Finding the area of ​​the spherical layer

In this publication, we will consider formulas that can be used to calculate the surface area of ​​a spherical layer (slice of a ball): spherical, bases and total.

Definition of a spherical layer

Spherical layer (or slice of a ball) – this is the part remaining between two parallel planes intersecting it. The picture below is colored yellow.

• R is the radius of the ball;
• r₁ is the radius of the first cut base;
• r₂ is the radius of the second cut base;
• h is the height of the spherical layer; perpendicular from the center of the first base to the center of the second.

Formula for finding the area of ​​a spherical layer

spherical surface

To find the area of ​​the spherical surface of the spherical layer, you need to know the radius of the ball, as well as the height of the cut.

S_{spheres district} = 2πRh

Grounds

The area of ​​the bases of the slice of the ball is equal to the product of the square of the corresponding radius by the number π.

S₁ = r₁²

S₂ = r₂²

Full surface

The total surface area of ​​a spherical layer is equal to the sum of the areas of its spherical surface and the two bases.

S_{full district} = 2πRh + πr₁² +πr₂² = π(2Rh + r₁² + r₂²)

Notes:

• if instead of radii (R, r₁ or r₂) given diameters (d), the latter should be divided by 2 to find the desired radius values.
• number value π when performing calculations, it is usually rounded to two decimal places – 3,14.