The surface area of a sphere is the area covered by the surface of the sphere, usually expressed in square metres or square centimetres. The surface area of a sphere can be calculated from its radius, which is 4πr^2.
The formula for the area of the sphere is 4πr^2, where r is the radius of the sphere.
Here are some examples of how to calculate the area of a sphere:
1. If you are baking a spherical cake with a diameter of 10 cm, you need to calculate the surface area of the sphere to find out how much icing is needed to cover the entire surface of the sphere. Using the formula 4πr^2, the surface area of this sphere is 4π×5^2=314.16cm^2.
2. When designing a spherical pool with a diameter of 20 cm, it is necessary to calculate the surface area of the sphere to determine how many tiles are needed to cover the entire surface of the sphere. Using the formula 4πr^2, the surface area of the sphere is 4π×10^2=1256.64cm^2.
3. If you are making a spherical lamp shade with a diameter of 15cm, you need to calculate the surface area of the sphere to find out how much paper or cloth you need to cover the entire surface of the sphere. According to the formula 4πr^2, the surface area of this sphere is 4π×7.5^2=706.86cm^2.
The area formula of the sphere is different from the volume formula of the sphere, as shown in the following figure:
Cut the upper hemisphere of a sphere with a radius of r into n equal parts, each of which is equally high.
Treat each part as a cylinder, where the radius is equal to the radius of its base circle.
Then the lateral area s(k) of the kth cylinder from bottom to top is 2πr(k)*h.
Where h=r/n and r(k)=√[r^2-(kh)^2].
s(k)=√[r^2-(kr/n)^2]2πr/n.
=2πr^2√[1/n^2-(k/n^2)^2]
So, when n approaches infinity, the sum of s(1)+s(2)+…+s(n) is equal to the surface area of the hemisphere, which is 2πr^2 multiplied by 2, resulting in the surface area of the entire sphere, which is 4πr^2.
Sphere surface area formula:
The formula for calculating the surface area of a sphere is S=4R^2π. If it’s a hemisphere, only half of the surface area of the sphere and the area of the bottom circle need to be calculated, resulting in S=1/2S.
Sphere + S base = 2πR^2 + πR^2 = 3πR^2.
Sphere surface area formula:
Let the radius of the sphere be R. The surface area of the sphere is uniquely determined by the radius, so its surface area is a function of R.
The surface area of a sphere refers to the area enclosed by the spherical surface, including the spherical surface and the space enclosed by it.
Relevant knowledge points to learn: “Formula for the Area of a Circle”, “Formula for the Area of an Ellipse”, “Formula for the Surface Area of a Sphere”, “Formula for the Area of a Torus”, “Formula for the Surface Area of a Cylinder”, “Formula for the Lateral Area of a Cylinder”, “Formula for the Surface Area of a Cone”, “Formula for the Lateral Area of a Cone”.
One common proof method involves cutting a sphere into many small triangles, then unfolding these triangles into a flat shape, and finally calculating the area of this flat shape, which represents the surface area of the sphere. This proof process requires the application of the formula for the area of a triangle, knowledge of plane geometry, and other related concepts.
