How to Lay Out a Cone Shape

Written by paul dohrman
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A cone is a popular shape for artificial Christmas trees, party hats and other decorations, so being able to draw a sewing pattern or template for a cone of specified height and width can be helpful for festive occasions. The pattern can also be used for geometry instruction in early grades. The three-dimensional cone shape is constructed from a two-dimensional circle, with essentially a pie wedge cut out. The size of the circle and the discarded wedge determines the cone's height and width.

Skill level:

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Things you need

  • Pencil
  • Compass
  • Paper
  • Scissors
  • Ruler
  • Tack and string (optional)
  • Protractor
  • Calculator

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  1. 1

    Decide on a height (H) and width (W) for the cone you want.

  2. 2

    Use H and W to calculate the radius, R, of the circle from which the cone is to be made. The Pythagorean theorem gives the formula for R as √(0.25W^2+H^2), where √ indicates the square root of the quantity in brackets. The caret ^ indicates exponentiating. So W^2 is W squared.

  3. 3

    Mark the centre of the circle on the material that you're making the cone out of.

  4. 4

    Draw a circle on the cone material using either a compass or a string. Use the compass by placing the sharp end at the centre of the circle and placing the pencil end on the material a distance R from the centre. Then trace out a circle by rotating the compass, moving the pencil around while keeping the point stationary. Alternatively, tack the string to the centre, tie the pencil a distance R away, and rotate the pencil around the tack.

  5. 5

    Cut out the traced circle.

  6. 6

    Draw a straight line from the centre to the edge with the ruler.

  7. 7

    Calculate the size of the pie wedge to cut out of the circle with the following formula: D = 360*√[1-(H/R)^2], where D is the number of degrees of the portion of the circle you want to make into your cone.

  8. 8

    Place the eye of the protractor at the centre of the circle. Place its baseline over the line you just drew on the circle. Find D degrees on the protractor. Put a mark next to the angle. (Protractors go up to only 180 degrees. If D is greater, just subtract D from 360 and find that angle on the protractor instead.)

  9. 9

    Draw a line from the circle centre through the mark you just made, and extend the line with a straightedge or ruler to the edge of the circle.

  10. 10

    Draw tabs that extend into the region to be cast off, so the two pieces of the cone can be joined. The opposite edge will overlap and attach to the tabs, to be drawn in a shape you judge fit for the material you're working with. This will keep the cone from unravelling.

  11. 11

    Cut along the two straight edges and around the tabs you just drew.

  12. 12

    Join the two edges of the portion of the circle you intend to keep as the cone, attaching the tabs underneath the opposite edge. The two right-angled corners created by cutting out the wedge will just touch each other when the two edges are joined. And this is your cone shape.

Tips and warnings

  • Because you were careful to make the radius the same distance from the circle centre all the way around, your cone will sit flat on its circular edge. This is because any line drawn down from the cone top to the edge will be perpendicular to the edge, the same property a circle has between its centre and its edge.

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