Math help please?

The yellow surface area is:

yellow = (7 * 6)/2

yellow = 21 cm²

The blue surface area is:

blue = (π.r²) * (α/360) → where r is the radius of the circle, in your case: r = OA = 7

blue = (49π) * (α/360) → where: tan(α) = AB/OA = 6/7 → α ≈ 40.6 °

blue ≈ 49π * (40.6/360)

blue ≈ 17.36 cm²

The surface area of the shaded region is:

= yellow - blue

≈ 21 - 17.36

≈ 3.6386 cm²

To be more precise:

blue = (49π) * (α/360) → where: tan(α) = AB/OA = 6/7

blue = 49π * atan(6/7) / 360

blue = (49/360).π.atan(6/7)

The surface area of the shaded region is:

= yellow - blue

= 21 - (49/360).π.atan(6/7)

= [7560 - 49π.atan(6/7)]/360

 The figure shows a right-angled triangle OAB. 

 AOC is a minor sector enclosed in the triangle. 

 If OA = 7 cm, AB = 6 cm, 

 calculate the area and perimeter 

 of the shaded region.

 The area of the triangle OAB = 21 cm^2

 OB = √85 = 9.22

 CB = 2.22

 Angle BOA = 40.60°

 The area of the minor sector AOC:

 49 pi (40.6/360) = 17.3608... cm^2

 The area of the shaded region = 3.6302... cm^2

 Length of arc AC = 14 pi (40.6/360) = 4.96 cm

 The perimeter of the shaded region 

 = (6 + 2.22 + 4.96) cm

 = 13.18 cm

Let x is angle in vertex O

x = arc tan (6/7) = 0.70863 rad

Now length of arc AC is

AC= OA * x = 7 * 0.70863 = 4.96

Use Pythagorean Theorem to find

OB = sqrt (6^2+7^2) = sqrt (85) = 9.22

BC= OB-OC=9.22-7 = 2.22

Perimeter of shaded region ABC is

 AB+BC+AC = 6 + 2.22 + 4.96 = 13.18

Area of sector AOC relates to area of circle with radius OA the same as angle x relates to 2Pi (full circle). Hence the area of AOC is

(x/(2Pi))*(OA^2 * Pi) = x * OA^2 / 2 = 0.70863 * 7^2 / 2 = 17.36

Area of triangle OAB is

OA*AB / 2 = 7 * 6 / 2 = 21

Area of shaded region is difference between the two

21 - 17.36 = 3.64