Explanation:
deca means ten.
a decagon has 10 sides and vertexes.
since the distance from the center to every vertex is equal (4 ft), this means it is a regular decagon (all sides are equal, and also all internal angles are the same).
the described scenario splits the tabletop virtually into 10 congruent isoceles triangles (the legs have the same length : 4 ft, and their 2 baseline angles have the same size). the sides of the decagon are the baselines of these triangles.
we know the inner angles at the top of each triangle, because they are 1/10 of the full circle around the table :
360 / 10 = 36°.
just by remembering that the sum of all angles in a triangle is always 180°, we know also the 2 baseline angles :
180 = 36 + 2×angle
2×angle = 144
each baseline angle is 144/2 = 72°
the area of a triangle is
baseline × height / 2
we can find these 2 lengths by using trigonometric functions of the known angles, and Pythagoras.
the height to the baseline of an isoceles triangle splits the baseline into 2 halves.
so, we have a right-angled triangle with 4 ft being the Hypotenuse, the height and half of the baseline being the legs.
and the angle at the top (center of the table) is 36/2 = 18°.
the third angle is still 72°.
half of the baseline is sin(18)×4 ft.
the height is cos(18)×4 ft
so, the area of one of these isoceles triangles is
2×sin(18)×4 × cos(18)×4 / 2 = sin(18)×cos(18)×16 =
= 4.702282018... ft²
the area of the whole tabletop is then the sum of all 10 triangles :
10×4.702282018... = 47.02282018... ft² ≈ 47 ft²