How to Calculate Angles for a Tapered Box

Updated July 20, 2017

Woodworkers, craftsmen, carpenters and artisans all bring to mind workers skilled in working with their hands -- artists in fact. These artists are skilled in working with materials such as metal, clay, wood or other natural materials. They use raw materials to create beautiful, functional, art, furniture, tools or other items that day-to-day existence necessitates. Woodworking dates back to the earliest days of human history. The ancient Egyptians first developed the art of veneering and used varnishes. The craft of woodworking emerged in China between 722 B.C. and 481 B.C. Today, woodworking is a skilled profession as well as a somewhat expensive hobby. Creating beautiful functional objects requires both knowledge as well as skill.

Make an accurate two-dimensional drawing of the desired tapered three-dimensional box. Include all dimensions.

Draw a right triangle on the two-dimensional drawing of the box using the tapered edge as the hypotenuse.

Apply trigonometry to determine the needed angle. For example if the tapered angle you wish to calculate is the hypotenuse of the previously drawn triangle and you know the lengths of the opposite and adjacent sides you can solve for the tangent of the tapered angle using the trigonometric relationship of tangent = opposite / adjacent.

Make an accurate two-dimensional drawing of the desired tapered three-dimensional box. Include all dimensions.

Calculate the slope of the tapered edge or edges. Slope is defined as rise over run and is measured from the horizontal plane. A slope of 0 degrees corresponds to a horizontal line and a slope of 90 degrees corresponds to a vertical line.

Calculate the corner angle. This is the angle that is measured between the two pieces -- it is 90 degrees for a four-sided box and 48.9 degrees Cor a six-sided box.

Calculate the end angle using the following equation:

end angle = ATan ( cos ( slope) x Tan ( corner angle / 2)).

Where ATan = arctangent, Tan = tangent, cos = cosine. End angles refer to relationship to a square end. The term refers to the angle at the end of the piece laying in the horizontal plane. This is the angle you would set your crosscut fence to.

Calculate the bevel angle using the formula:

bevel angle = ASin ( sin (slope ) x Sin (corner angle/2)). Where ASin = arcsine, Sin = sine, corner angle = angle where two pieces meet. The bevel angle is measured from the vertical. Forty-five degrees means a piece is cut with a 1:1 bevel. This is the angle you set your blade to.


There are tables to determine saw settings. Refer to the site given in references for a sample.

Things You'll Need

  • Scientific calculator
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About the Author

Yvonne Bernard began her writing career in 1994. She has written articles for “Design News” and published an engineering text book, “Environmental Engineering,” for McGraw-Hill. Bernard is also a member of several engineering societies. She has a Bachelor of Science in mechanical engineering from the University of Illinois.