Use this dynamic worksheet to answer the questions that follow and determine the equation for the area of any rhombus.

A rhombus is both a parallelogram and a kite. So, there are a couple of ways (without using trig.) that you can solve for the area of a rhombus.
Which equation you use will probably depend on what information you have.

You should have already completed the worksheets for parallelograms and kites. If not, go back and do that before proceeding.

There are two different sets of instructions below. Follow the steps for each and you should be able to derive the formula (either one) on your own.

A rhombus is a parallelogram

1. Let's start by clicking on the "Show variables" check box. Since we're going to derive the formula, let's make sure we're using the same terms. No other variables should be part of the equation you derive. When you're done, uncheck the "Show variables" check box.

2. Click on the "Show dimensions" check box to display the sizes of the base and height. Can you determine the area of the parallelogram?

4. Slide the slider all the way to the right. Now can you see how to determine the area? Remember how you determined the area of a rectangle. (If you don't remember, go back and review the worksheet for rectangles.)

5. Experiment by doing the following several times:
a) uncheck the "Show area" check box;
b) click on any of the points in blue and drag them around to change the shape of the rhombus;
c) slide the slider to see that no matter what the dimensions are, it's relation to a rectangle is always constant (a triangular portion can always be moved to create a rectangle);
d) click on the "Show dimensions" check box to see the dimensions of the rhombus you've created;
e) calculate the area on your own using the values of the variables 'b' (base) and 'h' (height);
f) check the "Show area" check box to verify;
g) click on the reset button in the upper right hand corner to reset this construction before repeating.

6. When you are satisfied, write down the equation for area in terms of base and height. Check your answer by checking the "Show equation" check box. If you have difficulty, raise your hand so your teacher can help.


A rhombus is a kite

1. Start by clicking on the "Show variables" check box. Since we're going to derive the formula, let's make sure we're using the same terms. No other variables should be part of the equation you derive. When finished, uncheck the "Show variables" check box.

2. Click on the "Show dimensions" check box to display the lengths of the diagonals. Can you determine the area of the rhombus?

3. Slide the Rotate slider all the way to the right. What is the shape of the entire figure? Can you express the formula for that new figure in terms of the variables already shown? (Hint: What is the length of the entire figure after rotation? What is its width?)

4. Slide the Rotate slider towards the middle. Can you see how much of the entire figure is made up of the kite? How do you know?

5. If you were able to determine the area of the entire figure, how much of that area is made from the rhombus?

6. Can you now determine the area of the rhombus? If not, you may find it helpful to review the rectangles worksheet. If so, continue to step 7 and test your conjecture.

7. Experiment by doing the following several times:
a) uncheck the "Show area" check box;
b) click on any of the points in blue and drag them around to change the shape of the rhombus;
c) slide the Rotate slider to see that a kite's relation to a rectangle of the same dimensions is constant;
d) click on the "Show dimensions" check box to see the dimensions of the rhombus you've created;
e) calculate the area on your own using the values of the variables "d_1" (first diagonal) and "d_2" (second diagonal);
f) check the "Show area" check box to verify;
g) click on the reset button in the upper right hand corner to reset this construction before repeating.

8. When you are satisfied, write down the equation for area in terms of base and height. Check your answer by checking the "Show equation" check box. If you have difficulty, raise your hand so your teacher can help.

Take the quiz!

Go to: Parallelograms, Kite, Quadrilateral Test, Real World Problem

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