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Surface Area and Volume Calculation for Spheres, Lecture notes of Physics

Formulas and examples for calculating the surface area and volume of spheres. It explains the concepts of radius, diameter, chord, and great circles, and demonstrates how to find the surface area and volume using these measurements. three examples with solutions.

What you will learn

  • What is a great circle of a sphere and how does it relate to the surface area?
  • How does the surface area of a sphere change when the radius doubles?
  • How can you approximate the volume of a sphere using pyramids?
  • What is the formula for finding the surface area of a sphere?
  • What is the formula for finding the volume of a sphere?

Typology: Lecture notes

2021/2022

Uploaded on 09/12/2022

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Surface Area and
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Surface Area and

Volume of

Spheres

Finding the Surface Area of a Sphere

A circle is described as a locus of points in a plane that are a given distance from a point. A sphere is the locus of points in space that are a given distance from a point.

Finding the Surface Area of a Sphere

  • A diameter is a chord that contains the center. As with all circles, the terms radius and diameter also represent distances, and the diameter is twice the radius.

Surface Area of a Sphere

  • The surface area of a sphere with radius r is S = 4 r 2 .

S = 4r

2

2

= 16 in.

2

S = 4r

2

2

= 64 in.

2 The surface area of the sphere in part (b) is four times greater than the surface area of the sphere in part (a) because 16 • 4 = 64 So, when the radius of a sphere doubles, the surface area DOES NOT double.

More...

  • If a plane intersects a sphere, the intersection is either a single point or a circle. If the plane contains the center of the sphere, then the intersection is a great circle of the sphere. Every great circle of a sphere separates a sphere into two congruent halves called hemispheres.

Solution: Begin by finding the radius of the sphere. C = 2r 13.8 = 2r 13.8 2 r 6.9 = r = r

Solution: Using a radius of 6.9 feet, the surface area is: S = 4r 2 = 4(6.9) 2 = 190.44 ft. 2

So, the surface area of the sphere is

190.44  ft.

Ex. 3: Finding the Surface Area of a Sphere

Finding the Volume of a Sphere

  • Imagine that the interior of a sphere with radius r is approximated by n pyramids as shown, each with a base area of B and a height of r, as shown. The volume of each pyramid is 1/ Br and the sum is nB.

More... V  n(1/3)Br = 1/3 (nB)r  1/3(4r 2 )r =4/3r 2 Each pyramid has a volume of 1/3Br. Regroup factors. Substitute 4r 2 for nB. Simplify.

Volume of a Sphere

  • The volume of a sphere with radius r is S = 4 r 3 . 3

Solution:

  • To find the volume of the slug, use the formula for the volume of a cylinder. V = r 2 h = ( 2 )(2) = 2 cm 3 To find the radius of the ball bearing, use the formula for the volume of a sphere and solve for r.

More... V =4/3r 3 2  = 4/3r 3 6  = 4r 3 1.5 = r 3 1.14  r Formula for volume of a sphere. Substitute 2 for V. Multiply each side by 3. Divide each side by 4. Use a calculator to take the cube root. So, the radius of the ball bearing is about 1.14 cm.