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    2-Dimensional Geometry Somerset Math Club 2016-2017    A   B   C   D   E   G   F   H   Area Formulas   1.   Circles: A=    2  a.   Circular Sectors: A=    2 *  / 360 (in degrees) 2.   Triangles:  =  ℎ / 2  3.   Rectangles/Parallelograms/Squares: A=b*h a.   Trapezoids: A=(   1    2 )∗  ℎ / 2  Challenge: Can you think of a way to explain why this is the area formula for a trapezoid?   Hint: Try rearranging parts of the trapezoid.   Challenge Question   ABCD is a trapezoid with AB parallel to DC. E and G and midpoints of AD and BC, respectively. If the area of trapezoid ABCD is 100, what is the area of the quadrilateral, EFGH?   Cir cles   The    2-Dimensional Geometry Somerset Math Club 2016-2017   re are 360 degrees in a full circle. However, radians can also be used instead of degrees. There are 2   radians in 360 degrees because there are 2   radiuses in a circumference. A radian is the central angle that is formed by a sector of the circumference of length r. Convert the following angles from degrees to radians in terms of  : 1.   60° 2.   90° 3.   180° 4.   360° Convert the following angles from radians to degrees:   1.    / 3  2.    / 2  3.     4.   2  5.   360 Why do we use radians?   First, a degree is an arbitrary definition: we define 360 degrees to be the number degrees in a full circle, but any number could have been defined as the number of degrees in a circle. The radian is a measurement defined in terms of a property of a circle, making it less arbitrary. It is also a dimensionless unit, so it can be squared or square rooted without issues with units.   Second, using radians give a much more easily used formula for the area of a sector of a circle or the length of an arc on a circle. The formula for the length of an arc with a central angle    2-Dimensional Geometry Somerset Math Club 2016-2017   b   c    in degrees, which can be proven using conversion from radians to degrees, is r    while the area of a circular sector with that central angle is r   2 ∗  / 2 .   Overlapping/Tangent Circles   If the radius of both circle A and circle C is 6 cm, what is the area that the two circles have in common (the green area)? If the radius of circle A is 6 cm and the radius of circle C is 3 cm, what is the area that the two circles have in common? If the radius of the smallest circle is 1 cm, the radius of the 2nd smallest circle is 2 cm, and the radius of the largest circle is 3 cm, what is the area of the triangle formed by connecting the centers of the circles?   Right Triangles   The area of a right triangle is ab/2. This is based on the area of a rectangle. By the Pythagorean Theorem, a  2 + b 2 = c  2   In an acute triangle, a  2 + b 2 > c  2  while in an obtuse triangle, a  2 + b 2 < c  2   The following are the trigonometric functions in a right    2-Dimensional Geometry Somerset Math Club 2016-2017   a   b   c   triangle.   sin    =    ℎ  sin is the abbreviation for sine.   cos  =    ℎ  cos is the abbreviation for cosine.   tan  =         tan is the abbreviation for tangent.   Using trigonometry, we can also find a formula for the area of non-right triangles. Given that the angle between sides a and b is  , the area of the triangle is a*b*sin  /2. The formula for the area of any triangle, given its three side lengths, is √   (  −  )(  −  )(  −  ) where s=  +  + 2   Triangle Problems   If a triangle with an area of 84 has side lengths of 13, 14, and 15, what are the three angles of the triangle? (Use a calculator)   What is the area of a triangle with side lengths 5, 7, and 9?   If two angles of a triangle are equal, what type of triangle is it? If all three angles of a triangle are equal, what type of triangle is it?   What is tangent in terms of sine and cosine? What is sin 30? What is cos 30? If the angles of a triangle are 30 degrees, 60 degrees, and 90 degrees, what is the ratio of the lengths of the sides of the triangle? If the angles of a triangle are 45 degrees, 45 degrees, and 90 degrees, what is the ratio of the lengths of the sides of the triangle?  
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