Cone

Scott Solomon

Cone = Three-Dimensional Shape with Circular Base Connecting to a Point

Cone                         coneinverted


Basics
r = radius = distance halfway across circular base of cone
d = diameter = distance all the way across circular base of cone
h = height
s = slant height = distance along cone’s side
Apex = point of cone
Base = circular portion of cone
SA = Surface Area = area covering surface, including base, of cone (measured in square units)
LA = Lateral Area = area covering surface, excluding base, of cone (measured in square units)
V = Volume = space occupied by cone (measured in cubic units)


Formulas (Mathematics Formula Sheet)
SA = πrs + πr²

V = \bf\displaystyle\frac{1}{3}πr²h


Formulas (NOT on Mathematics Formula Sheet)
LA = πrs

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


Question
A solid highway cone has a radius of 6 inches, height 8 inches, and slant height 10 inches.  What is its surface area?

Answer
96π square inches

Answer Process
SA = πrs + πr²
SA = π × 6 × 10 + π × 6² = 96π

Input Display Comment
clear blinker clears screen
π × 6 × 10 + π × 6 square π*6*10+π*6² SA = πrs + πr²
enter 96π Answer
Cone


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

Answer
301.5928947 square inches

Answer Process
See Toggle.
96π ⇔ 301.5928947

Input Display Comment
clear blinker clears screen
π × 6 × 10 + π × 6 square π*6*10+π*6² SA = πrs + πr²
enter 96π
toggle 301.5928947 Answer
Cone


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

Answer
301.6 square inches

Answer Process
See Rounding.
301.5928947 → 301.6


Question
A solid highway cone has a radius of 6 inches, height 8 inches, and slant height 10 inches.  To the nearest tenth of a square foot, what is its surface area?

Answer
2.1 square feet

Answer Process
See Measurement.
Square Feet = Square Inches ÷ 144 square inches per square foot
Square Feet = 301.5928947 ÷ 144 = 2.094395102 = 2.1 after rounding

Input Display Comment
clear blinker clears screen
π × 6 × 10 + π × 6 square π*6*10+π*6² SA = πrs + πr²
enter 96π
toggle 301.5928947
301.5928947 ÷ 144 301.5928947÷144 Square Feet = Square Inches ÷ 144 square inches per square foot
enter 2.094395102 Answer
(before rounding)
Cone


Question
A solid toy cone has a radius of 3 inches, height 4 inches, and slant height 4 inches.  To the nearest hundredth of a cubic inch, what is its volume?

Answer
37.70 cubic inches

Answer Process
V = \bf\displaystyle\frac{1}{3}πr²h

V = \bf\displaystyle\frac{1}{3} x π x 3² x 4 = 37.69911184 after toggle = 37.70 after rounding

Input Display Comment
clear blinker clears screen
fraction 1 down 3
×
π × 3 square × 4		 \bf\displaystyle\frac{1}{3}*π*3² *4 V = \bf\displaystyle\frac{1}{3}πr²h
enter 12π
toggle 37.69911184 Answer
(before rounding)
Cone


Question
A solid highway cone has a diameter of \bf\displaystyle6\frac{1}{2} inches, height \bf\displaystyle4\frac{1}{4} inches, and slant height \bf\displaystyle5\frac{3}{4}inches.  To the nearest hundredth of a cubic inch, what is its volume?

Answer
47.01 cubic inches

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

r = \bf\displaystyle\frac{1}{2}\times6\frac{1}{2} = \bf\displaystyle\frac{13}{4}

V = \bf\displaystyle\frac{1}{3}πr²h

V = \bf\displaystyle\frac{1}{3}\times\pi\times\frac{13}{4}^{2}\times5\frac{3}{4} = \bf\displaystyle\frac{2873\pi }{192} = 47.00935257 after toggle = 47.01 after rounding

Input Display Comment
clear blinker clears screen
fraction 1 down 2 right ×
2nd mixed number 6 right 1 down 2 right		 \bf\displaystyle\frac{1}{2}*6\frac{1}{2} r = \bf\displaystyle \frac{1}{2}d
enter \bf\displaystyle\frac{13}{4} r
fraction 1 down 3 × π ×
fraction 13 down 4 right square
×
2nd mixed number 4 right 1 down 4 right		 \bf\displaystyle\frac{1}{3}\times\pi\times\frac{13}{4}^{2}\times5\frac{3}{4} V = \bf\displaystyle\frac{1}{3}πr²h
enter \bf\displaystyle\frac{2873\pi }{192}
toggle 47.00935257 Answer
(before rounding)
Cone


Question
A solid highway cone has a radius of 3 inches, height 4 inches, and slant height 5 inches.  To the nearest tenth of a square inch, what is its lateral area?

Answer
47.1 square inches

Answer Process
LA = πrs
LA = π × 3 × 5 = 47.1238898 after toggle = 47.1 after rounding

Input Display Comment
clear blinker clears screen
π × 3 × 5 π*3*5 LA = πrs
enter 15π
toggle 47.1238898 Answer
(before rounding)
Cone


Practice – Questions
1.  A solid highway cone has a radius of 5 inches, height 12 inches, and slant height 13 inches.  To the nearest tenth of a square inch, what is its surface area?

2.  With reference to Question 2, what is the cone’s surface area to the nearest tenth of a square foot?

3.  A cotton candy cone has a radius of 5 inches, height 12 inches, and slant height 13 inches.  To the nearest hundredth of a cubic inch, what is its volume?

4.  A cotton candy cone has a diameter of \bf\displaystyle10\frac{1}{2} inches, height \bf\displaystyle12\frac{1}{4} inches, and slant height \bf\displaystyle13\frac{3}{4} inches.  To the nearest hundredth of a cubic inch, what is its volume?

5.  A cotton candy cone has a radius of 5 inches, height 12 inches, and slant height 13 inches.  To the nearest tenth of a square inch, what is its lateral area?


Practice – Answers
1.  282.7 square inches

2.  2.0 square feet

3.  314.16 cubic inches

4.  353.58 cubic inches

5.  204.2 square inches

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