Write an article based on this "Label the vertices of your rhombus, if they are not already labeled. Notice that the two diagonals of your rhombus create four congruent triangles. Identify the 90 degree angle of your triangle. Determine the measurement of angle EAB{\displaystyle EAB}. Determine the measurement of the missing angle. Determine the length of one side of your triangle. Set up a sine or cosine ratio. Solve the ratio to find the length of the hypotenuse. Multiply the length of the hypotenuse by four. Write your final answer."
It doesn't matter which variables you give them. The vertices (singular vertex) are the corners of the rhombus. For example, you might label the vertices A{\displaystyle A}, B{\displaystyle B}, C{\displaystyle C}, and D{\displaystyle D}. Outline one of these triangles. You will use it to find the length of one side of the rhombus. Since the triangles are congruent, it doesn't matter which one you outline; however, for simplicity you should outline a triangle that shares a known angle of the rhombus. For example, I know that angle DAB{\displaystyle DAB} of the rhombus is 70 degrees, so I would outline a triangle that includes point A. The two diagonals of a rhombus are perpendicular, so the central angle of your triangle will be 90 degrees. If this angle is not already labeled, label it E{\displaystyle E}. Remember that the diagonals of a rhombus bisect its vertices. So, if you know the measurement of angle DAB{\displaystyle DAB} of the rhombus, divide it in half to find the measurement of angle EAB{\displaystyle EAB} of the triangle. Label the degrees for this angle on your triangle. This method will not work if you do not know the measurement of at least one vertex of your rhombus. For example, you know angle DAB{\displaystyle DAB} of the rhombus is 70 degrees, so the angle EAB{\displaystyle EAB} of the triangle is half that, or 35 degrees. Remember, the interior degrees of a triangle will add up to 180. So, if you know the measurement of two angles, you can subtract to find the measurement of the third angle. Label the degrees for this angle on your triangle. For example, you know that angle AEB{\displaystyle AEB} is 90 degrees, and angle EAB{\displaystyle EAB} is 35 degrees. To find the third angle, sum the two angles you already know, then subtract that sum from 180.90+35=125{\displaystyle 90+35=125}180125=55{\displaystyle 180-125=55}So, the measurement of angel ABE{\displaystyle ABE} is 55 degrees. To do this, divide the length of the diagonal that the side runs along by 2. Label the side length on your triangle. Since the diagonals of a rhombus bisect each other, you know that the length on either side of their intersection will be equal. This method will not work if you do not know the length of at least one diagonal of your rhombus. For example, if you know that diagonal AC{\displaystyle AC} is 16 centimeters, you can divide 16 in half to find the length of side AE{\displaystyle AE} of your triangle. 162=8{\displaystyle 16\div 2=8}, so side AE{\displaystyle AE} is 8cm{\displaystyle 8cm}. Whether you use sine or cosine will depend on which side and angle measurements of your triangle you know. For more information, read Use Right Angled Trigonometry. If you know the length of the side opposite to your angle, use sine. Set up the ratio sin()=Oppositeh{\displaystyle \sin( heta )={
rac {Opposite}{h}}}, where {\displaystyle heta } is the measurement of the angle, "Opposite" is the length of the opposite side, and h{\displaystyle h} is the length of the hypotenuse. If you know the length of the side adjacent to your angle, use cosine. Set up the ratio cos()=Adjacenth{\displaystyle \cos( heta )={
rac {Adjacent}{h}}}. Where {\displaystyle heta } is the measurement of the angle, "Adjacent" is the length of the adjacent side, and h{\displaystyle h} is the length of the hypotenuse. For example, if you know that angle EAB{\displaystyle EAB} of your triangle is 35 degrees, and the adjacent side is 8 centimeters, you should use cosine:cos(35)=8h{\displaystyle \cos(35)={
rac {8}{h}}} The length of the hypotenuse is also the length of one side of your rhombus, so you need this measurement to find the perimeter of the rhombus. For example:cos(35)=8h{\displaystyle \cos(35)={
rac {8}{h}}}.819=8h{\displaystyle .819={
rac {8}{h}}}.819h=8{\displaystyle .819h=8}.819h.819=8.819{\displaystyle {
rac {.819h}{.819}}={
rac {8}{.819}}}h=9.768{\displaystyle h=9.768}So, the length of the hypotenuse, side AB{\displaystyle AB} is about 9.768. Since the hypotenuse is also the side of the rhombus, to find the perimeter of the rhombus, you need to plug the value of h{\displaystyle h} into the formula for the perimeter of a rhombus, which is P=4S{\displaystyle P=4S}, where S{\displaystyle S} equals the length of one side of the rhombus. In this case, it is the same value that we found for h{\displaystyle h}. For example:P=4S{\displaystyle P=4S}P=4(9.768){\displaystyle P=4(9.768)}P=39.072{\displaystyle P=39.072} Your answer will be approximate since you rounded the sine or cosine measurement. Don't forget to include the correct unit of measurement. For example, a rhombus that has angle DAB{\displaystyle DAB} measuring 70 degrees, and diagonal AC{\displaystyle AC} measuring 16 centimeters long, the perimeter is about 39 centimeters.