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What if you know the area of the parallelogram, can you find the height?


How do you find the height of a Parallelogram if it is not given?

The area of a parallelogram is found by multiplying the base times the height, but there is a slight twist.

The height is not the side length like you might use in a rectangle, but instead it is the altitude.

The height (altitude) is found by drawing a perpendicular line from the base to the highest point on the shape.

The Pythagorean Theorem can also be used to find the height of the parallelogram if height is not given. In order to use this method you need to know the base length of the right triangle. In the problem below it is the distance from a to b. 

Once the height is calculated you can use the area of a parallelogram formula,
base * height

Find the area of a parallelogram with a base of 12 units and a slant height of 5 units
Step 1. Find the height (altitude) using the Pythagorean Theorem.
​When using the Pythagorean Theorem the C^2 is always the hypotenuse of the right triangle.
a^2 +b^2 =c^2
3^2 + h^2 =5^2  (see the right triangle)
9+ h^2 = 25
h^2 = 16
Take the square root each side
​√h^2 = √16
height of parallelogram =4

Step 2. Now use,  area= base * height
A=12 * 4 = 48 units^2

Step 2. The side of the parallelogram becomes the hypotenuse in the right triangle
Therefore 6=2x 
Divide both sides by 2
6/2  = 2x/x 
x=3

Step 3. Now that I know x I can find the height using 30-60-90 rules.  The parallelogram height is the length of the long leg.

Long leg=3√x 
 3√3

Step 4. Use area equals base * height or A =b*h

10 * 3√3 = 30√3 = 51.9615 units^2 area of parallelogram

Area parallelogram without height given

a   b                   c
  3          9
​12
5
5
3
One can calculate the area of a parallelogram using vectors. 

The area of a parallelogram is equal to the magnitude of the cross product.  

Follow the link for a very helpful video that explains how to find the area from two vectors

Parallelogram area from two vectors

Area parallelogram given diagonals

The diagonals of a parallelogram do not define the area of a parallelogram so one can not use:  ½ d1*d2  again do not use   ½ d1 * d2

Common Core Standard  6.G.1 , 7.G.6    6th Grade Math    7th Grade Math
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Problem 1. What is the area of a parallelogram with a base of 8 units and sides of 5 units and a height of 4 units? 

​​This problem is straight forward because height is given. Just use base times height






Step 1. Multiply the base of 8 units times the height of 4 units.

Step 2. 8*4 = 32 units squared
Problem 2. What is the area of a parallelogram that has a side of 6 units, a base of 10 units, and an angle measure of 60 degrees?







This problem is a little tricky because the height is not given. Because the parallelogram has an angle of 60 degrees you can create a 30-60-90 triangle to find the height.

Step 1. Find the altitude. If you draw a vertex straight down it creates a triangle. See picture below. The triangle is a 30-60-90 triangle. I can use the 30-60-90 rules to find the height of the parallelogram.

The rules of a 30-60-90 are as follows:
 parallelogram 6x10
30 60 90 triangle
60°
​x
x√3
2X
parallelogram 6x10
60°

The formula for area of a Parallelogram equals b x h ( base x height )

parallelogram 5x8

​x

How do you find the area of a Parallelogram ?

Finding the area of a parallelogram explained
There are at least two ways to find the height (altitude) of a parallelogram.

Method one:  Use the Pythagorean Theorem and the side length of the parallelogram and the distance of the base to the height.
Method two:If you are given an angle measure that creates a special right triangle you can use the rules of the special right triangle to find the height.
parallelogram forms a right triangle
x√3
2X
In this case the angle measure creates a 30-60-90 triangle and the height is the long leg.

Step 1. Find the length of the short leg using, hypotenuse = 2x = 8  
Divide each side by 2
2x/2 = 8/2
x =4
Length of short leg (x)= 4

Step 2. Plug x into the long leg formula  
 4 * √3 = 4* 1.7320508 = 6.928 units^2 = height (altitude)
8
Video works out " How to find the height of the parallelogram given area. "
Drawing a line from the corner to the base creates a 30-60-90 triangle
Short leg =x 

Long leg = x√3

Hypotenuse = 2x



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altitude of a parallelogram
Area/Perimeter Formulas            
Perimeter Paralleogram
What is a paraleogram?