Explanation:
the stop sign is a regular octagon (8 equal sides and vertices and equal inner angles).
it is usually mounted with flat sides on the top and the bottom (not with a vertex at the top and bottom).
so, when it is said here the sign is 790mm tall, this means the diameter from the middle of a side to the middle of the opposite side (and not from a vertex to the opposite vertex).
the corresponding diameter of the inner red area is
790 - 2×20 = 750 mm
because the white border had to be subtracted twice (to and bottom).
we have to assume that the letters "STOP" (written in white) are considered part of the inner red area, as we have no information about their sizes.
such a regular octagon can be virtually split into 8 congruent isoceles triangles (both legs are equally long).
their baselines are the sides of the octagon.
and the inner top angles of these triangles are
360/8 = 45°
remember, the sum of all angles in a triangle is always 180°. where the 2 baseline angles in an isoceles triangle are equal.
so we have
180 = 45 + 2×angle
2×angle = 135°
each inner baseline angle is 135/2 = 67.5°
the diameter as described above represents the heights of 2 opposing triangles.
so, one height is
750/2 = 375 mm
the area of one such triangle is
baseline × height / 2 = baseline × 375 /2
to get the baseline we have to imagine that the height splits the baseline into 2 halves. and the height, half of the baseline and one isoceles leg create a right-angled triangle with the top angle also cut in half :
45/2 = 22.5°
per law of sine
a/sin(A) = b/sin(B) = c/sin(C)
with a, b, c being the sides, A, B, C being the corresponding opposite angles
we know
375/sin(67.5) = baseline/2 / sin(22.5)
baseline = 2×sin(22.5)×375/sin(67.5) =
= 310.6601718... mm
the area of one of these 8 isoceles triangles is then
(2×sin(22.5)×375/sin(67.5)) × 375 / 2 =
= 375²×sin(22.5)/sin(67.5) = 58,248.78221... mm²
the total area of the red area of the sign (= the sum of all 8 triangles) is then
8×58,248.78221... = 465,990.2577... mm²