Content

• To step
• Method 1 of 3: Using radius formulas
• Method 2 of 3: Define key concepts
• Method 3 of 3: Finding the radius as the distance between two points
• Tips

The radius of a sphere (abbreviated as the variable r or R.) is the distance from the exact center of the sphere to a point on that sphere's surface. As with circles, the radius of a sphere is often an essential metric for calculating the diameter, circumference, area and volume of a sphere. However, you can also work backwards from the diameter, circumference, etc. to find the radius of the sphere. Use the formula that is appropriate for the data you have.

To step

Method 1 of 3: Using radius formulas

1. Determine the radius if you know the diameter. The radius is half a diameter, so you use the formula r = D / 2. This is identical to the method of calculating the radius of a circle where the diameter is given.
   • If you have a sphere with a diameter of 16 cm, you calculate the radius with 16/2 = 8 cm. If the diameter is 42, then the radius is 21.
2. Determine the radius if you know the circumference. Use the formula C / 2π. Since the circumference equals πD, which in turn equals 2πr, calculate the radius by dividing the circumference by 2π.
   • If you have a sphere with a circumference of 20 m, you will find the radius with 20 / 2π = 3.183 m.
   • You can use the same formula to convert between the radius and the circumference of a circle.
3. Calculate the radius if you know the volume of the sphere. Use the formula ((V / π) (3/4)). The volume of a sphere is derived from the equation V = (4/3) πr. By solving the equation for r, you get ((V / π) (3/4)) = r, so it becomes clear that the radius of a or sphere is equal to the volume divided by π, times 3/4 , to the 1/3 power (or cube root).
   • If you have a sphere with a volume of 100 cm, you get the radius as follows:
     • ((V / π) (3/4)) = r
     • ((100 / π) (3/4)) = r
     • ((31.83) (3/4)) = r
     • (23.87) = r
     • 2,88 = r
4. Determine the radius of the surface. Use the formula r = √ (A / (4π)). You calculate the area of ​​a sphere with the equation A = 4πr. Solving the equation for r gives √ (A / (4π)) = r, which means that the radius of a sphere is equal to the square root of its area divided by 4π. You can also power (A / (4π)) to 1/2 for the same result.
   • If you have a sphere with an area of ​​1200 cm, you calculate the radius as follows:
     • √ (A / (4π)) = r
     • √ (1200 / (4π)) = r
     • √ (300 / (π)) = r
     • √ (95.49) = r
     • 9.77 cm = r

Method 2 of 3: Define key concepts

1. Know the basic dimensions of a sphere. The radius (r) is the distance from the exact center of the sphere to any point on the surface of the sphere. In general, you can find the radius of a sphere if you know its diameter, circumference, volume or area.
   • Diameter (D): the length of the line through the center of a sphere & ndash; double the radius. The diameter is the length of a line through the center of the sphere, from one point on the outside of the sphere to a corresponding point directly opposite it. In other words, the greatest possible distance between two points on the sphere.
   • Circumference (C): the one-dimensional distance around the sphere at its widest point. In other words, the circumference of the circular cross-section of a sphere, the plane of which runs through the center of the sphere.
   • Volume (V): the three-dimensional space within the sphere. It is the "space occupied by the sphere".
   • Surface (A): the two-dimensional space on the outer surface of the sphere. The amount of flat space that covers the outside of the sphere.
   • Pi (π): a constant expressing the ratio of the circle's circumference to the circle's diameter. The first 10 digits of Pi are always 3,141592653, although this is usually rounded to 3,14.
2. Use different measurements to determine the radius. You can use the diameter, circumference, volume, and area to calculate the radius of a sphere. If you know the length of the radius, you can calculate any of these numbers. So, to find the radius, you can reverse the formulas for calculating these parts. Learn the radius formulas to calculate diameter, circumference, area and volume.
   • D = 2r. As with circles, the diameter of a sphere is twice the radius.
   • C = πD or 2πr. As with circles, the circumference of a sphere is equal to π times its diameter. Since the diameter is twice the radius, we can also say that the circumference is twice the radius times π.
   • V = (4/3) πr. The volume of a sphere is the radius to cubic power (r x r x r), times π, times 4/3.
   • A = 4πr. The area of ​​a sphere is the radius to the power of two (rxr) times π, times 4. Since the circumference of a circle is πr, it can also be said that the area of ​​a sphere is equal to four times the area of a circle, as formed by its circumference.

