The Central Idea
The angle sum of a geodesic triangle depends on the geometry of the space containing it.
In the three standard constant-curvature geometries:
- Euclidean geometry: the angle sum equals \(180^\circ\);
- spherical geometry: the angle sum is greater than \(180^\circ\);
- hyperbolic geometry: the angle sum is less than \(180^\circ\).
Symbolically, if a triangle has interior angles \(\alpha\), \(\beta\), and \(\gamma\), then
\[ \alpha+\beta+\gamma \begin{cases} =\pi, & \text{flat geometry},\\[4pt] >\pi, & \text{positive curvature},\\[4pt] <\pi, & \text{negative curvature}. \end{cases} \]Why the Euclidean Rule Is Not Universal
The statement
\[ \alpha+\beta+\gamma=180^\circ \]is a theorem of Euclidean geometry. It follows from Euclidean assumptions, including the behavior of parallel lines.
Once those assumptions change, the theorem need not remain true.
Non-Euclidean geometry does not make Euclidean geometry incorrect. It shows that the Euclidean angle-sum theorem belongs to a particular geometric setting.
What Counts as a Triangle in a Curved Space?
In a curved geometry, a triangle is formed from three appropriate geodesic segments joining three vertices.
The sides are therefore not chosen merely because they look straight in an external picture. They must be intrinsically straight paths in the geometry being studied.
On a sphere, triangle sides are arcs of great circles. In the hyperbolic plane, triangle sides are hyperbolic geodesic segments.
How Angles Are Measured
At a vertex of a curved-space triangle, each geodesic side has a tangent direction.
The interior angle is the angle between those tangent directions in the local tangent geometry at the vertex.
This is important because a curved side may appear bent in a drawing even though it is geodesic in the underlying geometry.
The Euclidean Benchmark
For every ordinary Euclidean triangle,
\[ \boxed{\alpha+\beta+\gamma=\pi} \]or equivalently,
\[ \boxed{\alpha+\beta+\gamma=180^\circ}. \]The shape and size of the triangle do not affect this sum.
A Familiar Euclidean Example
Consider a triangle with angles
\[ 40^\circ,\qquad 60^\circ,\qquad 80^\circ. \]Their sum is
\[ 40^\circ+60^\circ+80^\circ=180^\circ. \]Changing the side lengths while remaining in Euclidean geometry may change the individual angles, but their total remains \(180^\circ\).
Spherical Triangles
A spherical triangle is formed by three arcs of great circles on a sphere.
Because a sphere has positive Gaussian curvature, a nondegenerate spherical triangle has
\[ \boxed{\alpha+\beta+\gamma>\pi}. \]In degrees,
\[ \boxed{\alpha+\beta+\gamma>180^\circ}. \]A Spherical Triangle with Three Right Angles
Start at the North Pole. Travel south along one meridian to the equator, turn \(90^\circ\), travel one quarter of the way around the equator, and then return to the North Pole along another meridian.
The resulting spherical triangle has three right angles:
\[ 90^\circ,\qquad 90^\circ,\qquad 90^\circ. \]Therefore,
\[ \boxed{90^\circ+90^\circ+90^\circ=270^\circ}. \]Its angle sum exceeds the Euclidean value by \(90^\circ\).
Spherical Excess
The amount by which a spherical triangle's angle sum exceeds \(\pi\) is called its spherical excess.
Define
\[ \boxed{E=\alpha+\beta+\gamma-\pi}. \]For a nondegenerate spherical triangle,
\[ E>0. \]Example: Computing Spherical Excess
Suppose a spherical triangle has angles
\[ 80^\circ,\qquad70^\circ,\qquad60^\circ. \]The angle sum is
\[ 80^\circ+70^\circ+60^\circ=210^\circ. \]Therefore, the spherical excess is
\[ \boxed{E=210^\circ-180^\circ=30^\circ}. \]In radians,
\[ \boxed{E=\frac{\pi}{6}}. \]Spherical Excess and Area
For a sphere of radius \(R\), the Gaussian curvature is constant:
\[ K=\frac{1}{R^2}. \]For a spherical triangle whose angles are measured in radians,
\[ \boxed{A=R^2E}. \]Since
\[ E=\alpha+\beta+\gamma-\pi, \]we obtain
\[ \boxed{A=R^2(\alpha+\beta+\gamma-\pi)}. \]Why Radians Matter in the Area Formula
The expression
\[ A=R^2E \]requires the spherical excess \(E\) to be measured in radians.
