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

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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.