Now that you have deeper understanding of the topic, you are ready to solve the
problems below.
Let the students bring several round containers or lids and record the diameter and
circumference in a table.
If diameter is the input and circumference is the output, what's the function rule?
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Answer:
dahilan ng paglaki ng populasyon ng isang bansa o
rehiyon?
2. Naapektuhan ba ng paglaki ng populasyon ang likas na yaman ng isang lugar o
bansa?
3. llarawan mo ang sitwasyon ng populasyon ng Pilipinas.
Objectives
o Identify some basic parts of a circle, such as the radius and diameter
o Calculate the circumference of a circle
o Calculate the area of a circle
In this article, we will consider a geometric figure that does not involve line segments, but is instead curved: the circle. We will apply what we know about algebra to the study of circles and thereby determine some of the properties of these figures.
Introduction to Circles
Imagine a point P having a specific location; next, imagine all the possible points that are some fixed distance r from point P. A few of these points are illustrated below. If we were to draw all of the (infinite number of) points that are a distance r from P, we would end up with a circle, which is shown below as a solid line.
Thus, a circle is simply the set of all points equidistant (that is, all the same distance) from a center point (P in the example above). The distance r from the center of the circle to the circle itself is called the radius; twice the radius (2r) is called the diameter. The radius and diameter are illustrated below.
Obviously, as we increase the diameter (or radius) of a circle, the circle gets bigger, and hence, the circumference of the circle also gets bigger. We are led to think that there is therefore some relationship between the circumference and the diameter. As it turns out, if we measure the circumference and the diameter of any circle, we always find that the circumference is slightly more than three times the diameter. The two example circles below illustrate this point, where D is the diameter and C the circumference of each circle.
Again, in each case, the circumference is slightly more than three times the diameter of the circle. If we divide the circumference of any circle by its diameter, we end up with a constant number. This constant, which we label with the Greek symbol π (pi), is approximately 3.141593. The exact value of π is unknown, and it is suspected that pi is an irrational number (a non-repeating decimal, which therefore cannot be expressed as a fraction with an integer numerator and integer denominator). Let's write out the relationship mentioned above: the quotient of the circumference (C) divided by the diameter (D) is the constant number π.
We can derive an expression for the circumference in terms of the diameter by multiplying both sides of the expression above by D, thereby isolating C.
Because the diameter is twice the radius (in other words, D = 2r), we can substitute 2r for D in the above expression.
Thus, we can calculate the circumference of a circle if we know the circle's radius (or, consequently, its diameter). For most calculations that require a decimal answer, estimating π as 3.14 is often sufficient. For instance, if a circle has a radius of 3 meters, then its circumference C is the following.
The answer above is exact (even though it is written in terms of the symbol π). If we need an approximate numerical answer, we
The symbol ≈ simply means "approximately equal to."
Practice Problem: A circle has a radius of 15 inches. What is its circumference?
Solution: Let's start by drawing a diagram of the situation. This approach can be very helpful, especially in situations involving circles, where the radius and diameter can easily be confused.
Because we are given a radius, we must either calculate the circumference (C) using the expression in terms of the radius, or we must convert the radius to a diameter (twice the radius) and use the expression in terms of the diameter. For simplicity, we'll use the former approach.
This result is exact. If we need an approximate decimal result, we can use π ≈ 3.14.
The Area of a Circle
Just as calculating the circumference of a circle more complicated than that of a triangle or rectangle, so is calculating the area. Let's try to get an estimate of the area of a circle by drawing a circle inside a square as shown below. The area of the circle is shaded.
Let's draw a vertical diameter and a horizontal diameter in the circle; we'll label these diameters as having length D. Note by comparison with the square, the square must therefore have sides of length D as well.
We know that a square (which is a rectangle whose length and width are equal) with sides of length D has the following area subscript in the case of the area of the circle):
Because the circle of diameter D obviously has a smaller area than the square with sides of length D, we know that the circle's area must be less than D2. By inspection, we can guess that the area Acircle of the circle is approximately three-fourths that of the square. Thus,
Notice that the number π once again appears. Let's now compare this exact result with our guess from above. We'll simply do some rearranging of the expression, keeping in mind that the radius (r) is equal to half the diameter (D)-in other words, D = 2r.