If the excess is given in degrees, convert first:
\[ E_{\text{rad}} =E_{\text{deg}}\frac{\pi}{180^\circ}. \]Angle measures in radians are dimensionless, so multiplying by \(R^2\) produces units of area.
Example: Area from Spherical Excess
Suppose a spherical triangle on a sphere of radius \(R\) has spherical excess
\[ E=\frac{\pi}{6}. \]Then
\[ A=R^2\left(\frac{\pi}{6}\right). \]Therefore,
\[ \boxed{A=\frac{\pi R^2}{6}}. \]Area of the Three-Right-Angle Triangle
The triangle with three right angles has excess
\[ E=\frac{\pi}{2}. \]Its area is therefore
\[ A=R^2\frac{\pi}{2}. \]Thus,
\[ \boxed{A=\frac{\pi R^2}{2}}. \]Since the full sphere has area \(4\pi R^2\), this triangle occupies
\[ \frac{\pi R^2/2}{4\pi R^2}=\frac18 \]of the sphere.
Larger Spherical Triangles Have Larger Excess
On a sphere of fixed radius,
\[ E=\frac{A}{R^2}. \]Therefore, larger spherical triangles have greater spherical excess.
Unlike Euclidean geometry, the angle sum of a spherical triangle depends on its size as well as its shape.
Small Spherical Triangles Look Nearly Euclidean
If a spherical triangle occupies a very small region compared with the radius of the sphere, its area \(A\) is small relative to \(R^2\).
Since
\[ E=\frac{A}{R^2}, \]its spherical excess is correspondingly small.
Thus,
\[ \alpha+\beta+\gamma\approx\pi. \]This is one reason Euclidean geometry works extremely well for many local measurements on Earth.
Hyperbolic Triangles
In the hyperbolic plane, geodesic triangles behave in the opposite direction from spherical triangles.
A nondegenerate hyperbolic triangle has
\[ \boxed{\alpha+\beta+\gamma<\pi}. \]In degrees,
\[ \boxed{\alpha+\beta+\gamma<180^\circ}. \]Hyperbolic Defect
The amount by which a hyperbolic triangle's angle sum falls short of \(\pi\) is called its angular defect or hyperbolic defect.
Define
\[ \boxed{D=\pi-(\alpha+\beta+\gamma)}. \]For a nondegenerate hyperbolic triangle,
\[ D>0. \]Example: Computing Hyperbolic Defect
Suppose a hyperbolic triangle has angles
\[ 50^\circ,\qquad60^\circ,\qquad40^\circ. \]Its angle sum is
\[ 50^\circ+60^\circ+40^\circ=150^\circ. \]The defect is
\[ \boxed{D=180^\circ-150^\circ=30^\circ}. \]In radians,
\[ \boxed{D=\frac{\pi}{6}}. \]Hyperbolic Defect and Area
Consider a hyperbolic plane with constant curvature
\[ K=-\frac{1}{R^2}. \]For a geodesic triangle, its area is
\[ \boxed{A=R^2D}, \]where \(D\) is measured in radians.
Equivalently,
\[ \boxed{A=R^2\left[\pi-(\alpha+\beta+\gamma)\right]}. \]Example: Hyperbolic Area from the Angles
Suppose a hyperbolic plane has curvature
\[ K=-\frac{1}{R^2} \]and a triangle has angle sum
\[ \alpha+\beta+\gamma=\frac{5\pi}{6}. \]Then
\[ D=\pi-\frac{5\pi}{6}=\frac{\pi}{6}. \]Therefore,
\[ \boxed{A=\frac{\pi R^2}{6}}. \]Larger Hyperbolic Triangles Have Larger Defect
For fixed curvature \(-1/R^2\),
\[ D=\frac{A}{R^2}. \]As the area increases, the sum of the angles decreases farther below \(\pi\).
This is the negative-curvature counterpart of spherical excess.
