10.6 Areas & 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
24º
13
27
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.