Question

Calculate the arc length S and area A of the following sector a=30∘ i=90 cm

Answer 1

The arc length (S) is 15π cm, and the sector area (A) is 45²π cm², calculated by converting the central angle from degrees to radians and applying the formulas for arc length and sector area.

Step-by-step explanation:

To calculate the arc length (S) and the area (A) of the sector with an angle a=30° and radius i=90 cm, you follow these steps:

1. Convert the angle a from degrees to radians. Since there are 2π radians in 360 degrees, you use the conversion formula: radians = (degrees × π) / 180. For a = 30°: radians = (30 × π) / 180 = π / 6.
2. Use the formula for the arc length As = r × angle in radians, where r is the radius of curvature (or simply, the radius of the circle). The radius is given as 90 cm, thus S = 90 cm × (π / 6) = 15π cm.
3. For the area A of a sector, the formula is A = (Ρr²)/2, where Ρ is the central angle in radians and r is the radius. So, A = ((π / 6) ×90 cm²)/2 = (90² × π) / 12 = 45²π cm².

So the arc length is 15π cm and the sector area is 45²π cm².