Small Hyperbolic Triangles Also Look Nearly Euclidean
If the area is small compared with the curvature scale \(R^2\), then
\[ D=\frac{A}{R^2} \]is small.
Consequently,
\[ \alpha+\beta+\gamma\approx\pi. \]Local geometry approaches Euclidean behavior at sufficiently small scales on smooth curved spaces.
An Ideal Hyperbolic Triangle
Hyperbolic geometry permits a limiting object called an ideal triangle. Its vertices lie at the ideal boundary of the hyperbolic plane rather than at ordinary interior points.
Each interior angle approaches zero:
\[ \alpha=\beta=\gamma=0. \]Its defect is therefore
\[ D=\pi. \]In curvature \(-1/R^2\), every ideal triangle has area
\[ \boxed{A=\pi R^2}. \]This striking result shows how strongly hyperbolic area and angle defect are linked.
Three Geometries Side by Side
| Geometry | Curvature | Triangle angle sum | Deviation from \(\pi\) |
|---|---|---|---|
| Euclidean | Zero | \(=\pi\) | None |
| Spherical | Positive | \(>\pi\) | Spherical excess |
| Hyperbolic | Negative | \(<\pi\) | Hyperbolic defect |
Angle Sum as a Curvature Signal
Triangle angle sums provide an intrinsic way to detect curvature.
If a sufficiently small geodesic triangle has an angle sum slightly above \(\pi\), that behavior signals positive curvature in the region.
If the sum is slightly below \(\pi\), it signals negative curvature.
If the sum is exactly \(\pi\) throughout a flat region, it is consistent with zero Gaussian curvature there.
The Sign Pattern
It is useful to remember the correspondence
\[ \boxed{ \begin{array}{ccl} K>0 &\Longleftrightarrow& \alpha+\beta+\gamma>\pi,\\[4pt] K=0 &\Longleftrightarrow& \alpha+\beta+\gamma=\pi,\\[4pt] K<0 &\Longleftrightarrow& \alpha+\beta+\gamma<\pi \end{array}} \]for the standard constant-curvature models and, locally, for sufficiently small geodesic triangles in regions where curvature has a consistent sign.
From Constant Curvature to Variable Curvature
Real surfaces need not have the same curvature at every point.
A surface may have one region with positive curvature, another with negative curvature, and still another that is nearly flat.
For such surfaces, a triangle's angle excess or defect reflects the curvature accumulated over the region it encloses rather than a single constant value.
The Geodesic-Triangle Form of Gauss–Bonnet
For a geodesic triangle \(T\) on a smooth oriented surface, under the usual conditions for a simple triangular region,
\[ \boxed{ \alpha+\beta+\gamma-\pi =\iint_T K\,dA }. \]Here \(K\) is the Gaussian curvature and \(dA\) is the surface-area element.
This formula unifies the Euclidean, spherical, and hyperbolic cases.
Recovering the Spherical Formula
On a sphere of radius \(R\),
\[ K=\frac{1}{R^2}. \]Since \(K\) is constant,
\[ \iint_T K\,dA =\frac{1}{R^2}\iint_T dA =\frac{A}{R^2}. \]Therefore,
\[ \alpha+\beta+\gamma-\pi =\frac{A}{R^2}, \]which rearranges to
\[ \boxed{A=R^2(\alpha+\beta+\gamma-\pi)}. \]Recovering the Hyperbolic Formula
In a hyperbolic plane with
\[ K=-\frac{1}{R^2}, \]Gauss–Bonnet gives
\[ \alpha+\beta+\gamma-\pi =-\frac{A}{R^2}. \]Rearranging,
\[ \boxed{ A=R^2\left[\pi-(\alpha+\beta+\gamma)\right] }. \]Integrated Curvature Matters
The quantity
\[ \iint_T K\,dA \]is the total Gaussian curvature accumulated over the triangular region.
If curvature varies from point to point, the angle sum is governed by this integrated total.
A triangle that spans both positively and negatively curved regions can therefore have contributions that partially cancel.
Why Geodesic Sides Matter in Gauss–Bonnet
The simple triangle formula
\[ \alpha+\beta+\gamma-\pi=\iint_T K\,dA \]assumes the triangle's sides are geodesics.
