Print this Page
11.7 – Spheres
Key Terms
 Pythagorean Theorem – The theorem that relates the side lengths of a right triangle.
 The theorem states that the square of the hypotenuse equals the sum of the squares of the legs:
 Axis – The diameter of a sphere.
 Diameter – A line segment that contains the center of the circle and has endpoints on the circle.
 This term also refers to the length of this line segment; the diameter of a circle is twice the radius.
 Sphere – A threedimensional figure consisting of all points in space that are the same distance from a given point.
Review
Remember 
 Surface Area
 BA = base area
 LA = lateral area
 p = perimeter of base
 h = height of solid
 r = radius
 s = slant height
 Volume

 Surface area measures the total area of all the faces of a solid.
 Volume measures the total amount of space inside a solid.

Notes
Surface Areas of Cylinders, Cones, and Spheres 
 The SA of a cone is less than the SA of a sphere, which is less than the SA of a cylinder.

Volumes of Cylinders, Cones, and Spheres 
 The volume of a cone is less than the volume of a sphere, which is less than the volume of a cylinder.

Surface Area of a Sphere 
 How to Find the Surface Area of Any Sphere
 Step 1: Write the surface area formula.
 Step 2: Find the radius of the sphere.
 Step 3: Square the radius.
 Step 4: Multiply the result by .


 SA (Sphere) = LA (Cylinder)

Volume of a Sphere 
 How to Find the Volume of Any Sphere
 Step 1: Write the volume formula.
 Step 2: Find the radius of the sphere.
 Step 3: Cube the radius.
 Step 4: Multiply the result by .


Pythagorean and Spheres 
 The crosssectional areas of the cylinder and the sphere are found using the same general formula.
 The Pythagorean theorem can be used to find the radius, a, of any crosssection of the sphere using b and r.

 To isolate “a” (the radius), subtract from both sides.


Volume of the Dugout Cylinder 
 Try to imagine removing (digging out) the two cones from the inside of the cylinder.
 What’s left over is the dugout cylinder!
 is the crosssectional area of a cylinder, and is the crosssectional area of an hourglass, or doublecone.
 If you then subtract (the doublecone) from (the cylinder), you can make a new shape, called a dugout cylinder, whose crosssectional area equals that of the sphere:
 To find the volume of a dugout cylinder:
 Step 1: Find the volume of each, the cylinder and doublecone.
 For this cylinder: and , so V=Bh is
 For this double cone, multiply the volume of a cone by 2:
 Note, the height of EACH cone is r, not 2r.
 Step 2: Subtract the volume of the doublecone from the cylinder.
Notice that the cylinder has the formula .
We can turn the 2 into a fraction, rewritten as ; and, ,
which is why the formula for a dugout cylinder is

Examples
Important!
Practice (Apex Study 11.7)
 Try practice problems on Pg 13
 Mandatory: write and answer problems on Pg 14
 1 Quiz
Permanent link to this article: http://newvillagegirlsacademy.org/math/?page_id=4938