nyuuzyou/wikis huggingface dataset:
Vitali Set Existence Theorem/Proof 1

== Theorem ==
== Proof ==
=== Lemma ===
Let $\map \mu X$ denote the Lebesgue measure of a set $X$ of real numbers.
We have that:
:$(1): \quad$ $\map \mu X$ is a countably additive function
:$(2): \quad$ $\map \mu X$ is translation invariant
:$(3): \quad$ From Measure of Interval is Length, $\map \mu {\closedint a b} = b - a$ for every closed interval $\closedint a b$.
For all real numbers in the closed unit interval $\mathbb I = \closedint 0 1$, define the relation $\sim$ such that:
:$\forall x, y \in \mathbb I: x \sim y \iff x - y \in \Q$
where $\Q$ is the set of rational numbers.
That is, $x \sim y$ their difference is rational.
By the , $\sim$ is an equivalence relation.
For each $x \in \mathbb I$, let $\eqclass x \sim$ denote the equivalence class of $x$ under $\sim$.
Let us use the Axiom of Choice to create a set $M \subset \mathbb I$ containing exactly $1$ element from each equivalence class.
Hence:
:for each $x \in \R$ there exists a unique $y \in M$ and $r \in \Q$ such that $x = y + r$.
Let:
:$M_r = \set {y + r: y \in M}$
for each $r \in \Q$.
Thus $\R$ is partitioned into countably many disjoint sets:
:$(1): \quad \ds \R = \bigcup \set {M_r: r \in \Q}$
 $M$ were Lebesgue measurable.
First, $\map \mu M = 0$ is impossible, because from $(1)$ this would mean $\map \mu \R = 0$.
Suppose $\map \mu M > 0$.
Then:
So $\map \mu M > 0$ is also impossible.
Hence $M$ is not Lebesgue measurable.
Category:Vitali Set Existence Theorem
Vitali Set Existence Theorem/Proof 2

== Theorem ==
== Proof ==
We construct such a set.
For $x, y \in \hointr 0 1$, define the sum modulo 1:
:$x +_1 y = \begin {cases} x + y & : x + y < 1 \\ x + y - 1 & : x + y \ge 1 \end {cases}$
Let $E \subset \hointr 0 1$ be a measurable set.
Let $E_1 = E \cap \hointr 0 {1 - x}$ and $E_2 = E \cap \hointr {1 - x} 1$.
By Measure of Interval is Length, these disjoint intervals are measurable.
By Measurable Sets form Algebra of Sets, so are these intersections $E_1$ and $E_2$.
So:
:$\map m {E_1} + \map m {E_2} = \map m E$
We have:
:$E_1 +_1 x = E_1 + x$
By Lebesgue Measure is Translation Invariant:
:$\map m {E_1 +_1 x} = \map m {E_1}$
Also:
:$E_2 +_1 x = E_2 + x - 1$
and so:
:$\map m {E_2 +_1 x} = \map m {E_2}$
Then we have:
:$\map m {E +_1 x} = \map m {E_1 +_1 x} + \map m {E_2 +_1 x} = \map m {E_1} + \map m {E_2} = \map m E$
So, for each $x \in \hointr 0 1$, the set $E +_1 x$ is measurable and:
:$\map m {E + x} = \map m E$
Taking, as before, $x, y \in \hointr 0 1$, define the relation:
:$x \sim y \iff x - y \in \Q$
where $\Q$ is the set of rational numbers.
By Difference is Rational is Equivalence Relation, $\sim$ is an equivalence relation.
As this is an equivalence relation we can invoke the fundamental theorem on equivalence relations.
Hence $\sim$ partitions $\hointr 0 1$ into equivalence classes.
By the axiom of choice, there is a set $P$ which contains exactly one element from each equivalence class.
 $P$ is measurable.
Let $\set {r_i}_{i \mathop = 0}^\infty$ be an enumeration of the rational numbers in $\hointr 0 1$ with $r_0 = 0$.
Let $P_i := P +_1 r_i$.
Then $P_0 = P$.
Let $x \in P_i \cap P_j$.
Then:
:$x = p_i + r_i = p_j + r_j$
where $p_i, p_j$ are elements of $P$.
But then $p_i - p_j$ is a rational number.
Since $P$ has only one element from each equivalence class:
:$i = j$
The $P_i$ are pairwise disjoint.
Each real number $x \in \hointr 0 1$ is in some equivalence class and hence is equivalent to an element of $P$.
But if $x$ differs from an element in $P$ by the rational number $r_i$, then $x \in P_i$ and so:
:$\ds \bigcup P_i = \hointr 0 1$
Since each $P_i$ is a translation modulo $1$ of $P$, each $P_i$ will be measurable if $P$ is, with measure $\map m {P_i} = \map m P$.
But if this were the case, then:
:$\ds m \hointr 0 1 = \sum_{i \mathop = 1}^\infty m \paren {P_i} = \sum_{i \mathop = 1}^\infty \map m P$
Therefore:
:$\map m P = 0$ implies $m \hointr 0 1 = 0$
and:
:$\map m P \ne 0$ implies $m \hointr 0 1 = \infty$
This contradicts Measure of Interval is Length.
So the set $P$ is not measurable.
== Sources ==
* : $\S 1.1$
Category:Vitali Set Existence Theorem
Viviani's Theorem