If the boundary curves themselves bend intrinsically, additional boundary curvature terms appear in the more general Gauss–Bonnet theorem.
Using geodesic edges removes those side-curvature contributions and makes the relationship especially clean.
Angle Sum Is Intrinsic
A creature confined to a surface could construct geodesic triangles, measure their interior angles, and infer information about curvature without observing the surface from outside.
That makes triangle angle sums an intrinsic geometric measurement.
No external three-dimensional viewpoint is required.
The Cylinder Provides an Important Contrast
A cylinder looks curved when embedded in three-dimensional space, but its Gaussian curvature is
\[ K=0. \]Locally, a cylinder can be unrolled onto a flat plane without stretching.
Consequently, sufficiently ordinary geodesic triangles on the cylinder obey the Euclidean angle sum
\[ \boxed{\alpha+\beta+\gamma=\pi}. \]This reinforces the difference between extrinsic bending and intrinsic curvature.
Curvature Scale
In constant-curvature geometry, the parameter \(R\) sets a natural length scale.
For a sphere,
\[ K=\frac{1}{R^2}, \]while a standard hyperbolic plane may be scaled so that
\[ K=-\frac{1}{R^2}. \]Triangles much smaller than this curvature scale tend to look nearly Euclidean. Effects become more pronounced when the triangle spans a larger portion of the curved space.
Same Angles, Different Euclidean Intuition
In Euclidean geometry, the three angles determine a triangle only up to similarity. Triangles with the same angles can have many different sizes.
In spherical or hyperbolic geometry with fixed curvature, the angle data also determine the area through excess or defect.
Curvature therefore links angle information to absolute size in a way that does not occur in Euclidean geometry.
Why Euclidean Similarity Behaves Differently
Euclidean triangles can be scaled by any positive factor while preserving all three angles.
On a sphere of fixed radius, scaling a geodesic triangle changes its area and therefore changes its spherical excess.
Likewise, in a hyperbolic plane of fixed curvature, changing the size of a triangle changes its defect.
As a result, there is no direct analogue of arbitrary Euclidean similarity among geodesic triangles in a fixed nonzero constant-curvature geometry.
A Dimensionless Measure of Curvature Effect
In the constant-curvature cases, angle excess or defect can be written as
\[ |E|\text{ or }D=\frac{A}{R^2}. \]The ratio \(A/R^2\) is dimensionless.
It compares the triangle's area with the natural area scale established by curvature.
Small values correspond to nearly Euclidean behavior; larger values reveal stronger curvature effects.
Estimating Curvature from a Small Triangle
If curvature is approximately constant across a small geodesic triangle, Gauss–Bonnet suggests
\[ \alpha+\beta+\gamma-\pi\approx KA. \]Therefore,
\[ \boxed{ K\approx \frac{\alpha+\beta+\gamma-\pi}{A} }. \]This gives a conceptual way to estimate local Gaussian curvature using angle and area measurements made within the surface.
Example: Estimating Positive Curvature
Suppose a small geodesic triangle has area \(0.020\) square units and angle excess
\[ E=0.005\text{ radians}. \]If curvature is nearly constant over the triangle,
\[ K\approx\frac{0.005}{0.020}=0.25. \]Thus,
\[ \boxed{K\approx0.25}. \]The positive sign is consistent with the positive angle excess.
Example: Estimating Negative Curvature
Suppose a small geodesic triangle has area \(0.040\) square units and its angle sum is
\[ \pi-0.012. \]Then
\[ \alpha+\beta+\gamma-\pi=-0.012. \]Therefore,
\[ K\approx\frac{-0.012}{0.040}=-0.3. \]So
\[ \boxed{K\approx-0.3}. \]Why the Effect Can Be Hard to Notice
Curvature-induced angle differences may be extremely small when the triangle is small compared with the curvature scale.
Measurement uncertainty can then be larger than the angle excess or defect itself.
Detecting curvature experimentally therefore requires both sufficiently large geometric regions and sufficiently precise measurements.
Geodesy and Large-Scale Measurement
On Earth, survey triangles spanning modest local regions are often modeled approximately with Euclidean geometry.
Over larger regions, Earth's curvature becomes relevant, and spherical or ellipsoidal geodesy provides more accurate models.
