (12.5 x 8)

= 1035.6 / 100

= 10.356 feet

To learn more about an octagon, please check: link