Sphere

Scott Solomon

Sphere = 3-Dimensional Round Shape with All Surface Points Equidistant from Center

Sphere


Basics
r = radius = distance from surface to center
d = diameter = distance all the way across sphere through its center
SA = Surface Area = area covering surface of sphere (measured in square units)
V = Volume = space occupied by sphere (measured in cubic units)


Formulas (Mathematics Formula Sheet)
SA = 4πr²

V = \bf\displaystyle\frac{4}{3}πr³


Formulas (NOT on Mathematics Formula Sheet)
r = \bf\displaystyle \frac{1}{2}d


Question
A ball has a radius of 7 centimeters.  What is its surface area?

Answer
196π square centimeters

Answer Process
SA = 4πr²
SA = 4 × π × 7² = 196π

Input Display Comment
clear blinker clears screen
4 × π × 7 square 4*π*7² SA = 4πr²
enter 196π Answer
Sphere

 

Question
With reference to the question above, what is the surface area of the ball in decimal form?

Answer
615.7521601 square centimeters

Answer Process
See Toggle.
196π ⇔ 615.7521601

Input Display Comment
clear blinker clears screen
4 × π × 7 square 4*π*7² SA = 4πr²
enter 196π
toggle 615.7521601 Answer
(Toggle)
Sphere


Question
With reference to the question above, what is the area of the sphere to the nearest tenth?

Answer
615.8 square centimeters

Answer Process
See Rounding.
615.7521601 → 615.8


Question
A ball has a diameter of \bf\displaystyle14\frac{1}{2} centimeters.  To the nearest tenth of a square centimeter, what is its surface area?

Answer
660.5 square centimeters

Answer Process
r = \bf\displaystyle \frac{1}{2}d

r = \bf\displaystyle\frac{1}{2}\times14\frac{1}{2} = \bf\displaystyle\frac{29}{4}

SA = 4πr²
SA = \bf\displaystyle 4\times\pi\times\frac{29}{4}^{2} = \bf\displaystyle\frac{841\pi }{4} = 660.5198554 after toggle = 615.8 after rounding

Input Display Comment
clear blinker clears screen
fraction 1 down 2 right
×
2nd mixed number 14 right 1 down 2 right		 \bf\displaystyle\frac{1}{2}*14\frac{1}{2}

r = \bf\displaystyle \frac{1}{2}d

 
enter \bf\displaystyle\frac{29}{4} r
4 × π × fraction 29 down 4 right square 4*π*\bf\displaystyle{{\frac{29}{4}}^{2}} SA = 4πr²
enter \bf\displaystyle\frac{841\pi }{4}
toggle 660.5198554 Answer
(before rounding)
Sphere


Question
A ball has a radius of 7 centimeters.  To the nearest hundredth of a cubic centimeter, what is its volume?

Answer
1436.76 cubic centimeters

Answer Process
V = \bf\displaystyle\frac{4}{3}πr³

V = \bf\displaystyle\frac{4}{3} x π x 7³ = \bf\displaystyle\frac{1372\pi }{3} = 1436.75504 after toggle = 1436.76 after rounding

Input Display Comment
clear blinker clears screen
fraction 4 down 3 right
× π ×
7 power 3		 \bf\displaystyle\frac{4}{3}*π*7³ V = \bf\displaystyle\frac{4}{3}πr³
enter \bf\displaystyle\frac{1372\pi }{3}
toggle 1436.75504 Answer
(before rounding)
Sphere


Question
A ball has a diameter of \bf\displaystyle14\frac{1}{2} centimeters.  To the nearest hundredth of a cubic centimeter, what is its volume?

Answer
1596.26 cubic centimeters

Answer Process
r = \bf\displaystyle \frac{1}{2}d

r = \bf\displaystyle\frac{1}{2}\times14\frac{1}{2} = \bf\displaystyle\frac{29}{4}

V = \bf\displaystyle\frac{4}{3}πr³

V = \bf\displaystyle\frac{4}{3}\times\pi\times\frac{29}{4}^{3} = \bf\displaystyle\frac{24389\pi }{48} =1596.256317 after toggle = 1596.26 after rounding

Input Display Comment
clear blinker clears screen
fraction 1 down 2 right ×
2nd mixed number 14 right 1 down 2 right		 \bf\displaystyle\frac{1}{2}*14\frac{1}{2} r = \bf\displaystyle \frac{1}{2}d
enter \bf\displaystyle\frac{29}{4} r
fraction 4 down 3 × π ×
fraction 29 down 4 right
power 3		 \bf\displaystyle\frac{4}{3}\times\pi\times\frac{29}{4}^{3} V = \bf\displaystyle\frac{4}{3}πr³
enter \bf\displaystyle\frac{24389\pi }{48}
toggle 1596.256317 Answer
(before rounding)
Sphere


Practice – Questions
1.   A model globe has a radius of 6 inches.  What is its surface area?

2.   With reference to Question 1, what is the globe’s surface area to the nearest tenth?

3.  A model globe has a diameter of \bf\displaystyle12\frac{1}{4} inches.  To the nearest tenth of a square inch, what is its surface area?

4.  A model globe has a radius of 6 inches.  To the nearest hundredth of a cubic inch, what is its volume?

5.  A model globe has a diameter of \bf\displaystyle12\frac{1}{4} inches.  To the nearest hundredth of a cubic inch, what is its volume?


Practice – Answers
1.  144π square inches

2.  452.4 square inches

3.  471.4 square inches

4.  904.78 cubic inches

5.  962.51 cubic inches

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