Askjagden's Guide to Geometry: Areas of Circles and Regular Polygons
How do you find the area of a regular polygon? First of all, you can start by learning the definition of a regular polygon. A polygon is a two-dimensional shape that has many angles. Although the word "many" usually means two or more, in this case, it means three or more, as you cannot have a shape with two angles. ("Poly-" means "many," and "gon" means "angles.") A regular polygon is a polygon with equal sides and angles. To find the total number of degrees of the interior angles in a polygon, the formula (n - 2)(180) can be used. The total degrees of the exterior angles of a polygon is always 360 degrees.
Now, how can you find the area of a regular polygon? The formula for it is this: A = 1/2aP. "A," obviously, stands for "area." "a" stands for the "apothem," and "P" stands for "perimeter." What is the apothem? The apothem is the shortest distance from the center of a regular polygon to a side. How was this formula come up with? First of all, a regular polygon can be divided into many congruent polygons, depending on the number of sides the polygon has. For example, a pentagon has five sides, so it can be divided into 5 congruent triangles. Therefore, the area of a regular polygon is equal to the (number of sides the polygon has) x (1/2bh). 1/2bh, by the way, is the area of a triangle. The height in this case is the apothem of the regular polygon, so "h" can be substituted by "a." The number of bases times the base is equal to the perimeter, so (number of sides the polygon has) x b can be substituted by "P." Hence, the equation for the area of a regular polygon is A = 1/2aP.
Try a problem: find the area of an octagon that has a perimeter of 80 cm. Also, if a line is drawn from the center of the octagon to one vertex of the octagon, that line's length is 13 cm. Now, the apothem is the distance from the center of the regular polygon to the center of a base of the polygon. The line indicated in the problem is the distance from the center to a vertex of the polygon. Notice that when you draw this all out, you will see a right triangle when the apothem and line indicated in the problem is drawn. The height is the apothem, and the other leg's length can be determined through careful inference. The reason you need this information is because you can find the apothem's length through the Pythagorean theorem. The problem states that the polygon's perimeter is 80 cm. Therefore, each side length is 10 cm, because there are 8 sides in an octagon, and 80/8 is 10. Now, in the right triangle, the other leg's length is half the side's length, which is 10 cm. Half of that is 5 cm. Now you know that the hypotenuse is 13 cm, and that the base of the triangle is 5 cm. This makes a Pythagorean triple: a 5-12-13 triangle. Hence, the apothem's length is 12 cm. Now you know the apothem's length and the perimeter of the polygon. Now just plug and slug: A = 1/2(12)(80) = 6(80) = 480. The polygon's area is 480 cm^2.
That was the area of a regular polygon. What about the circle? You can liken the area of a regular polygon to a circle's. You can think of the radius of a circle as the apothem of a regular polygon. The perimeter of a circle is known as the circumference of a circle, and the formula for finding the circumference of a circle is C = 2(pi)(r), "r" being the radius. Now just plug it into the area of a regular polygon: 1/2aP = 1/2(r)(2(pi)(r)) = pi(r)^2. That is how you find the area of a circle.
Try a problem: find the area of a circle that has a diameter of 10 meters. The radius is half the diameter, so the radius is 5 meters. The radius, which is, in this case, 5, squared is 25 meters^2. That times pi is 25pi, so the area of a circle is 25pi m^2.
Have fun with geometry!
