10.6Areas & Trigonometry

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.

A barn is in the shape of a regular octagon with a 56 diameter. Find the area of the barn floor.

We know that we can divide the octagon into 8 congruent isosceles triangles. The vertex angle is 45 degrees. (360 divided by 8 = 45).

The base angle is (180 * (8-2)) = 1080. Now 1080 divide by 8 = 135 degrees. Finally, divide 135 by 2 = 67.5 degrees for the base angles. Now use trigonometry to find the

We need to divide the diameter in half to find the radius of each side.

56 divided by 2 = 28.

Now use trigonometry to find the height and base of each angle.

Sin 67.5 = a a = 26.

28

Next use either the Pythagorean Theorem or cosine to find ½ the base:

262 + b2 = 282 b = 10.7. Now multiply by 2 and get the base of 21.4

Find the perimeter 8 * 21.4 = 171 ft.

Finally A = ½ ap so A = 1.2 (26)(171) A = 2223 ft2

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 pentagon base can be divided into 5 congruent triangles with vertex angle of 72 degrees and base angles of 54 degrees.

We get the base angle by taking the number of degrees in a pentagon [180 *(number of sides – 2)], so 180 * 3 = 540. Since this is a regular polygon, divide 540 by 5 = 108. Now we divide the angle by 2 = 54.

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.