Surface Area of a Conical Frustum
![Surface Area = (pi * (R1+R2) * sqrt[(R1-R2)² + H²]) + (pi * R1²) + (pi * R2²)](equations/0041_equation.gif)
Where:
π = a Greek symbol for pi; 3.14159...
r1 = radius of the top circle
r2 = radius of the base of the shape (the bottom circle)
h = height of the shape
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A conical frustum is a three-dimensional cone whose top point has been cut off. Another way to think of it is as a cylinder whose bottom circle (base) is larger than its top circle. Megaphones from the old days (ones that cheerleaders sometimes use) are in the shape of a conical frustum.
The surface area of a conical frustum is the total area that makes up its exterior. For example, imagine taking the shape apart and laying out all the shapes that make it up. If you found all took their areas and added them together, you'd get its surface area.
The first part of the equation that's in parenthesis (from π until h²) is the surface area of the body of the shape. To find this, you must know the radius of both end circles and how high the shape is.
The second portion of parenthesis (πr1²) is the area of the shape's top circle. (Keep in mind that this makes sense because the equation for the area of a circle is πr².) After you find this part of the equation, you can add it to the area of the shape's base (its bottom circle) and you will have the total surface area of conical frustum's ends.
The third portion of parenthesis (πr2²) is the area of the base of the shape (the bottom circle.) Add this to the second and first set of parenthesis and you will have the total surface area.
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Although π (pi) has an infinite amount of numbers after the decimal point, people commonly use 3.14 as its value when solving an equation that uses it. Our calculator, however, uses a more accurate version of pi with thirteen digits after the decimal point (3.1415926535898.) This means that if the same values that I use for the example in this help file are plugged into our calculator, the answer will be slightly different (and more accurate.)
So why do I use a less accurate number if it gives a less accurate answer? I use the rounded version because that is what most teachers will have you use for pi in school. In some cases, you may get counted off for giving a different answer than your math book (or your teacher) has.
Also, it is much easier to learn and remember the rounded value. Not to mention, if I used the number with thirteen decimals in each example, it would take much longer to work through and the long string of digits after the decimal might very well become confusing.
Example
Find the surface area of a conical frustum whose height is 7 inches, top circle has a radius of 3 inches, and base has a radius of 6 inches.
After everything is plugged in, we strictly follow the order of operations (PEMDAS) so that all math is done in the correct order. Click the previous link to learn all about order of operations.
Also, make sure to keep track of long decimals for square roots of numbers and other calculations until you've found the answer to your equation. After you have the answer, then you can round off the decimals.
(If your teacher(s) tells you to round off every number from each separate calculation while in the process of solving an equation, though, listen to him/her. Always follow your teacher’s instructions before you follow mine.)
Do the math in each set of minor Parenthesis:
3 + 6 = 9
3 - 6 = -3
3² = 9
6² = 14
Plug these values in:
(multiply that to 28.26):
28.26(7.6157731058639082856614110271583) = 215.22174797171404815279147562749
Now we have:
(215.22174797171404815279147562749) + 28.26 + 43.96
Do the addition:
215.22174797171404815279147562749 + 28.26 = 243.48174797171404815279147562749
(243.48174797171404815279147562749) + 43.96 = 287.44174797171404815279147562749
Round the answer to four decimal places:
287.4417
The surface area of our conical frustum is 287.4417 square inches.
Why Square Inches?
Note that my answer is in square inches (square in or in²). When calculating the area of a three-dimensional shape, the units of your answer will always be squared (square feet, square inches, etc.) This is because when assigning one number for the measurement of any shape (with one, two, or three dimensions), both the length and the width need to be accounted for.
Therefore, the value for the surface area of a conical frustum tells how many single-unit squares fit into the shape. Single unit squares are used because they account for both length and width. For example, if the surface area of a three-dimensional shape is 24 square feet, that means that twenty-four squares with the area of one square foot can fit onto that shape's surface.
Works Cited
Weisstein, Eric W. "Conical Frustum." MathWorld. 1999. Wolfram Research, Inc. 12
July 2006. <http://mathworld.wolfram.com/ConicalFrustum.html>.
Pontuti, Marcus. "Volume Of A Conical Frustum." Microgeometry. 2005. Marcus
Pontuti. 12 July 2006. <http://microgeometry.netfirms.com/content/frustumcone.html>.
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