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A triangle has content , equal to half base times height when the chosen side is nonzero
Statement
Every triangle is Jordan measurable and has content .
If , then in the convention of Base and perpendicular height for a chosen side of a plane figure,
Facts & Assumptions
Given: Vertices and the triangle of Parallelograms and triangles in .
A region between continuous graphs is compact and Jordan measurable, and its content equals its graph area (Riemann area between continuous graphs equals Jordan content).
A linear map with matrix sends a bounded Jordan set to a bounded Jordan set with content , including singular (A linear endomorphism of sends bounded Jordan sets to bounded Jordan sets and scales their content by the absolute determinant).
For bounded and , and have equal inner and outer contents (Jordan inner content, outer content, measurability, and content are translation invariant).
For , ( for in ).
If on and is integrable, then (The second fundamental theorem: if is differentiable on with and is integrable, then ).
The Riemann integral is linear (Integrable functions on form a set closed under sums and scalar multiples, and ).
Proof
The standard triangle is the region , ; [L1], [L5], [L6], and [L7] give .
Let have columns and . Then , so [L2], [L3], and step 1.1 give , with the singular cases included.
If , apply [L4] with and in step 2.1 to obtain the half-base-times-height formula.
Depends on
- Parallelograms and triangles in $\mathbb{R}^2$
- Base and perpendicular height for a chosen side of a plane figure
- Jordan inner content, outer content, measurability, and content are translation invariant
- A linear endomorphism of $\mathbb R^n$ sends bounded Jordan sets to bounded Jordan sets and scales their content by the absolute determinant
- Riemann area between continuous graphs equals Jordan content
- The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G' = f$ and $f$ is integrable, then $\int_a^b f = G(b)-G(a)$
- For a natural $n \ge 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $\iota(n)\,x^{\,n-1}$; for $n = 0$ it is the constant $1$, with derivative $0$; for a natural $n \ge 1$ the function $x \mapsto x^{-n}$ is differentiable at every $x \ne 0$ with derivative $-\iota(n)\,x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term
- Integrable functions on $[a,b]$ form a set closed under sums and scalar multiples, and $\int_a^b(\lambda f+\mu g) = \lambda\int_a^b f + \mu\int_a^b g$
- $\lVert v\rVert_2\,d(w,\mathbb{R}v)=|\det[v\ w]|$ for $v\ne0$ in $\mathbb{R}^2$
Used by
- A triangle has zero Jordan content if and only if its vertices are collinear Corollary
- Schwarz lanterns can have mesh tending to zero while their polyhedral areas diverge Counterexample
- One triangle computed by both the determinant and base--height formulas Example
- Three collinear vertices give a triangle of zero content Example
- A simple polygon is Jordan measurable and its content is the sum of the contents of its triangles Theorem
- The shoelace formula for the area of a counterclockwise simple polygon Theorem
Dependency tree · two levels
60 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- W. F. Trench, Introduction to Real Analysis, §7.3 (standard reference, not scraped)
- M. E. Taylor, Introduction to Analysis in Several Variables, Proposition 3.1.10 (standard reference, not scraped)