Area of sphere formula

A spherical segment is the solid defined by cutting a Call the and the height of the segment (the distance from the plane to the top of . Let the and , respectively. Call the distance from the center to the start of the segment , and the height from the bottom to the top of the segment . Call the , and the height above the center . Then , More things to try: • • • References Beyer, W.H. Harris, J.W. and Stocker, H. "Spherical Zone (Spherical Layer)."§4.8.5 in Kern, W.F. and Bland, J.R. "Spherical Segment."§36 in Smith, D.E. "Spherical Segment."§541 in Essentials of Plane and Solid Geometry. Boston, MA: Ginn and Co., p.542, 1923. Referenced on Wolfram|Alpha Cite this as: MathWorld--A Wolfram Web Resource. Subject classifications • • • • • • • • • • • • • • • • • • • • • Created, developed and nurtured by Eric Weisstein at Wolfram Research

• • • • • • What Is the Surface Area of a Sphere? The surface area of a sphere is the region or area covered by the outer, curved surface of the sphere. A sphere is a three-dimensional solid with every point on the surface at equal distances from the center. In simple words, any solid, round object shaped like a ball is a sphere. The radius of a sphere is the distance between the surface and the center of the sphere. While one endpoint of radius is on the surface, the other lies at the center of that sphere. Read on to know the exact definition of the surface area of a sphere, along with the derivation of the formula. Definition of the Surface Area of a Sphere The surface area of a sphere is defined as the region covered by the sphere’s outer surface in three-dimensional space. Since the sphere is curved, its curved surface area is the same as the total area of the sphere. It is expressed as: Surface area $(TSA) =$ Curved Surface Area $(CSA) = 4\pi r^$ square units Derivation of Surface Area of Sphere Now that you have understood the surface area of a sphere, it’s time to derive the formula. The Greek mathematician Archimedes discovered that the surface area of a sphere is the same as the lateral surface area of a cylinder such that the radius of the sphere is the same as the radius of the cylinder and the height of the cylinder is the same as the diameter of the sphere. According to Archimedes, a sphere can fit into a cylinder so that the height of the cylinder becomes th...

A sphere is a perfectly round geometrical 3-dimensional object. It can be characterized as the set of all points located distance \(r\) (radius) away from a given point (center). It is perfectly symmetrical, and has no edges or vertices. A sphere with radius \(r\) has a volume of \( \frac \pi r^3 \) and a surface area of \( 4 \pi r^2 \). A sphere has several interesting properties, one of which is that, of all shapes with the same surface area, the sphere has the largest volume. To prove that the surface area of a sphere of radius \(r\) is \(4 \pi r^2 \), one straightforward method we can use is calculus. We first have to realize that for a curve parameterized by \(x(t)\) and \(y(t\)), the \[ S = \int_a^b \sqrt \] Archimedes' Hat-Box Theorem Archimedes' hat-box theorem states that for any sphere section, its lateral surface will equal that of the cylinder with the same height as the section and the same radius of the sphere. Let us recall our last proof section. After revolving the semicircle around the \(x\)-axis, we will obtain a sphere's surface area, and if we cut just a partial section with parallel bases, the new surface area will be demonstrated in the image below: From the image, the section's lateral surface area is colored light blue with 2 circular bases of different radii. In order to visualize the section's height better, this section will be rotated by 90 degrees, as shown below: Now inside the section, there are 2 variable angles, \(\angle a\) and \(\angle b...

Sphere Shape r = radius V = volume A = surface area C = circumference π = pi = 3.1415926535898 √ = square root Calculator Use This online calculator will calculate the 3 unknown values of a sphere given any 1 known variable including radius r, surface area A, volume V and circumference C. It will also give the answers for volume, surface area and circumference in terms of PI π. A sphere is a set of points in three dimensional space that are located at an equal distance r (the radius) from a given point (the center point). Units: Note that units are shown for convenience but do not affect the calculations. The units are in place to give an indication of the order of the results such as ft, ft 2 or ft 3. For example, if you are starting with mm and you know r in mm, your calculations will result with A in mm 2, V in mm 3 and C in mm. Sphere Formulas in terms of radius r: • Volume of a sphere: • V = (4/3) πr 3 • Circumference of a sphere: • C = 2 πr • Surface area of a sphere: • A = 4 πr 2 \[ C \approx 1.77245\sqrt \] Sphere Calculations: Use the following additional formulas along with the formulas above. • Given the radius of a sphere calculate the volume, surface area and circumference Given r find V, A, C • use the formulas above • Given the volume of a sphere calculate the radius, surface area and circumference Given V find r, A, C • r = cube root(3V / 4 π) • Given the surface area of a sphere calculate the radius, volume and circumference Given A find r, V, C • r = √(A ...