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Properties of Spheres in Math

  • A sphere is a 3D set of points in which all points are an equal distance from a center point.
  • Perfectly symmetrical
  • No vertices, corners, or edges, , therefore it is not polyhedra
  • The radius of a sphere is the distance from the center to the points on the surface of the sphere.
  • The volume of a sphere is the amount of space enclosed by the sphere.
  • The formula for volume of a sphere and was first discovered by Archimedes and is 4/3πr3
  • Compared to other shapes, a sphere has the smallest surface area for its volume.

Common Core Standard  G.GMD.3
a sphere cut into sections showing small circle great circle
Small Circle
Great Circle
great circle of a sphere
Small Circle

The Great Circle of a Sphere

Cutting a Sphere into Sections
If a sphere is cut into sections on the plane, then each of these sections is a circle

The section that passes through the center of the circle is the Great Circle

The radius of the Great circle is equal to the radius of the Sphere.

The diameter of the Great circle is equal to the diameter Sphere.

The shortest distance between two points on the sphere is along the segment of the great circle joining them.

The Great circle divides a sphere into two equal parts called hemispheres.

An example in Geography of a Great Circle is the equator.

All other sections are called Small circles.
The diameter of small circles is not equal to the diameter of the sphere.

Great Circle
4/3pi x 5^3

sphere with radius of 5 units
r =radius  In the picture it is the distance from A to B
Step 1
Step 2
Step 3

sphere with radius of 6 units


The formula for finding the volume of a sphere equals 4⁄3 πr^3 r =radius

Step 1. Plug 6 in the formula for the radius

Step 2. 4⁄3 π6^3 = 4⁄3 π216

Step 3. Multiply 4⁄3 x 216=( 4*216 )/3​

Step 4 864/3 = 288π units^3 = 904.32 units^3

Step by Step directions for solving the problem

Volume of a sphere formula equals 4⁄3 πr^3

= 523.6 units^3
properties of a sphere

What is the volume of a sphere with a radius of 5?

What is the volume of a sphere with a radius of 6?

Volume of a sphere applet

Plug 5 into the equation for volume of a sphere

Transcript Spheres
Welcome to MooMooMath. Today we are going to look at spheres and how to look at both the surface area and the volume of a sphere. Surface area is very simple, if you look at a circle or a sphere there is a great circle on the inside and I just highlighted it in red. Now they call it the great circle because it can be drawn inside a sphere. What you are going to do is find the area of that great circle which is just π x r^(2 )and the surface area is kind of like on a beach ball the amount of plastic you would need to create the ball and it is just four times the great circle. So if you can find the area of the circle you just multiply it by four. So you have 4 x π and in this case, the radius is 5 and this becomes 5 squared. So 5 squared is 25 and 25 times 4 is 100 and you just stick the π next to it so it becomes 100 π and it is a surface area so the units are always squared. Now let’s shift over here and look at volume, which is a similar formula. It is four-thirds times πr^3 and we will cube that. So all I need for this formula is the radius again. So the radius is five and we will cube it times π times 4⁄3 So 5 cubed is 5 times 5 times 5 which is 125 so 4⁄3 times 125 is 500 over 3 and don’t forget your π and your units are cubed because it is 3 dimensional so take your radius cube it and stick a 1 under it and multiple it by 4⁄3 and you get 500⁄(3 π units cubed) So let’s look at a cheat sheet 
Surface Area equals 4 x πr^2
Volume of a Sphere equals Four-thirds pi r cubed
In this example, our radius is 6 and six squared is 36 and 36 times 4 is 144 π units squared
Now to find the volume use 3 as the radius so 3 cubed is 27 and 27 times 4⁄3 is 36π units cubed so those are the two formulas for volume and surface area of a cube.