How to Analyze an Ellipse

Four Parts:The tutorialExplanatory Charts, Diagrams, PhotosVideo AssistanceHelpful Guidance

In this article, you'll learn how to analyze an Ellipse for its geometric elements and make a chart of your findings in Microsoft Excel.


  • Become familiar with the image to create:
    Image titled Analyze Ellipse Data and Graph

Part 1
The tutorial

  1. 1
    Accept an ellipse in standard formula format, i.e. or , where .
  2. Image titled Analyze an Ellipse Step 2
    Find the following elements, which you also memorize the formulas for per the following KEY:
    • The Center =
    • The Major Axis =
    • The Minor Axis =
    • The Vertices = ,
    • The Co-Vertices = ,
    • The Foci = , for c of
  3. Image titled Analyze an Ellipse Step 3
    Given the equation , analyze the ellipse it represents into the above elements:
    • Following the standard formula format, = because both numbers are subtracted in the standard format. Thus, the Center is at at .
    • , so look to see which is larger of the two denominators and find that , and , so and .
    • Per the KEY, find that the Major Axis = and thus = .
    • Per the KEY, find that the Minor Axis = and thus = .
    • Per the KEY, find that the Vertices = , and thus = , so the Vertices = and .
    • Per the KEY, find that the Co-Vertices = , and thus = , so the Co-Vertices = and .
    • You now should be able to sketch the graph with pencil and paper, given the above successful completion of finding the elements so far.
    • Lastly per the KEY, find that the Foci = , for of . You know that and , so and . Therefore the Foci lie at and , or otherwise stated, at and . Congratulations, you've completed all the elements.

Part 2
Explanatory Charts, Diagrams, Photos

  • (dependent upon the tutorial data above)
  1. Image titled Analyze an Ellipse Step 4
    Create the Excel Chart. Open a new workbook and start a new worksheet titled Ellipse.
    • Input x into cell A1 and y into cell B1 and Format Cell font red, align center horizontal.
    • Select between the A of Column A and the 1 of Row 1 and thus the entire worksheet at upper leftmost corner. Format Cells Number Number Decimal Places 4 and Font 9 or 10.
    • Look over your figures to see the extent of the ellipse to guide your x values series. The Minor Axis runs from to so you want to create a series of x values that range from -3 to 13. Select cell A2 and input -3 and select a cell about 40 cells down, say in A43, and input 13. Do Edit Fill Series Column Linear Trend OK.
    • For the y values of the ellipse, you need to restate the ellipse's formula in terms of what y equals all by itself. Switch the y term with the 1 term and arrive at . Multiply both sides by -100 and take the square root to arrive at . Bring the 2 to the left side to isolate y and arrive finally at . You can simplify to if you like -- Excel doesn't care much either way. Input that formula to B2 as =SQRT(100-(25/16)*(A2-5)^2)-2 where A2 is the X value, and select cell range B2:B43 and Edit Fill Down.
    • Notice that almost all the y values are positive. Clearly, you've only taken half the square roots. Copy A2:A43 to A44 and paste and then sort the column from largest values to smallest. That sets up a continuum with the 13 ending at A43 and decreases around back to -3 for the x values. For the y values, copy the formula in cell B43 and insert a minus sign in front of it so that it reads now =-SQRT(100-(25/16)*(A44-5)^2)-2 when you copy and paste it into cell B44 to B85.
    • Select A2:B85 and using the Chart Wizard or the Ribbon, select Charts All/Other, Scatter, Smooth Line Scatter and leave the chart on the Ellipse worksheet. Add vertical grid lines and stretch the chart until the grid squares are approximate squares.
    • Go to cell B2 and select the formula in the formula bar and copy it and then select cell C2 and type y and paste the formula in after it. Make it bold.
    • Go to cell D4 and type in the main formula and format font size 18 bold.
    • Copy cells A1:B1 to cells D6:E6 for the x y headers in red.
    • Input the Vertices pairs in cells D7:E8 and add the title Vertices (along Major Axis) in cell F7.
    • Input the Co-Vertices pairs in cells D10:E11 and add the title Co-Vertices (along Minor Axis) in cell F10.
    • Input the Center pair in cells D13:E13 and add the title Center in cell F13.
    • Input the Foci pairs in cells D15:E16 and add the title Foci in cell F15.
    • Select in the Plot Area of the Chart and do menu item Chart Add Data and in response to the data query/ies, respond with the pairs you just input. They may not "land correctly" and so require editing in the formula bar -- just edit them until they appear as given above, eg. Series 2 should be =SERIES("Vertices",Ellipse!$D$7:$D$8,Ellipse!$E$7:$E$8,2) and notice how "Vertices" was typed in the title at the beginning of the series description -- do that for each series so that they appear correctly in the Legend.
    • Format the Vertices with a thin brown line and Size 8 red markers, or blue ones for the Co-Vertices. Format the Center as a large Light Blue Dot and the Foci as Large Black squares.
    • Add data labels with both x and y if you prefer, as the chart above demonstrates, though it gets a bit crowded. Use font size 8 or 9.
    • Move the Legend to the top of the Chart.
    • Copy the Chart and Data with Grabber and save paste to a Paster worksheet to save your work and then save the workbook.
      Image titled Analyze an Ellipse Step 4

Part 3
Video Assistance

Part 4
Helpful Guidance

  1. 1
    Make use of helper articles when proceeding through this tutorial:


Article Info

Categories: Mathematics | Microsoft Excel