TT-2+Area+of+Oblique+Triangles

=TT-2 The Area Of An Oblique Triangle= (Chapter 6 in the text book - pg 408)

toc Welcome to TT-2! This section will teach you how to find the area of any triangle, how to find the included angle when given the area of a triangle, how to find the area of a parallelogram, how to find the areas of any regular polygon, and how to find the area of segments, and the area of a quadrilateral that does have a right angle.

Vocabulary
What is an **__oblique triangle__**?

An **__oblique triangle__** is a triangle that does not have a right angle (either acute or obtuse).

But, first let's learn standard notation for an oblique triangle.

In standard notation, the angles of a triangle are labeled A, B, and C, and their opposite sides are labeled a, b, and c, as shown below.



Area of Any Triangle
To solve for the area any triangle use the formula below. (where K = area)

K = 1/2 (side) (other side) (sine of the included angle)

Example:
Find the area of a triangle if two sides measure 16 inches and 7 inches, and the included angle is 62 °.

K = 1/2 (side) (other side) (sine of the included angle) K = 1/2 (16) (7) (sin(62) ) K = 49.5 inches^2

Included Angle
To solve for the included angle when given the area of the triangle you use the same formula as before.

K = 1/2 (side) (other side) (sine of the included angle)


 * Be sure to remember that sine gives you the reference angle and you must take the supplement of the reference angle because there are always two possibilities when solving for the included angle (either acute or obtuse).**

Example:
Find the measure of an included angle of a triangle of the area of the triangle is 72 square units and two sides measure 12 and 17 units.

K = 1/2 (side) (other side) (sine of the included angle) 72 = 1/2 (12) (17) (sin x) 72 = 102 (sin x) 72 / 102 = sin x sin^-1 = (72/102) x = 44.9°


 * find the supplement**

180° - 44.9° = 135.1°

x = 44.9° or 135.1 °

Area of A Parallelogram
When solving for the area of a parallelogram, split the parallelogram into two triangles. Solve for the area of one triangle and then multiply that area by two to get the of the parallelogram. Once again you still use the same formula.

K = 1/2 (side) (other side) (sine of the included angle)

Example:
A parallelogram has sides that are 8 and 12 units long with an included angle of 146 °. Find the area of the parallelogram.

*First draw the picture*



Now, remember to split the parallelogram into two triangles.

Now use the formula to find the area of one of the triangles.

K = 1/2 (side) (other side) (sine of the included angle) K = 1/2 (8) (12) (sin 146) K = 26.8 units ^2

26.8 x 2 = 53.6 units ^2

Area of A Regular Polygon
When finding the area of a polygon you divide the polygon into triangles and find the area of one triangle and multiply that area by the number of triangles. To solve this you are going to use the same equation.

K = 1/2 (side) (other side) (sine of the included angle)

Example:
Find the area of an octagon inscribed in a circle of radius 14 meters.

We can divide the octagon up into eight equal triangles.



Now we can work with one triangle.

K = 1/2 (side) (other side) (sine of the included angle) K = 1/2 (14) (14) (sin45) K = 69.3

Now we multiply the area of one triangle by eight to get the area of all eight triangles (the area of the entire octagon).

69.3 x 8 = 554.4 meters ^2

Area of Segments
To find the area of a segment of a circle, first you find the area of the circle using the formula...

A = pi r(^2)

Once you know the area of the circle, you multiply the area of the circle by how many degrees are in the designated section over how many degrees are in the circle.

Use the equation "K = 1/2 (side) (other side) (sine of the included angle)" to find the area of the triangle.

Once you have the area of that section of the circle you subtract the area of the triangle to get the area of the segment.

Example:
Find the area of a segment of a circle that has a radius of 12 cm and a central angle of 75 °.



A = pi r(^2) A = pi 12(^2) A = 452.4

425.4 x (75/360) = 94.3

K = 1/2 (side) (other side) (sine of the included angle) K = 1/2 (12) (12) (sin75) K = 69.6

94.3 - 69.6 = 24.7 cm ^2

Area of Quadrilateral Without Right Angle
To find the area of a quadrilateral without a right angle, you must split the quadrilateral into two triangles, find the area of each, and then add them together.

You will have to use SOH CAH TOA to find the angle measures & the pythagorean theorem to find the hypotenuse.

Example:
Find the area of the quadrilateral. Split the quadrilateral up into two triangles. K = 1/2 (side) (other side) (sine of the included angle) K = 1/2 (4) (6) (sin90) K = 12 u ^2

Now use SOH CAH TOA to find the angle measures (how to break up 112 °).

tan(x) = 6/4 tan^-1(6/4) = 56.3 °

112 ° - 56.3 ° = 55.7 °

Now use the Pythagorean Theorem to find the hypotenuse.

4^2 + 6^2 = = 7.2 Let's find the area of the other triangle.

K = 1/2 (11) (7.2) (sin55.7) K = 32.7 u ^2

Now let's add the two areas together.

12 + 32.7 = 44.7 u ^2

Helpful Links:
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@http://mathcentral.uregina.ca/QQ/database/QQ.09.05/carlin1.html

@http://mathworld.wolfram.com/CircularSegment.html

@http://www.tpub.com/math1/18c.htm

@http://www.wikihow.com/Find-the-Area-of-a-Quadrilateral