Answer:
- a = 54, b = 24
- area = (54√3 -24π) cm²
Step-by-step explanation:
You want the shaded area in the figure shown.
Trapezoid
The length AC in the figure can be found from ...
AC = (12 cm)·cos(π/3) = 6 cm
This means the bases of trapezoid OADB are AD = 6 cm, OB = 12 cm.
The length OC in the figure can be found from ...
OC = (12 cm)·sin(π/3) = 6√3 cm
How that we have the bases and height of the trapezoid, we can find its area:
A = 1/2(b1 +b2)h
A = 1/2(6 cm + 12 cm)(6√3 cm) = 54√3 cm² . . . . trapezoid area
Sector
The white sector in trapezoid OADB has central angle θ = π/3 radians and radius r = 12 cm. Its area is ...
A = 1/2r²θ
A = 1/2(12 cm)²(π/3) = 24π cm² . . . . sector area
Shaded area
The shaded area is the difference between the trapezoid area and the sector area:
shaded area = (trapezoid area) - (sector area)
shaded area = 54√3 - 24π cm²
In the required form, a=54, b=24.
