Circle Formulas: Area and PerimeterDate: 03/19/2003 at 16:52:32 From: Leo Subject: Math formulas I get confused with the different formulas, such as Pi r squared or Pi d (Pi times radius squared or Pi times diameter). I don't remember which formula goes to which problem. Date: 03/21/2003 at 13:45:56 From: Doctor Dotty Subject: Re: Math formulas Hi Leo, Thanks for the question! The best way is to actually understand what each equation is actually doing. CIRCUMFERENCE: It was discovered that the ratio between the diameter of a circle and the circumference (distance around the outside) was constant. What this means is that you can multiply the diameter of a circle by a special number that is just over 3 to find the distance around the outside. The amazing thing is that this number is the same, no matter how big the circle is! This number was given the name 'Pi' and is about 3.14159. So circumference = Pi * diameter The diameter is twice the radius (as the radius measures from the centre to the outside of a circle, and the diameter measures straight across the circle through the centre) so this equation can be written as: circumference = Pi * 2 * radius Now, can you see where that all comes from? It is just multiplying a length (the radius) by a number to get a bigger length (the circumference) AREA: Here is a circle, with a sector (the shape of a slice of pizza) drawn in. * * * * . * * . . * * . . * * . . * * . . r * * .. * * . * * * * * * * * * * * * * * * * Let's call the angle between the two radii x, and the curved length at the end b. Imagine that the angle x gets extremely small. As it gets smaller and smaller, b gets closer and closer to a straight line. By the time the angle is nearly 0, b is very close to a straight line. So that makes the sector into a triangle. Now imagine splitting the circle into lots of sectors like the first, and putting the resulting triangles into a row. Here's a simplified drawing: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ |\ |\ |\ |\ |\ | | \ | \ | \ | \ | \ | | \ | \ | \ | \ | \ | | \ | \ | \ | \ | \ | | \ | \ | \ | \ | \ | |_ _ _\|_ _ _\|_ _ _\|_ _ _\| _ _ \| This gives a rectangle with the same area as the circle. Now, looking at the original sector in the circle we can see that the sides of this rectangle are equal to the radius of the circle. The side along the top is made from all the little bits of the circumference. When all the sectors are in here, then the length of the top side of the rectangle is half of the circumference. Let's call the circumference c, and call the area 'a'. We know that the area of a rectangle (and therefore the area of the circle) is: area = height * width So: a = r * c/2 But we know that c = Pi * d [from the circumference bit above] So: a = r * (Pi * 2 * r)/2 a = r * 2(Pi * r)/2 a = r * (Pi * r) a = r * Pi * r a = Pi * r^2 [where ^2 means squared] Can you see where that comes from now? Another way to remember that this equation regards the area is that the radius is squared. The radius could be measured in metres, so the result will be measured in metres squared. The only thing you measure in metres squared is area. Does that make sense? If I can help any more with this problem or any other, please write back! - Doctor Dotty, The Math Forum http://mathforum.org/dr.math/ |
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