find the perimeter of a rhombus of each side measuring a unit.
Answers
Hey mate here your answer
1
Set up the formula for perimeter of a rhombus. Since, by definition, all four sides of a rhombus are the same length, the formula is {\displaystyle P=4S}P=4S, where {\displaystyle P}P equals the perimeter, and {\displaystyle S}S equals the length of one side.[2]
You could also use the formula {\displaystyle P=S+S+S+S}P=S+S+S+S to find the perimeter, since the perimeter of any polygon is the sum of all its sides.[3]
If you know that not all sides are the same length, then you are not working with a rhombus, and you cannot use this formula.
If you don’t know the length of any side of the rhombus, you cannot use this method.
A square is a special type of rhombus, with four 90-degree angles.
2
Plug in the side length of the rhombus. Make sure you are substituting for the variable {\displaystyle S}S.
For example, if you know one side of the rhombus is 4 meters long, your formula will look like this: {\displaystyle P=4(4)}P=4(4).
3
Solve for P{\displaystyle P}P. To do this, multiply {\displaystyle S}S by 4.
For example:
{\displaystyle P=4(4)}P=4(4)
{\displaystyle P=16}P=16
So, the perimeter of the rhombus is {\displaystyle 16m}16m.
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Method 2 of 3:
Using Length of Diagonals
1
Notice that the two diagonals of your rhombus create four congruent triangles. 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.
2
Identify the 90 degree angle of your triangle. The two diagonals of a rhombus are perpendicular, so the central angle of your triangle will be 90 degrees. [4]
3
Label the hypotenuse of your triangle. The hypotenuse is the side opposite a 90 degree angle.[5]Traditionally, the hypotenuse is labeled {\displaystyle c}c.
The hypotenuse of your triangle is one side of the rhombus. So, if you find the length of {\displaystyle c}c, you will know the length of one side of the rhombus.
4
Label the other two sides of your triangle. Traditionally, these are labeled {\displaystyle a}a and {\displaystyle b}b.
5
Find the length of side a{\displaystyle a}a. To do this, divide the length of the diagonal that {\displaystyle a}a 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.[6] Since side {\displaystyle a}a is half the length of the diagonal, you can find its length by dividing the diagonal length in half.
For example, if side {\displaystyle a}a runs along a diagonal that is 12 meters long, you can find the length of side {\displaystyle a}a by calculating:
{\displaystyle a={\frac {12}{2}}}a={\frac {12}{2}}
{\displaystyle a=6}a=6
6
Find the length of side b{\displaystyle b}b. To do this, divide the length of the diagonal that {\displaystyle b}b runs along by 2. Label the side length on your triangle.
For example, if side {\displaystyle b}b runs along a diagonal that is 16 meters long, you can find the length of side {\displaystyle b}b by calculating:
{\displaystyle b={\frac {16}{2}}}b={\frac {16}{2}}
{\displaystyle b=8}b = 8
7
Set up the Pythagorean Theorem. The theorem states that {\displaystyle a^{2}+b^{2}=c^{2}}a^{{2}}+b^{{2}}=c^{{2}}. This is a basic geometric formula for finding the side lengths of a right triangle.
8
Plug in the known side lengths of your triangle into the Pythagorean Theorem. Make sure you substitute for {\displaystyle a}a and {\displaystyle b}b, but the order doesn’t matter due to the commutative property.
For example, if {\displaystyle a=6}a=6 and {\displaystyle b=8}b = 8, your equation will look like this: {\displaystyle 6^{2}+8^{2}=c^{2}}6^{{2}}+8^{{2}}=c^{{2}}.
9
Solve for c{\displaystyle c}c. To do this, you need to square {\displaystyle a}a and {\displaystyle b}b, add, then find the square root of the sum.
For example:
{\displaystyle 6^{2}+8^{2}=c^{2}}6^{{2}}+8^{{2}}=c^{{2}}
{\displaystyle 36+64=c^{2}}36+64=c^{{2}}
{\displaystyle 100=c^{2}}100=c^{{2}}
{\displaystyle {\sqrt {100}}={\sqrt {c^{2}}}}{\sqrt {100}}={\sqrt {c^{{2}}}}
{\displaystyle 10=c}10=c
10
Multiply c{\displaystyle c}c by four. Since the hypotenuse is also the side of the rhombus, to find the perimeter of the rhombus, you need to plug the value of {\displaystyle c}c into the formula for the perimeter of a rhombus, which is {\displaystyle P=4S}P=4S, where {\displaystyle s}s equals the length of one side of the rhombus. In this case, it is the same value that we found for {\displaystyle c}c.
For example: {\displaystyle P=4S}P=4S
{\displaystyle P=4(10)}P=4(10)
{\displaystyle P=40}P=40
11
Write your final answer. Don’t forget to include the correct unit of measurement.
For example, a rhombus that has diagonals measuring 12 and 16 meters long has a perimeter of 40 meters.