Remembering Area FormulasDate: 12/23/2001 at 13:47:55 From: Summer Subject: Areas Hi, I was wondering if there was a good way to help me memorize the formulas for areas of different shapes. I am having a lot of trouble doing that and I need help. Is there a funny saying, or something? Summer Date: 12/23/2001 at 15:13:07 From: Doctor Ian Subject: Re: Areas Hi Summer, You only need to memorize things that you don't understand, so feeling like you need to memorize something is a sign that you still don't understand it. And when you memorize things, it's easy to recall them incorrectly, by dropping a minus sign, or mixing up the numerator and denominator, or in a hundred other ways. For example, if you're trying to 'memorize' the associative, commutative, and distributive properties, you would be better off being able to reconstruct them from simple examples: Properties, Defined and Illustrated http://mathforum.org/dr.math/problems/rad.10.10.1.html Or if you're trying to 'memorize' the rules for working with exponents, you would be better off being able to reconstruct them from a few simple examples: Properties of Exponents http://mathforum.org/dr.math/problems/crystal2.01.22.01.html Understanding examples like these is kind of like keeping a spare key hidden under the ceramic frog next to your back door... you can use it to let yourself in if you ever lock yourself out of the house. If you'd like to write to me and tell me about some things that you're trying to memorize, I can try to help you understand them in a way that will make memorization unnecessary. Would you like to try that? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ Date: 12/23/2001 at 15:57:24 From: Doctor Achilles Subject: Re: Areas Hi Summer, Thanks for writing to Dr. Math. I remember that I once learned the piano scales with "Every Good Boy Does Good" or something like that, but it never really stuck for me. I don't know of any funny saying or anything that can help you remember area formulas. But I think there is a better way to remember them than funny sayings. First there are two things you have to memorize. Only two, though! 1) The area of a rectangle is width * length. 2) The area of a circle is pi * radius * radius. Then, when you see any other area problem, just use those two facts to figure out the area. A square is easy enough to figure out. A square is just a special type of rectangle in which width = length. So you just multiply side * side and you've got it. For another example, you can find the area of a triangle: h |\ e | \ i | \ g | \ h | \ t | \ ------- base The area will be 1/2 * base * height. How do I remember that? Well, I imagine this rectangle: ------- h | | e | | i | | g | | h | | t | | ------- base and I remember that the area will be base * height. Then I look at this triangle again with an imaginary rectangle drawn around it: ........ h |\ . e | \ . i | \ . g | \ . h | \ . t | \. ------- base And the diagonal line cuts the rectangle in half. That means that the area of the triangle is half the area of the rectangle. What about this triangle? h /\ e / \ i / \ g / \ h / \ t / \ ------------ base Here I imagine another rectangle around it like this: .............. h . /\ . e . / \ . i . / \ . g . / \ . h . / \ . t ./ \. ------------ base Again, the area of the rectangle is base * height. To convince yourself that the area of the triangle is half of that, draw the figure above on a piece of paper and cut out the rectangle. Then, cut the triangle out and save the two scrap pieces. If you put the two scrap pieces next to each other, they will add up to be the size of the triangle. What about other strange shapes? Here's a fun one, the parallelogram: -------------- h \ \ e \ \ i \ \ g \ \ h \ \ t \ \ -------------- base To do this one, imagine a rectangle like this: --------------....... h \ . \ . e \ . \ . i \ . \ . g \ . \ . h \ . \ . t \. \. -------------- base The area of that rectangle is again base * height. If you draw it and cut it out, you will see that the little area on the left that was left out of the imaginary rectangle is exactly the same size as the little area on the right that we added to the imaginary rectangle. So the parallelogram's area is also base * height (Notice the difference here between base * height and side * side). Here's one of the trickiest, the isosceles trapezoid: base1 -------- h / \ e / \ i / \ g / \ h / \ t / \ -------------------- base2 For this one, imagine a rectangle like this: base1 --------....... h /. \ . e / . \ . i / . \ . g / . \ . h / . \ . t / . \. -------------------- base2 Again, just like with the parallelogram, the area we left out on the left is equal in size to the area we added on the right. That means that the area of the trapezoid is the same size as the area of the rectangle we're imagining. So far so good, but what is the area of the rectangle? Here's where trapezoids get a little bit tricky: the height of the imaginary rectangle is the same as the height of the trapezoid. But the base is bigger than base1 and smaller than base2. How big is it? To help us figure out, let's look at the figure again: base1 --------!!!!!!! h /. \ . e / . \ . i / . \ . g / . \ . h / . \ . t / . \. --------------------- base2 The !'s represent the part of the base that we added to base1. So the base of our imaginary rectangle is base1 + the length of the !'s. So how long is that? Look at this figure: base1 --------!!!!!!! h /. \ . e / . \ . i / . \ . g / . \ . h / . \ . t / . \. #######-------------- base2 The #'s represent the part of the base that we cut out of base2. So the base of our imaginary rectangle is base2 - the length of the #'s. Here's the crucial point: The length that we cut off of base2 is EXACTLY EQUAL to the length we added to base1. (Note that this is true only for an isosceles trapezoid. For other trapezoids, we end up with the same formula, but we have to do a little more work to get there.) So the base of the imaginary rectangle equals some number that is halfway between base1 and base2. How do you find a number halfway between? Just take the average. So the base of our imaginary rectangle is: base2 + base1 --------------- 2 So the area of our imaginary rectangle is: base2 + base1 height * --------------- 2 And since the imaginary rectangle is the same size as the real trapezoid, that must be the area of the trapezoid. Of course on a test you won't be able to go through ALL of that from scratch. So what you should do is practice making imaginary rectangles until you can do it quickly, then you should be able to go through without much trouble. Plus, if you learn the reasons why areas equal what they do, instead of learning a funny rhyme, then if you ever forget the formulas completely, you can always figure them out again; but if you ever forgot the rhyme, you'd be stuck with nowhere to go. Learning math takes a lot of practice, but if you try to learn why things are the way they are, instead of just memorizing formulas with no reason behind them, you'll do better and you'll enjoy it more. There are a lot more types of figures in geometry out there than I just told you about. Here's my last bit of advice: whenever you come to a new figure and you want to learn the area or volume or whatever, try to imagine a simpler figure drawn around it and try to understand how the size of the new figure is different from the simple one you imagined. That way you can understand the area of new figures without having to memorize a single thing you don't know already. And if you ever come a figure you don't get, just write back and I'll try to help you understand it. If there are other postulates and theorems that you're stuck on, try to understand why they are true and that should help you learn them. Feel free to write back if you have other questions about those as well. - Doctor Achilles, The Math Forum http://mathforum.org/dr.math/ Date: 12/23/2001 at 20:15:26 From: Summer Subject: Areas I just wanted to say thank you! Your answers were both quite prompt and VERY indepth. I keep re-reading your advice and it makes so much sense to me! Thank you so much! I am going to be sure to tell all my friends who also need help in this area to be sure and contact you. I appreciate your taking the time to answer. Summer |
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