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When the values of $y$ are free at both fixed endpoint locations, integration by parts leaves the boundary term $[F_{y^{\prime }}\eta {]}_a^b$. Since the endpoint variations are arbitrary, stationarity requires

A catenary is a curve of the form $y=a\cosh ((x-x_0)/a)+y_0$. It is the equilibrium shape of a uniform flexible chain in a uniform gravitational field.

When a variational integrand $f(z,z^{\prime })$ has no explicit dependence on the independent variable, every extremal satisfies

The integral momentum equation balances momentum flux, pressure, body force, and forces on a control volume.

For steady inviscid flow, pressure, kinetic energy density, and conservative potential are constant along a streamline.

For $\mathbf{u}=\nabla \varphi +\beta \nabla \alpha$, the densitygives incompressibility and material conservation of $\alpha$ and $\beta$. Differentiating the Clebsch representation then yields the Euler momentum equation with

For time-dependent irrotational flow with $u=\nabla \varphi$, constant mass density, and no body force, the Euler equations for an inviscid fluid integrate toA time-dependent change of gauge in $\varphi$ may set $C(t)=0$.

This is the unsteady potential-flow version of Bernoulli's principle, with a time-dependent potential gauge.

A lattice polygon has all vertices in the integer lattice; triangulation shows that its area is a half-integer.

A spherical cap is the part of a sphere cut off by a plane.

A regular polygon has equal side lengths and equal interior angles. Its vertices on a circle are scaled and rotated roots of unity.

A right angle is an angle of $\pi /2$ radians; two nonzero vectors meet at a right angle exactly when their inner product is zero.

The perpendicular bisector of a line segment is the line through its midpoint at a right angle to the segment.

An ellipse is the locus of points whose distances from two fixed foci have constant sum. If the focal distance is $d$ and the sum is $c>d$, its semimajor and semiminor axes are $c/2$ and $\sqrt{c^2-d^2}/2$.

A cyclic quadrilateral has all four vertices on one circle. Among quadrilaterals with fixed cyclically ordered side lengths, a cyclic quadrilateral has maximal area.

This is the planar case of the isoperimetric inequality. Every bounded planar domain with a sufficiently regular boundary of length $L$ and area $A$ satisfiesEquality holds exactly for circular domains.

The open unit disk in the Euclidean plane is

The Borsuk conjecture asserted that every bounded subset of ${\mathbb{R}}^d$ with positive diameter can be partitioned into $d+1$ subsets of strictly smaller diameter. It is false in sufficiently high dimension.

A triangular prism is the orthogonal Cartesian product of a triangle with a line segment.

A triangle is the convex hull of three noncollinear points: the three vertices must not lie on one line. Its boundary consists of the three joining line segments; depending on context, "triangle" can mean this boundary or the filled region it encloses. Here a triangle's area refers to the filled region. The noncollinearity assumption excludes a collapsed shape with zero area, for which ordinary interior angles are not defined.

For example, the positions $(0,0)$, $(4,0)$ and $(0,3)$ form a right triangle with a right angle at the origin. Taking the side of length $4$ as its base gives perpendicular height $3$, so its area is $4\cdot 3/2=6$. The height is measured at a right angle to the chosen base; it is not generally the length of another side. In this example it happens to be a side because the triangle is a right triangle.

By contrast, side lengths $3,4,6$ define a valid triangle: each side is shorter than the sum of the other two, as required by the strict triangle inequality. But $3^2+4^2=25\ne 36=6^2$, so this is not a right triangle and the Pythagorean theorem cannot calculate its longest side from the other two. The law of cosines instead gives the opposite interior angle $C$ throughThis example separates the definition of a triangle from the additional assumption of a right angle.

Every triangle in the Euclidean plane has interior angles summing to $\pi$ radians, a result proved in Euclid's Elements I.32.^[ref] It lets two measured angles determine the third, and underlies the angle-angle criterion for similar triangles. The qualification "Euclidean plane" matters: the earlier spherical triangle demonstrates a different sum.

A sphere is the set of points at one fixed positive Euclidean distance from a center.