If you need calculate the area of a sphere online, thanks to our calculator you can get it just by entering the value of the radius of this figure.
To calculate the area of a sphere manually, you only have to multiply by four the squared radius of the sphere and all this by the PI number, so you will get the value of the sphere. surface.
If what you want is to know the volume of this figure, you can do it with our tool for calculate the volume of a sphere.
What is the area of a sphere?
If you are not clear about the area of a figure with volume such as the sphere, in the representation above these lines you can see an area with darker shading that represents the area of the sphere. surface area occupied by the sphere.
The area of a sphere is defined as four times its radius squared by π. In the following section we will see how it is calculated from its formula.
How to calculate the area of a sphere
To calculate the area of a sphere we have to apply the following formula mathematics:
Sphere area = 4πr2
where r will be the measure of the radius.
Solved exercise to find the area of a sphere
For example, let's calculate the area of a sphere inside a cylinder 3 meters high.
As can be seen in the figure, the three meters correspond to the diameter of the sphere but we need the radius which is half that value, i.e. 1.5 meters.
Now we apply the formula and we are left with the following:
Sphere area = 4π1,52 = 9π = 28.27 m2
As you can see, solving the problem is simple and the only difficulty lies in remembering to what is the formula for calculating the area of the sphere?.
Area of a sphere inscribed in a cube
If we have a sphere inscribed in a cube, it is the same case that we have seen before with the cylinder. We only have to keep in mind that the measure of the side of the cube will be the diameter of the sphere.
For example, if we are asked to calculate the area of a sphere inscribed in a cube of side 10cm, the first thing we will do will be to calculate the radius r:
radius = side of the cube / 2 = 5 cm
Now we apply the formula to get the surface of a sphere:
A = 4π52 = 100π = 314.15 cm2
Area of a sphere with integrals
To calculate the surface of a sphere by means of double or surface integrals we will use the following expression:
To understand the demonstration, we will do it on an example in which we are going to find the area of a sphere given by the equation x2 + y2 + z2 = a2
The sphere is a figure that can be parametrized with the following parametric equations:
- x = a sinΦ cosθ
- y = a sinΦ sinθ, with θ ∈ (0, 2π),Φ ∈ (0, π)
- z = a cos Φ
which define the position vector in this way:
R (Φ,θ) = (a sinΦ cosθ, a sinΦ senθ , a cos Φ)
The surface differential in this type of parameterized surfaces is defined as:
dS = |||RΦ x Rθ || dΦdθ
We do the partial derivatives and we have:
RΦ = (a cosΦ cosθ, a cosΦ senθ , -a sinΦ)
Rθ = (-a sinΦ sinθ, a sinΦ cosθ, 0)
Now we solve the vector product and calculate its magnitude:
Finally, we are left with the area of the sphere calculated by double or surface integrals es:
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