Bolt hole patterns and right triangle trigonometry
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Bolt hole Patterns and Right Triangle Trigonometry. Development of the formulas: 2r sin( ½ θ ) and 2r sin θ. Lightly inscribe the circular radius. The center of each bolt hole will pass through this circle with given radius r. The radius is often referred to as the distance from center.

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Bolt hole Patterns and Right Triangle Trigonometry

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Bolt hole Patterns and Right Triangle Trigonometry

Development of the formulas:

2r sin( ½ θ) and 2r sinθ


Lightly inscribe the circular radius.

The center of each bolt hole will pass through this circle with given radius r.

The radius is often referred to as the distance from center

Bolt Hole Patterns

r


r

Finding the angle between two consecutive holes

  • Count the number of holes

  • Take 360° and divide by the number of holes.

  • This equals the angle between any two consecutive holes

  • In our example….

  • 360° ÷ 8 = 45°

  • Therefore <θ = 45°

θ


r

Checking the distance between two consecutive holes

Bisect angle θ

This creates a right triangle

With hypotenuse equal to r and one angle equal to θ divided by 2

θ

½ θ

r

x

Sin ½ θ = x divided by r

x = r sin ½ θ

Distance between hole centers = 2x

2x = 2r sin ½ θ


r

Checking the distance between two nonconsecutive holes

Draw altitude CX on ΔABC

This creates right triangle CXB

Let θ = <XCB

C

A

θ

r

Sin θ = XB divided by r

XB = r sin θ

AB =2(r sin θ) or 2r sinθ

X

B


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