- Perimeter of the shaded portion:
The perimeter of the shaded portion can be calculated by adding the lengths of all the sides of the shape. First, we need to find the length of the side BC, which is the same as the length of AP. Since AP is given as 21 cm, BC is also 21 cm.
Now, we can calculate the perimeter by adding all the sides: BC + CD + DA + AB.
P = 21 cm + 21 cm + 21 cm + 21 cm
P = 84 cm
- Area of the shaded portion:
To find the area of the shaded portion, we can divide the shape into two parts: a triangle and a sector of a circle.
1. Triangle ABC:
Since angle O is given as 60 degrees, we can see that triangle ABC is an equilateral triangle. Therefore, all sides are equal to 21 cm.
We can use the formula for the area of an equilateral triangle:
Area = (sqrt(3) / 4) * side^2
Area = (sqrt(3) / 4) * 21^2
Area = (sqrt(3) / 4) * 441
Area = 110.25 * sqrt(3)
2. Sector of a circle:
The sector is formed by the arc BC and the radii AB and AC. The angle of the sector is 60 degrees, which is one-sixth of the total circumference of the circle.
Circumference of the circle = 2πr
60 degrees is one-sixth of 360 degrees, so the circumference of the circle is 6 times the length of the arc BC.
6 * BC = 6 * 21 cm = 126 cm
Now, we can find the radius of the circle using the formula for the circumference:
2πr = 126 cm
r = 126 / (2π)
r ≈ 20 cm
Now, we can find the area of the sector:
Area = (θ/360) * πr^2
Area = (60/360) * π * 20^2
Area = (1/6) * π * 400
Area = 200π/3
Therefore, the total area of the shaded portion is the sum of the area of the triangle and the area of the sector:
Total Area = (110.25 * sqrt(3)) + (200π/3)