Method 3 of 3: Finding the radius as the distance between two points

1. Find the coordinates (x, y, z) of the center of the sphere. One way to think about the radius of a sphere is as the distance between the center of the sphere and any point on its surface. Because this is true, you can use the coordinates of the center and a point on the surface of the sphere to determine the radius of the sphere by calculating the distance between the two points using a variation of the standard distance formula. To start, find the coordinates of the center of the sphere. Note that a sphere is three-dimensional, it will be an (x, y, z) point instead of an (x, y) point.
   • This is easier to understand with an example. Suppose a sphere is given with as center (-1, 4, 12). In the next few steps, we are going to use this point in determining the radius.
2. Find the coordinates of a point on the surface of the sphere. Then you need to determine the (x, y, z) coordinates of a point on the surface of the sphere. This is possible each point on the surface of the sphere. Because by definition all points on the surface of a sphere are equidistant from the center, you can use any point to determine the radius.
   • In the context of our example exercise, we make that the point (3, 3, 0) on the surface of the sphere. By calculating the distance between this point and the center, we can find the radius.
3. Determine the radius with the formula d = √ ((x₂ - x₁) + (y₂ - y₁) + (z₂ - z₁)). Now that you know the center of the sphere and a point on the surface of the sphere, you can find out the radius by calculating the distance between them. Use the three-dimensional distance formula d = √ ((x₂ - x₁) + (y₂ - y₁) + (z₂ - z₁)), where d is the distance, (x₁, y₁, z₁) represents the coordinates of the center, and (x₂, y₂, z₂) represents the coordinates of the point on the surface to determine the distance between the two points.
   • In our example, we substitute (4, -1, 12) for (x₁, y₁, z₁) and (3, 3, 0) for (x₂, y₂, z₂), solving this as follows:
     • d = √ ((x₂ - x₁) + (y₂ - y₁) + (z₂ - z₁))
     • d = √ ((3 - 4) + (3 - -1) + (0 - 12))
     • d = √ ((- 1) + (4) + (-12))
     • d = √ (1 + 16 + 144)
     • d = √ (161)
     • d = 12.69. This is the radius of our sphere.
4. In general, know that r = √ ((x₂ - x₁) + (y₂ - y₁) + (z₂ - z₁)). In a sphere, every point on the surface has the same distance from the center of the sphere. Taking the above three-dimensional distance formula and replacing the variable "d" with the variable "r" of the radius, we get an equation that allows us to find the radius at any given center point (x₁, y₁, z₁) and any corresponding point on the surface (x₂, y₂, z₂).
   • By squaring both sides of this equation, we get: r = (x₂ - x₁) + (y₂ - y₁) + (z₂ - z₁). Note: This is essentially the same as the standard equation for a sphere (r = x + y + z), assuming the center is equal to (0,0,0).

Tips

• The order of the operations is important. If you are not sure how the calculation rules work, and your calculator supports parentheses, make sure to use them.
• This article was created because this topic was in high demand. However, if you are trying to understand spatial geometry for the first time, it is probably better to start with the other side: calculating the properties of a sphere when the radius is given.
• Pi or π is a Greek letter that indicates the ratio of the diameter of a circle to its circumference. It is an irrational number and cannot be written as a ratio of real numbers. There are many approximations, and 333/106 returns pi to four decimal places. Today most people remember approximation 3.14 which is usually accurate enough for everyday purposes.