Triangle angle behavior is one conceptual route from local surveying to global geometry.
Astronomy and the Geometry of Space
The same general idea has a broader conceptual role in astronomy and cosmology: geometric measurements over large scales can be used to test whether spatial relationships behave as flat, positively curved, or negatively curved models predict.
Actual physical cosmology requires substantially more structure than the elementary geometric models in this lesson, but triangle-based reasoning illustrates the underlying mathematical strategy.
Computer Graphics and Geometric Modeling
Curved meshes and surfaces are frequently approximated by collections of small triangles.
In discrete differential geometry, angle behavior around vertices can be used to approximate curvature.
This provides a computational counterpart to the smooth relationship between angle excess, area, and Gaussian curvature.
Common Error: Assuming Every Triangle Has Sum \(180^\circ\)
The rule
\[ \alpha+\beta+\gamma=180^\circ \]is not a definition of a triangle.
It is a theorem in Euclidean geometry. In curved geometries, triangles are still triangles even though their angle sums may differ.
Common Error: Using Arbitrary Curves as Triangle Sides
The standard angle-sum relationships in this lesson concern geodesic triangles.
If the sides are arbitrary curved paths, their own geodesic curvature contributes to the geometry, and the simple formulas no longer apply without modification.
Common Error: Measuring the Picture Instead of the Geometry
A model such as the Poincaré disk may depict hyperbolic geodesics as circular arcs in the Euclidean drawing.
The geometry is determined by the model's metric and angle rules, not by treating the page itself as the hyperbolic plane.
Always distinguish the representation from the intrinsic geometry being represented.
Common Error: Confusing Degrees and Radians
The area formulas
\[ A=R^2E \]and
\[ A=R^2D \]require \(E\) and \(D\) in radians.
A defect of \(30^\circ\) must therefore be converted to
\[ \frac{\pi}{6} \]before substitution.
Common Error: Thinking More Curvature Means Every Triangle Has the Same Excess
On a sphere of fixed curvature,
\[ E=\frac{A}{R^2}. \]The excess depends on both curvature and the triangle's area.
Two triangles on the same sphere can therefore have different angle sums because they enclose different areas.
Common Error: Equating Visible Bending with Positive Angle Excess
A cylinder is visibly bent in three-dimensional space, yet its intrinsic Gaussian curvature is zero.
Geodesic triangle angle sums detect intrinsic curvature, not merely how a surface appears to bend from an external viewpoint.
Common Error: Applying a Constant-Curvature Formula Everywhere
The formulas
\[ A=R^2E \]and
\[ A=R^2D \]assume the relevant spherical or hyperbolic constant-curvature model.
On a surface with variable curvature, use the broader relation
\[ \alpha+\beta+\gamma-\pi=\iint_T K\,dA. \]A Reliable Procedure for Classifying by Angle Sum
- Confirm that the sides form a geodesic triangle in the geometry being studied.
- Add the three interior angles.
- Compare the total with \(\pi\), or \(180^\circ\).
- If the sum is greater, identify a positive excess.
- If the sum is equal, identify the Euclidean or intrinsically flat case.
- If the sum is smaller, identify a positive defect.
- Use radians before applying curvature-area formulas.
A Reliable Procedure for Spherical Area
- Add the three interior angles.
- Convert all angles to radians if necessary.
- Compute the spherical excess \(E=\alpha+\beta+\gamma-\pi\).
- Use the sphere's radius \(R\).
- Compute \(A=R^2E\).
- Check that \(E>0\) for a nondegenerate spherical triangle.
A Reliable Procedure for Hyperbolic Area
- Add the three interior angles.
- Convert all angles to radians if necessary.
- Compute the defect \(D=\pi-(\alpha+\beta+\gamma)\).
- Identify the curvature scale from \(K=-1/R^2\).
- Compute \(A=R^2D\).
- Check that \(D>0\) for a nondegenerate hyperbolic triangle.
