10.2 Developing Formulas for Circles and Regular Polygons

10.2 Developing Formulas for Circles and Regular Polygons If you do not use the pi button, we will use 3.14 for pi. http://www.google.com/sear ch? hl=en&site=imghp&tbm=isch&source=hp&bi From the center of a regular polygon to a vertex is called a radius. Sometimes we need to divide a figure into triangles to find the area. When we do this, at times we may need to use trigonometry to find the needed lengths. We know that we can divide the octagon into 8 congruent isosceles triangles. The vertex angle is 45 degrees. (360 divided by 8 = 45). But to use trig, we need a right angle. The apothem forms a right angle with the side. So we will take 8 * 2 = 16. Now 360 divided by 16 = 22.5 °

Now use trigonometry to find the height and base of each angle. Let a = 5 inches. Use trig to find the base of the triangle. Tan 22.5 = x/5 which is 2.07. This is half the side, so 2.07 * 2 = 4.14. Perimeter = 4.14 * 8 = 33.12 A = ½ * 5 * 33.12 = 82.8 in2

A regular hexagon can be divided into 6 congruent equilateral triangles. 6 * 2 = 12. 360 divided by 12 = 30. So now we have a 30-60-90 triangle and you can use the special right triangles to solve. Find the area of a parallelogram: We will use trig to find the height. Remember, the height is perpendicular to the base. Opposite angles of a parallelogram are congruent also

Sin 24 = h h = 5.29 13 Now A = bh A = 27 * 5.29 = 142.8 un2 The vertex angle: There are 360 degrees in a circle. Now divide 360 by 5 = 72. Now we need to find the apothem

We will use tangent to find this since we know the angle and the adjacent side. Tan 54 = a 3

Now find the perimeter 6 * 5 = 30. Now the formula for volume of a regular polygon A = 1/2ap so A = ½ (4.1)(30) = 922.3 cm3

Without using trigonometry, we must know two of these three lengths: the apothem, the distance from the center to a vertex, or the length of a side. Using trigonometry, we need to know only one of these lengths, because we can always know the angles of the congruent triangles that form the regular polygon.