Do you know about - Geometry for Beginners - How to Find the Area of a Circle
Geometry Help! Again, for I know. Ready to share new things that are useful. You and your friends.Welcome to Geometry for Beginners. This article returns to the idea of finding area, but this time the outline will be a circle rather than a polygon. Terms we have used previously for finding area - like base and height - do not apply to circles, so new terminology becomes necessary. In addition, we need to understand some concepts we have never encountered before to understand the derivation of the formula.
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Note: Some mathematicians do, in fact, think a circle to be a polygon - a polygon with an infinite amount of sides. The idea of "infinite amount of sides" comes from Calculus, but a few mental images can help Geometry students understand the basic idea. Get a piece of paper if your quality to visualize images in your mind is as weak as mine. Now, draw (either on paper or on your mental whiteboard) a triangle. With the triangle and all the other figures, try to make all the sides equal in length. Now, move to the right of the triangle and draw a quadrilateral of similar size. Move right again, and draw a pentagon. Then draw a hexagon and an octagon. This is commonly sufficient figures to see the pattern that as the amount of sides increases, the polygon becomes more and more circular.
In Calculus, we think what the "end result" would be if we could continue to increase the amount of sides of a polygon forever. We call this end succeed the "limit." For our situation, a polygon with an infinite amount of sides would have a circle as its limit.
In addition to comprehension this limit concept, we also need to quote the meaning of pi before we can understand the method for area of a circle. Remember that the irrational amount pi is the ratio of the circumference of a circle (distance around) to its diameter (distance across straight through the center). Also, remember that circumference is equivalent to the perimeter of polygons and has two inherent formulas: C = (pi)d or C = 2(pi)r. Now we are ready to find the area of circles.
We already know that area is measured with squares; and, for rectangles, those squares are easy to see and count. Unfortunately, squares don't fit into circles nicely. To understand the area method for circles, we need good mental image skills and a good comprehension of the "limit" idea mentioned earlier in this article.
On your "paper" draw a circle with a diameter of 1 to 2 inches. Now, divide this circle into 4 equal parts by drawing another diameter perpendicular to the original diameter. You should now be able to see 4 shapes like pieces of pizza. Now, take those 4 pieces and fit them side by side but alternating point up and then point down. We now have a parallelogram-type outline having two bumps or curves on both the top and lowest and a rather steep lean to the side.
Now we are going to do the same type of limit process we discussed earlier. Look back at your circle with 4 parts. Draw two more diameters to divide each part in half. You should now see eight pie-shaped pieces that are the same "height" as before, but are more narrow. Take these eight pieces and fit them side by side, again alternating point up and point down. Again, we have that parallelogram-type shape, but now the lean to the side is decreased. Said in a separate way, the sides are becoming more vertical. In addition, the top and lowest now have four bumps or curves each, but the curves are flatter.
As we continue to divide the circle into more and more pie pieces and continue fitting the pieces together side by side as before, the resulting outline becomes a rectangle because the sides come to be vertical and the curves on the top and lowest flatten completely. The height of this resulting rectangle is actually the radius of the circle, r. The top and lowest of the rectangle come from the circumference. This means the base is one-half of the circumference, C.
The area of the circle is the same as the area of the rectangle. The rectangle area method can, thus, convert from A = bh to A = (1/2C)(r). Remembering the method for circumference, we can convert the area method even further. A = (1/2C)(r) becomes A = 1/2(2(pi)r)(r). By simplifying the multiplication, the succeed is A = (pi)r^2.
This circle area formula, A = (pi)r^2, can be used to find the area if we know whether the radius or diameter of the circle; or we can find what the radius or diameter must be for a given area.
For example: If the radius of a circle is 5 cm., find the area of the circle.
Solution: A = (pi)r^2 becomes A = (pi)5^2 or A = 25pi. The final form of the write back will depend on the teacher, the situation, or the subject. Sometimes, we want the write back in terms of pi because this is the Exact answer, but we mentally estimation for meaning using 3 as the value of pi. Thus, the circle has an exact area of 25pi sq. Cm. Which is about 75 sq. Cm. Other situations need a more literal, decimal value for the area, so we use the pi key on the calculator.
3 Final Cautions About Circles:
1. Answers with pi are Exact, while decimals are Approximations.
2. Radius and diameter are often confused. Using the wrong value is very easy. Think!
3. The circumference and area formulas are similar and easy to confuse. Think before you start working on with a formula!
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