Worked Example: Spherical Triangle in Degrees
A spherical triangle on a sphere of radius \(10\) has angles
\[ 100^\circ,\qquad50^\circ,\qquad60^\circ. \]The total is
\[ 100^\circ+50^\circ+60^\circ=210^\circ. \]The excess is
\[ E=30^\circ=\frac{\pi}{6}. \]Therefore,
\[ A=10^2\frac{\pi}{6} =\frac{50\pi}{3}. \]Hence,
\[ \boxed{A=\frac{50\pi}{3}}. \]Worked Example: Hyperbolic Triangle in Radians
In a hyperbolic plane with curvature
\[ K=-\frac{1}{25}, \]we have \(R=5\).
Suppose the triangle angles are
\[ \frac{\pi}{4},\qquad\frac{\pi}{3},\qquad\frac{\pi}{6}. \]Their sum is
\[ \frac{\pi}{4}+\frac{\pi}{3}+\frac{\pi}{6} =\frac{3\pi+4\pi+2\pi}{12} =\frac{3\pi}{4}. \]The defect is
\[ D=\pi-\frac{3\pi}{4}=\frac{\pi}{4}. \]Thus,
\[ A=25\frac{\pi}{4}. \]Therefore,
\[ \boxed{A=\frac{25\pi}{4}}. \]Worked Example: Finding an Unknown Spherical Angle
A spherical triangle has spherical excess \(20^\circ\) and two angles
\[ 70^\circ\quad\text{and}\quad80^\circ. \]Since the total angle sum is
\[ 180^\circ+20^\circ=200^\circ, \]the third angle is
\[ \gamma=200^\circ-70^\circ-80^\circ. \]Therefore,
\[ \boxed{\gamma=50^\circ}. \]Worked Example: Finding an Unknown Hyperbolic Angle
A hyperbolic triangle has defect \(24^\circ\), and two angles are
\[ 55^\circ\quad\text{and}\quad61^\circ. \]Its total angle sum is
\[ 180^\circ-24^\circ=156^\circ. \]Therefore,
\[ \gamma=156^\circ-55^\circ-61^\circ=40^\circ. \]Thus,
\[ \boxed{\gamma=40^\circ}. \]Worked Example: Inferring Curvature Sign
A small geodesic triangle has measured angles
\[ 60.02^\circ,\qquad59.99^\circ,\qquad60.01^\circ. \]Their sum is
\[ 180.02^\circ. \]The small positive excess is consistent with positive Gaussian curvature over the triangular region, assuming the excess is larger than the measurement uncertainty.
Practice 1: Classify the Geometry
A geodesic triangle has angles
\[ 65^\circ,\qquad55^\circ,\qquad60^\circ. \]Find the angle sum and identify which standard curvature case it matches.
Practice 2: Spherical Excess
A spherical triangle has angles
\[ 95^\circ,\qquad70^\circ,\qquad45^\circ. \]Find its spherical excess in degrees and radians.
Practice 3: Spherical Area
A spherical triangle lies on a sphere of radius \(12\) and has spherical excess
\[ E=\frac{\pi}{8}. \]Find its area.
Practice 4: Hyperbolic Defect
A hyperbolic triangle has angles
\[ 45^\circ,\qquad50^\circ,\qquad65^\circ. \]Find its defect in degrees and radians.
Practice 5: Hyperbolic Area
In a hyperbolic plane with curvature
\[ K=-\frac{1}{16}, \]a triangle has defect
\[ D=\frac{\pi}{5}. \]Find its area.
Practice 6: Find the Missing Spherical Angle
A spherical triangle has excess \(15^\circ\). Two angles are
\[ 80^\circ\quad\text{and}\quad65^\circ. \]Find the third angle.
Practice 7: Find the Missing Hyperbolic Angle
A hyperbolic triangle has defect \(30^\circ\). Two angles are
\[ 40^\circ\quad\text{and}\quad55^\circ. \]Find the third angle.
Practice 8: Local Curvature Estimate
A small geodesic triangle has area \(0.03\) and angle excess \(0.006\) radians. Assuming curvature is approximately constant over the region, estimate \(K\).
Practice 9: Explain the Cylinder
A student says, "A cylinder bends in space, so every geodesic triangle on it must have angle sum greater than \(180^\circ\)."
Explain the error.
Practice 10: Variable Curvature
A geodesic triangle encloses a region on which
\[ \iint_T K\,dA=-0.04. \]Find its angle sum in radians and determine whether it has an excess or a defect.
