The radius of a sphere (abbreviated as variable r or R) is the distance from the exact center of the sphere to some point on the outer edge. As with circles, the radius of the sphere is often essential information for calculating measurements such as diameter, circumference, surface area, or volume. However, it is also possible to calculate the radius of the sphere using the diameter, circumference, etc. Use the appropriate formula for the information you have.
Method 1 of 3: Using radius calculation formulas
Step 1. Find the radius with the help of the diameter
The radius measures exactly half the diameter. So the formula is r = D/2. This formula is identical to the method used to calculate the radius of a circle using its diameter.
- If you have a sphere with a diameter of 16 cm, find the radius by dividing 16/2, arriving at the final result of 8 cm. If the diameter is 42 cm, the radius will be 21 cm.
Step 2. Find radius with the help of the circumference
use the formula C/2π. Since the circle is equal to πD, which equals 2πr, dividing it by 2π will give the radius.
- If you have a sphere with a circumference of 20 m, find the radius by dividing 20/2π, getting the final result of 3.183 m.
- Use the same formula to convert between the radius and circumference of the circle.
Step 3. Find the radius with the help of the sphere's volume
Use the formula ((V/π)(3/4))1/3. The volume of the sphere can be found using the equation V = (4/3)πr3. Solving the variable r in this equation the result will be ((V/π)(3/4))1/3 = r, that is, the radius of the sphere is equal to the volume divided by π, times 3/4, all raised to the 1/3 power (or cubic root).
- If you have a sphere with a 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
Step 4. Find the radius with the help of the surface area
use the formula r = √(A/(4π)). The surface area can be found using the equation A = 4πr2. The formula √(A/(4π)) = r means that the radius of the sphere equals the square root of the surface area divided by 4π. You can also raise (A/(4π)) to the 1/2 power to get the same result.
- If you have a sphere with a surface area of 1200 cm2, find the radius as follows:
- √(A/(4π)) = r
- √(1200/(4π)) = r
- √(300/(π)) = r
- √(95, 49) = r
- 9, 77 cm = r
Method 2 of 3: Defining Key Concepts
Step 1. Identify the basic measurements of the sphere
The Lightning (r) is the distance from the exact center of the sphere to some point on its surface. Generally speaking, you can find the radius if you know the diameter, circumference, volume, or surface area of the sphere.
- Diameter (D): is the distance across the sphere - it is twice the radius. The diameter is equivalent to the length of a line passing through the center of the sphere: from one end outside the sphere to the corresponding point on the other side passing directly through the entire sphere. In other words, it can be said that it is the greatest distance between two points inside the sphere.
- Circumference (C): is the one-dimensional distance around the sphere at its widest point. In other words, it is the perimeter of a spherical section through the section whose plane passes exactly through the center of the sphere.
- Volume (V): is the three-dimensional space contained within the sphere. He is the "space that the sphere occupies".
- Surface area (A): is the two-dimensional area on the outer surface of the sphere. It is the amount of flat space that covers the outside of the sphere.
- Pi (π): a constant that expresses the relationship of the circumference to the diameter of a circle. The first ten digits of pi are always 3, 141592653, but it is usually rounded to 3, 14.
Step 2. Use various measurements to find the radius
You can use the following measurements to find the radius of a sphere: diameter, circumference, volume, and surface area. You can also calculate each of these measurements if you know the radius value. Therefore, to find the radius, just invert the formula for calculating these measurements. Learn the formulas that use radius to find distance, circumference, surface 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 the diameter. Since the diameter is twice the radius, it is also possible to say that the circumference is twice the radius times π.
- V = (4/3)πr3. The volume of the sphere is the cubic radius (twice itself), times π, times 4/3.
- A = 4πr2. The surface area of a sphere is the radius cubic (times itself), times π, times 4. Since the area of the circle is πr2, it is also possible to say that the surface area of a sphere is equivalent to four times the area of the circle formed by its circumference.
Method 3 of 3: Finding the radius as the distance between two points
Step 1. Find the coordinates (x, y, z) of the center point of the sphere
The radius of a sphere can be thought of as the distance between the sphere's center and any point on its surface. Since this is true, if you know the coordinates of the point at the center of the sphere and any other point on the surface, you can find the radius by calculating the distance between the two points with a variant of the basic distance formula. To start, find the coordinates of the sphere's center point. As spheres are three-dimensional, the coordinates are the points (x, y, x), not just (x, y).
- This process is easier to understand through an example. Therefore, consider a sphere centered around the (x, y, z) points (4, -1, 12). In the next steps, we'll use these points to find the radius.
Step 2. Find the coordinates of a point on the sphere's surface
Next, you will need to find the coordinates (x, y, z) of a point on the sphere's surface. It can be any point on the surface. Since the points on the surface of a sphere are equidistant from the center point by definition, any point will serve to find the radius.
- For the example shown, let's say we know the point (3, 3, 0) lies on the surface of the sphere. By calculating the distance between this point and the center point, it is possible to find the radius.
Step 3. Find the radius using the formula d = √((x2 - x1)2 + (y2 -y1)2 + (z2 - z1)2).
Now that we know the center of the sphere and a point on its surface, calculating the distance between the two will result in the radius measurement. Use the three-dimensional distance formula d = √((x2 - x1)2 + (y2 -y1)2 + (z2 - z1)2), where d equals the distance, (x1y1, z1) is equivalent to the coordinates of the center point, and (x2y2, z2) equals the coordinates of the surface point to find the distance between two points.
- In the example used, we will use (4, -1, 12) for (x1y1, z1) and (3, 3, 0) for (x2y2, z2), being resolved as follows:
- 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.
Step 4. Know that generally r = √((x2 - x1)2 + (y2 -y1)2 + (z2 - z1)2).
On the sphere, each point on the surface is the same distance from the center point. If we take the three-dimensional distance formula given above and replace the variable "d" with "r" for the radius, we have a formula that can find the radius if we know any center point (x1y1, z1) and any corresponding at the surface point (x2y2, z2).
- By squaring both sides of the equation, we have r2 = (x2 - x1)2 + (y2 -y1)2 + (z2 - z1)2. Note that this is basically the same as the r sphere equation.2 = x2 + y2 + z2 which assumes the center point of (0, 0, 0).
- The order in which the operations are done is relevant. If you're not sure how priorities work, and your calculator supports the parenthesis function, then use it.
- π or pi is a Greek letter that represents the relationship of the diameter and circumference of a circle. It is an irrational number and cannot be written as a ratio of real numbers. There are several approaches to this measurement. The 333/106 approximation gives pi four decimal places. Today, most people memorize the number 3, 14, which is usually accurate enough for everyday use.
- This article is published on demand. However, if you are trying to get acquainted with geometric figures for the first time, it is far better to start from the back: Calculating the properties of the sphere from the radius.