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<spacer> <spacer> Angles and Trigonometry > Angles

As described in the Definitions page, traditional taxicab geometry was defined to use Euclidean angles. But, in its most general form, the measure of an angle is defined as arc length along the unit circle in the geometry. If we use this definition for angular measure, then we quickly see that taxicab geometry will have its own measure for angles that is very different from Euclidean geometry since circles in taxicab geometry are very different than Euclidean circles.

As in Euclidean geometry, we can begin by defining a unit of angular measure.

DEFINITION: A t-radian is an angle whose vertex is the center of a unit taxicab circle and intercepts an arc of unit (taxicab) length (i.e. a taxicab length of 1). The taxicab measure of a taxicab angle is the number of t-radians subtended by the angle on the unit taxicab circle about the vertex (Figure 1).


FIGURE 1: A t-radian is measured as length along the unit taxicab circle.

Converting from Euclidean to taxicab angle measure

If we have an Euclidean angle θe in standard position, we can derive a formula for the taxicab measure of the angle.

THEOREM 1: An acute Euclidean angle θe in standard position has a taxicab measure of


where the subscript e indicates an Euclidean quantity or function.

Proof: The terminal side of the angle is a line passing through the origin with slope taneθe. So, the taxicab measure θ of the Euclidean angle is equal to the taxicab distance from (1,0) to the intersection of the lines y = -x + 1 and y = x taneθe. The x-coordinate of this intersection is


and thus the y-coordinate of the intersection is y0 = -x0 + 1. Therefore, the taxicab distance from (1,0) to the intersection of the lines is


If the angle is not in standard position but has a reference angle φe, a more general formula can be derived.

COROLLARY 2: If an acute Euclidean angle θe with Euclidean reference angle φe is contained entirely in a quadrant, then the angle has a taxicab measure of


To prove the corollary, find the taxicab measure of the angles (θe + φe) and φe in standard position. Then compute the difference.

From this corollary we can immediately see that the same angle (in a Euclidean sense) in different positions will have a different taxicab measure. Taxicab angles are translation invariant, but they are NOT rotation invariant.

The taxicab measure of other angles can also be found. Except for a few cases these formulas will be more complicated since angles lying in two or more quadrants encompass corners of the unit circle.

Right Angles

One of the most important angles in Euclidean geometry is the right angle whose measure is π / 2 radians or 90 degrees. In general, the same Euclidean angle will have different taxicab angle measure depending on its position. But, this is not the case for right angles.

THEOREM 3: The taxicab measure of any Euclidean right angle is 2 t-radians.


FIGURE 2: Taxicab right angles are precisely Euclidean right angles
and have a measure of 2 t-radians.

Proof: Without loss of generality, let θe be an Euclidean right angle encompassing the y-axis. As shown in Figure 2, split θ into two Euclidean angles αe and βe with reference angles π / 2 - αe and π / 2 - βe. Using the previous corollary and the facts that αe = π / 2 - βe and cos(π / 2 - αe) = sin αe, we see that the taxicab measure of the right angle is


Arc Length

Since all distances along a taxicab circle are scaled equally as the radius is changed, the familiar arc length theorem should be obvious and therefore we state it without proof.

THEOREM 4: The length s of the arc intercepted on a (taxicab) circle of radius r by the central angle with taxicab measure θ is given by s = r θ.


FIGURE 3: Arc length is the product of the central angle and the radius of the circle.

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References
[1] Thompson, Kevin and Tevian Dray. Taxicab Angles and Trigonometry, The Pi Mu Epsilon Journal, Worcester, MA. Vol. 11, No. 2 (Spring 2000), pp. 87-96.
[2] Akça, Ziya and Rüstem Kaya. On the Taxicab Trigonometry, Journal of Institute Of Mathematics and Computer Sciences, Vol. 10, No. 3, pp. 151-159.
[3] Dawson, Robert J. MacG. Crackpot Angle Bisectors!, Mathematics Magazine, Vol. 80, No. 1 (Feb 2007), pp. 59-64.
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