If θ=180^(∘), how much of the sphere is occupied by the lune? What about when θ=120^(∘) and θ = 90^(∘)? Find the relation between θ and the part occupied of the sphere. To simplify the formula, convert from degrees to radians.
If θ is in degrees, the area is θ360^(∘)* 4π r^2. If θ is in radians, the area is 2θ r^2.
If the angle between the two great circles is θ =180^(∘), the lune will be half of the sphere. Thus, its surface area is S_1=2π r^2.
When the angle between the half great circles is θ = 120^(∘), the lune will occupy one-third of the sphere, that is, its surface area is S_2 = 13* 4π r^2.
If θ =90^(∘), the lune occupies one-quarter of the sphere, which implies that its surface area is S_3 = π r^2.
Let's summarize the results obtained above.
Angle θ
Portion of the Sphere
Surface Area of the Lune
180^(∘)
1/2 = 180^(∘)/360^(∘)
1/2* 4π r^2 = 2π r^2
120^(∘)
1/3 = 120^(∘)/360^(∘)
1/3* 4π r^2
90^(∘)
1/4 = 90^(∘)/360^(∘)
1/4 * 4π r^2 = π r^2
From the table above, we can write the following relation between the surface area of the lune and the central angle.
S = θ/360^(∘) * 4π r^2
That way, we can find the area of any lune just by knowing the angle θ between the great circles and the radius of the sphere. Even more, we can simplify the formula above by converting the angle from degrees to radians.
θ/360^(∘) = θ/360^(∘) * 360^(∘)/2π = θ/2π
Let's substitute the latter expression into the formula above.
