Greenwood press

Mathematical Concepts
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Restructuring the angles
in an intersection to make
turning a vehicle easier.
initial street design
restructured street design

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The amount of space, 
s, saved by using right-angled spaces is 
s = −l cos α
for each row in the parking lot, where l is the length of the space and 
α is the
angle of the turn into the space. When the shape of a space is transformed from
a rectangle (right-angled) to a parallelogram (obtuse-angled), the extra horizon-
tal distance needed in a parking-lot row will be the amount of space that the last
car displaced from its previous perpendicular arrangement. In the obtuse-angled
situation, the length of the parking space is the hypotenuse of a right triangle
formed with the curb. The cosine of the angle between the curb and the parking
lines,
cos θ, is the ratio of the horizontal curb space, s, to the length of the park-
ing space, l. In an equation, this is written as 
cos θ =
s
l
.
Multiplying both sides of the equation by l will change it to 
s = l cos θ. The
angle against the curb and the car’s turning angle are supplementary, because the
curb and car’s path are parallel. The interior angles on the same side of the trans-
versal (the parking lines) are supplementary, so 
cos θ = − cos α. Substituting
this result into 
s = l cos θ generates the equation, s = −l cos α.
If the parking lines were at a 60° angle with the curb, the turning angle would
be 120°. Suppose the dimensions of a parking space are 8 feet by 20 feet. If the
lot is transformed from right-angled spaces to oblique-angled spaces, each row
would lose 
s = −20 cos 120
◦
= 10 feet, which is equivalent to a little more than
one space!