The radius of a sphere (abbreviated with the variable r) is the distance that separates the center of the solid from any point on its surface. Just as with the circle, the radius is often an essential data from which to start calculating the diameter, circumference, surface and / or volume of a sphere. However, you can also work backwards and use the diameter, circumference, etc. to figure it out. Use the most suitable formula in relation to the data in your possession.
Steps
Method 1 of 3: Using the Radius Calculation Formulas
Step 1. Find the radius from the diameter
The radius is half the diameter, so use the formula: r = D / 2. This is the same procedure that is used to find the value of the radius of a circle by knowing its diameter.
If you have a sphere with a diameter of 16 cm, then you can find its radius by dividing: 16/2 = 8 cm. If the diameter were 42 cm, the radius would be equal to 21 cm.
Step 2. Calculate the radius from the circumference
In this case, you need to use the formula: r = C / 2π. Since the circumference is equal to πD, that is, to 2πr, if you divide it by 2π you will get the radius.
- Suppose you have a sphere with a circumference of 20 m, to find the radius proceed to this calculation: 20 / 2π = 3, 183 m.
- This is the same formula you would use to find the radius of a circle from the circumference.
Step 3. Calculate the radius knowing the volume of the sphere
Use the formula: r = ((V / π) (3/4))1/3. The volume of a sphere is obtained with the equation: V = (4/3) πr3; you just solve for "r" and you get: ((V / π) (3/4))1/3 = r, which means that the radius of a sphere is equal to its volume divided by π, multiplied by ¾ and all raised to 1/3 (or under the cube root).
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If you have a sphere with the volume of 100 cm3, find the radius as follows:
- ((V / π) (3/4))1/3 = r;
- ((100 / π) (3/4))1/3 = r;
- ((31, 83)(3/4))1/3 = r;
- (23, 87)1/3 = r;
- 2, 88 cm = r.
Find the Radius of a Sphere Step 4 Step 4. Find the radius from the surface data
In this case, use the formula: r = √ (A / (4π)). The surface area of a sphere is obtained from the equation A = 4πr2. Solving it for "r" we arrive at: √ (A / (4π)) = r, ie the radius of a sphere is equal to the square root of its area divided by 4π. You can also decide to raise (A / (4π)) to the power of ½ and you will get the same result.
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Suppose you have a sphere with an area equal to 1200 cm2, find the radius like this:
- √ (A / (4π)) = r;
- √ (1200 / (4π)) = r;
- √ (300 / (π)) = r;
- √ (95, 49) = r;
- 9, 77 cm = r.
Method 2 of 3: Define Key Concepts
Find the Radius of a Sphere Step 5 Step 1. Identify the basic parameters of the sphere
The radius (r) is the distance that separates the center of the sphere from any point on its surface. Generally speaking, you can find the radius by knowing the diameter, circumference, surface and volume of the sphere.
- Diameter (D): is the segment that crosses the sphere, in practice it is equal to twice the radius. The diameter passes through the center and joins two points on the surface. In other words, it is the maximum distance that separates two points of the solid.
- Circumference (C): it is a one-dimensional distance, a closed plane curve that "wraps" the sphere at its widest point. In other words, it is the perimeter of the plane section obtained by intersecting the sphere with a plane that passes through the center.
- Volume (V): is the three-dimensional space contained by the sphere, that is the one occupied by the solid.
- Surface or area (A): represents the two-dimensional measure of the external surface of the sphere.
- Pi (π): is a constant that expresses the ratio between the circumference of a circle and its diameter. The first digits of pi are always 3, 141592653, although it is often rounded to 3, 14.
Find the Radius of a Sphere Step 6 Step 2. Use various elements to find the radius
In this regard, you can make use of the diameter, circumference, volume or area. You can also proceed in reverse and find all these values starting from that of the radius. However, to calculate the radius, you have to take advantage of the inverse formulas of those that allow you to arrive at all these elements. Learn formulas that use radius to find diameter, circumference, area and volume.
- D = 2r. Just like with circles, the diameter of a sphere is twice the radius.