Practice Answers
Practice 1
The sum is
\[ 65^\circ+55^\circ+60^\circ=180^\circ. \]Therefore, the triangle matches the flat Euclidean angle-sum case:
\[ \boxed{180^\circ}. \]Practice 2
The sum is
\[ 95^\circ+70^\circ+45^\circ=210^\circ. \]Thus,
\[ E=210^\circ-180^\circ=30^\circ. \]Converting to radians,
\[ \boxed{E=30^\circ=\frac{\pi}{6}}. \]Practice 3
Use
\[ A=R^2E. \]Therefore,
\[ A=12^2\frac{\pi}{8} =144\frac{\pi}{8} =18\pi. \]Hence,
\[ \boxed{A=18\pi}. \]Practice 4
The angle sum is
\[ 45^\circ+50^\circ+65^\circ=160^\circ. \]The defect is
\[ D=180^\circ-160^\circ=20^\circ. \]In radians,
\[ \boxed{D=\frac{\pi}{9}}. \]Practice 5
From
\[ K=-\frac{1}{R^2}=-\frac{1}{16}, \]we have \(R=4\). Therefore,
\[ A=R^2D =16\frac{\pi}{5}. \]Thus,
\[ \boxed{A=\frac{16\pi}{5}}. \]Practice 6
The total angle sum is
\[ 180^\circ+15^\circ=195^\circ. \]Therefore,
\[ \gamma=195^\circ-80^\circ-65^\circ=50^\circ. \]Hence,
\[ \boxed{\gamma=50^\circ}. \]Practice 7
The total angle sum is
\[ 180^\circ-30^\circ=150^\circ. \]Thus,
\[ \gamma=150^\circ-40^\circ-55^\circ=55^\circ. \]Therefore,
\[ \boxed{\gamma=55^\circ}. \]Practice 8
Use the local approximation
\[ K\approx\frac{E}{A}. \]Then
\[ K\approx\frac{0.006}{0.03}=0.2. \]Therefore,
\[ \boxed{K\approx0.2}. \]Practice 9
The student is confusing extrinsic bending with intrinsic Gaussian curvature. A circular cylinder has Gaussian curvature
\[ K=0. \]It can be unrolled locally into a plane without stretching, so its local intrinsic geometry is flat. Geodesic triangles therefore obey the Euclidean angle-sum relation in ordinary simply connected regions.
Practice 10
Gauss–Bonnet gives
\[ \alpha+\beta+\gamma-\pi=-0.04. \]Therefore,
\[ \boxed{\alpha+\beta+\gamma=\pi-0.04}. \]Since the sum is below \(\pi\), the triangle has a defect of
\[ \boxed{0.04\text{ radians}}. \]What to Remember
- Triangle angle sums depend on the geometry of the surrounding space.
- Euclidean geodesic triangles have angle sum exactly \(\pi\).
- Spherical geodesic triangles have angle sum greater than \(\pi\).
- Hyperbolic geodesic triangles have angle sum less than \(\pi\).
- Spherical excess is \(E=\alpha+\beta+\gamma-\pi\).
- Hyperbolic defect is \(D=\pi-(\alpha+\beta+\gamma)\).
- On a sphere of radius \(R\), \(A=R^2E\).
- For curvature \(-1/R^2\), hyperbolic area satisfies \(A=R^2D\).
- Angle excess and defect must be in radians when used in these area formulas.
- Small triangles on smooth curved spaces are approximately Euclidean.
- Angle sums provide an intrinsic signal of Gaussian curvature.
- For a geodesic triangle, \(\alpha+\beta+\gamma-\pi=\iint_T K\,dA\).
- Variable curvature contributes through its integral over the enclosed region.
- Extrinsic bending alone does not determine triangle angle sums.
- The cylinder is the standard example: visibly bent, but intrinsically flat.
Where We Go Next
Triangle angle sums reveal that curvature changes more than the appearance of a space. It changes measurable relationships among directions, area, and geodesic paths.
The next lesson, Distance on Spheres and Curved Surfaces, develops the corresponding idea for length: how distance is measured intrinsically along a curved surface, why straight-line chord distance is usually not the same as surface distance, and how geodesics determine shortest paths.