Explanation:
what is the area of the whole circle ?
pi×r²
what is the segment length of the whole circle ?
right, that is the circumference :
2×pi×r
and a whole circle stand for 360°.
if we have only a part of these 360°, then all the attributes of the circle apply only to the corresponding fraction of 360°.
the "segment area" is (I assume) the shaded area.
that is the area of the whole segment XZY minus the triangle area.
let's start with the arc length (the easier part) :
again, for 360° it is
2×pi×r
for 135° it is
2×pi×r × 135/360 = 2×pi×8 × 135/360 =
= 18.84955592... cm
for the segment area :
the area of the full segment is
pi×r² = pi×8² = 64pi
for 135° it is
64pi × 135/360 = 75.39822369... cm²
the inner triangle is an isoceles triangle (both legs are equally long : 8 cm).
when we have the 2 legs and the encoded angle, the area of the triangle is
area= (a x b x sin(c))/2, where a, b are the two legs, and c is the angle between them.
area of the triangle = 8×8×sin(135)/2 = 22.627417... cm²
the shaded area is then
75.39822369... - 22.627417... = 52.77080669... cm²