- C = πD or 2πr. Again, the formula is identical to the one used with circles; the circumference of a sphere is equal to π times its diameter. Since the diameter is twice the radius, the circumference can be defined as the product of π and twice the radius.
- V = (4/3) πr3. The volume of a sphere is equal to the cube of the radius (the radius multiplied by itself three times) by π, all multiplied by 4/3.
- A = 4πr2. The area of the sphere is equal to four times the radius raised to the power of two (multiplied by itself) by π. Since the area of a circle is πr2, you can also say that the area of a sphere is equal to four times the area of the circle defined by its circumference.
Method 3 of 3: Find the Radius as the Distance Between Two Points
Find the Radius of a Sphere Step 7 Step 1. Find the coordinates (x, y, z) of the center of the sphere
You can imagine the radius of a sphere as the distance that separates the center of the solid from any point on its surface. Since this concept coincides with the definition of radius, knowing the coordinates of the center and another point on the surface, you can find the radius by calculating the distance between them and applying a variation to the basic distance formula. To start, find the coordinates of the center of the sphere. Since you are working with a three-dimensional solid, the coordinates are three (x, y, z), rather than two (x, y).
The process is easier to understand thanks to an example. Consider a sphere centered at the point with coordinates (4, -1, 12). In the next few steps you will use this data to find the radius.
Find the Radius of a Sphere Step 8 Step 2. Find the coordinates of the point on the surface of the sphere
Now you have to identify the three spatial coordinates that identify a point on the surface of the solid. You can use any point. Since all the points that make up the surface of a sphere are equidistant from the center by definition, you can consider whichever you prefer.
Continuing with the previous example, consider the point with coordinates (3, 3, 0) lying on the surface of the solid. By calculating the distance between this point and the center you will find the radius.
Find the Radius of a Sphere Step 9 Step 3. Find the radius with the formula d = √ ((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2).
Now that you know the coordinates of the center and those of the point on the surface, you just have to calculate the distance to find the radius. Use the three-dimensional distance formula: d = √ ((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2), where d is the distance, (x1, y1, z1) are the coordinates of the center and (x2, y2, z2) are the coordinates of the point on the surface.
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Use the data from the previous example and insert the values (4, -1, 12) in place of the variables of (x1, y1, z1) and the values (3, 3, 0) for (x2, y2, z2); later solve like this:
- d = √ ((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2);
- d = √ ((3 - 4)2 + (3 - -1)2 + (0 - 12)2);
- d = √ ((- 1)2 + (4)2 + (-12)2);
- d = √ (1 + 16 + 144);
- d = √ (161);
- d = 12.69. This is the radius of the sphere.
Find the Radius of a Sphere Step 10 Step 4. Know that, in general, r = √ ((x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2).
In a sphere, all points lying on the surface are equidistant from the center. If you consider the three-dimensional distance formula expressed above and replace the variable "d" with "r" (radius), you get the formula for calculating the radius from the coordinates of the center (x1, y1, z1) and from those of any point on the surface (x2, y2, z2).
Raising both sides of the equation to a power of 2, we obtain: r2 = (x2 - x1)2 + (y2 - y1)2 + (z2 - z1)2. Note that this is practically identical to the basic equation of a sphere centered on the origin of the axes (0, 0, 0), i.e.: r2 = x2 + y2 + z2.
Advice
- Remember that the order in which the calculations are done is important. If you are unsure of the priorities with which you should perform the operations and you have a scientific calculator that allows the use of parentheses, make sure to enter them.
- π is a Greek letter that represents the ratio between the diameter of a circle and its circumference. It is an irrational number and cannot be written as a fraction of real numbers. However, there are some approximation attempts, for example 333/106 gives π with four decimal places. Currently, most people memorize the approximation of 3, 14, which is accurate enough for everyday calculations.
- This article tells you how to find the radius starting from other elements of the sphere. However, if you are approaching solid geometry for the first time, you should start with the reverse process: studying how to derive the various components of the sphere from the radius.
