Introduction to Working With Circles and Circular Figures
Lesson Summary
In this lesson, you will learn about the irrational number pi, or π. You will also learn to use formulas to find the circumference and area of circles, the surface area and volume of cylinders and spheres, and the volume of cones.
Before you begin to work with circles and circular figures, you need to know about the irrational number, π (pronounced "pie"). Over 2,000 years ago, mathematicians approximated the value of the ratio of the distance around a circle to the distance across a circle to be approximately 3. Years later, this value was named with the Greek letter π. The exact value of π is still a mathematical mystery. π is an irrational number. A rational number is a number that can be written as a ratio, a fraction, or a terminating or repeating decimal. Although its value has been computed in various ways over the past several hundred years, no one has been able to find a decimal value of π where the decimal terminates or develops a repeating pattern. Computers have been used to calculate the value of π to over fifty billion decimal places, but there is still no termination or repeating group of digits.
The most commonly used approximations for π are and 3.14. These are not the true values of π, only rounded approximations. You may have a π key on your calculator. This key will give you an approximation for π that varies according to how many digits your calculator displays.
Circumference of a Circle
Now that you know about the irrational number π, it's time to start working with circles. The distance around a circle is called its circumference. There are many reasons why people need to find a circle's circumference. For example, the amount of lace edge around a circular skirt can be found by using the circumference formula. The amount of fencing for a circular garden is another example of when you need the circumference formula.
Since π is the ratio of circumference to diameter, the approximation of π times the diameter of the circle gives you the circumference of the circle. The diameter of a circle is the distance across a circle through its center. A radius is the distance from the center to the edge. One-half the diameter is equal to the radius or two radii are equal to the length of the diameter.
Here is a theorem that will help you solve circumference problems:
Since π is approximately (not exactly) equal to 3.14, after you substitute the value 3.14 for π in the formula, you should use ≈ instead of =. The symbol ≈ means approximately equal to.
Examples:
Find the circumference of each circle. Use the approximation of 3.14 for π.
Notice that these two circles have the same circumference because a circle with a diameter of 10 cm has a radius of 5 cm. You pick which formula to use based on what information you are given—the circle's radius or its diameter.
Area of a Circle
To understand the area of a circle, take a look at the following figure. Imagine a circle that is cut into wedges and rearranged to form a shape that resembles a parallelogram.
Notice that this formula squares the radius, not the diameter, so if you are given the diameter, you should divide it by two or multiply it by one-half to obtain the radius. Some people will mistakenly think that squaring the radius and doubling it to get the diameter are the same. They are the same only when the radius is 2. Otherwise, two times a number and the square of a number are very different.
Examples:
Find the approximate area for each circle. Use 3.14 for π.
Surface Area of a Cylinder
When you are looking for the surface area of a cylinder, you need to find the area of two circles (the bases) and the area of the curved surface that makes up the side of the cylinder. The area of the curved surface is hard to visualize when it is rolled up. Picture a paper towel roll. It has a circular top and bottom. When you unroll a sheet of the paper towel, it is shaped like a rectangle. The area of the curved surface is the area of a rectangle with the same height as the cylinder, and the base measurement is the same as the circumference of the circle base.
Surface area of a cylinder = area of two circles + area of rectangle
= 2πr^{2} + bh
= 2πr^{2} + 2πrh
Examples:
Find the surface area of each cylinder. Use 3.14 for π.
Volume of a Cylinder
Similar to finding the volume of a prism, you can find the volume of a cylinder by finding the product of the area of the base and the height of the figure. Of course, the base of a cylinder is a circle, so you need to find the area of a circle times the height.
Examples:
Find the volume of each cylinder. Use 3.14 for π.
Volume of a Cone
A cone relates to a cylinder in the same way that a pyramid relates to a prism. If you have a cone and a cylinder with the same radius and height, it would take three of the cones to fill the cylinder. In other words, the cone holds one-third the amount of the cylinder.
Examples:
Find the volume of the cone. Use 3.14 for π.
Surface Area of a Sphere
A sphere is the set of all points that are the same distance from some point called the center. A sphere is most likely to be called a ball. Try to find an old baseball and take the cover off of it. When you lay out the cover of the ball, it roughly appears to be four circles. Recall that the formula for finding the area of a circle is A = πr^{2}.
Examples:
Find the surface area of the sphere. Use 3.14 for π.
Volume of a Sphere
If you were filling balloons with helium, it would be important for you to know the volume of a sphere. To find the volume of a sphere, picture the sphere filled with numerous pyramids. The height of each pyramid represents the radius (r) of the sphere. The sum of the areas of all the bases represents the surface area of the sphere.
Examples:
Find the volume of the sphere. Use 3.14 for π.
Practice problems for these concepts can be found at: Working With Circles and Circular Figures Practice Questions.
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