== Theorem ==
Let $T$ be an equilateral triangle.
Let $P$ be a point inside $T$.
Let $x$, $y$ and $z$ be the lengths of the perpendiculars dropped from $P$ to each of the three sides of $T$.
Then;
:$x + y + z = h$
where $h$ is the height of $T$.
== Proof 1 ==
== Proof 2 ==
Category:Viviani's Theorem
Category:Equilateral Triangles
Viviani's Theorem/Proof 1

== Theorem ==
== Proof ==
Let $T = \triangle ABC$ be an equilateral triangle whose vertices are $A$, $B$ and $C$.
Let $h$ be the height of $T$.
Let $a$ be the length of one side of $T$.
Let $P$ be a point inside $T$.
Let $\triangle APB$, $\triangle BPC$ and $\triangle CPA$ be the three triangles formed by joining $P$ to each of the three vertices $A$, $B$ and $C$ of $T$.
Let the heights of $\triangle APB$, $\triangle BPC$ and $\triangle CPA$ be $x$, $y$ and $z$.
:
By definition, these heights are the lengths of the perpendiculars dropped from $P$ to each of the three sides of $T$.
Let $A$ be the area of $T$.
By Area of Triangle in Terms of Side and Altitude:
:$A = \dfrac {a h} 2$
But we also have that the area of $T$ is also equal to the sum of the areas of each of $\triangle APB$, $\triangle BPC$ and $\triangle CPA$.
By Area of Triangle in Terms of Side and Altitude, these areas are equal to $\dfrac {a x} 2$, $\dfrac {a y} 2$ and $\dfrac {a z} 2$.
That is:
:$A = \dfrac {a h} 2 = \dfrac {a x} 2 + \dfrac {a y} 2 + \dfrac {a z} 2$
from which it follows that:
:$h = x + y + z$
Category:Viviani's Theorem
Viviani's Theorem/Proof 2

== Theorem ==
== Proof ==
Let $T = \triangle ABC$ be an equilateral triangle whose vertices are $A$, $B$ and $C$.
Let $h$ be the height of $T$.
Let $P$ be a point inside $T$.
Let $\triangle PDE$, $\triangle PFG$ and $\triangle PJH$ be three equilateral triangles constructed from $P$ to each side of $ABC$ as depicted.
Let the heights of $\triangle PDE$, $\triangle PFG$ and $\triangle PJH$ be $x$, $y$ and $z$.
Let $\triangle CGK$ be an equilateral triangle constructed also as depicted.
:
The result follows by inspection.
Category:Viviani's Theorem
Viète's Formulas

== Theorem ==
Let $P_n$ be a polynomial of degree $n$ with real or complex coefficients:
where $a_n \ne 0$.
Let $z_1, \ldots, z_n$ be the roots of $P_n$ (be they real or complex), not assumed distinct.
Then:
where $\map {e_k} {\set {z_1, \ldots, z_n} }$ denotes the elementary symmetric function of degree $k$ on $\set {z_1, \ldots, z_n}$.
Listed explicitly:
== Proof ==
First we note that from the Polynomial Factor Theorem:
:$\ds \map {P_n} x = a_n \prod_{k \mathop = 1}^n \paren {x - z_k}$
To shorten the exposition, let us define:
:$E_{\tuple {r, s} } := \map {e_r} {\set {z_1, \ldots, z_s} }$
for $r, s \in \Z_{\ge 1}$.
It is sufficient to consider the case $a_n = 1$, in which case:
:$\ds \map {P_n} x = \prod_{k \mathop = 1}^n \paren {x - z_k}$
The proof then proceeds by induction as follows.
Let $\map {\Bbb P} n$ be the statement:
=== Basis for the Induction ===
We have:
Hence $\map {\Bbb P} 1$ holds.
This is our basis for the induction.
=== Induction Hypothesis ===
Now it needs to be shown that if $\map {\Bbb P} n$ is true then it logically follows that $\map {\Bbb P} {n + 1}$ is true.
This is the induction hypothesis:
from which it is to be shown that:
=== Induction Step ===
This is our induction step:
Let it be assumed that $\map {\Bbb P} n$ holds for some $n \ge 1$.
Let $\map {Q_{n + 1} } x$ be an arbitrary polynomial of degree $n + 1$ with real or complex coefficients:
We can divide all the coefficients by $q_{n + 1}$ to force $\map {Q_{n + 1} } x$ to be monic.
Hence after renaming those coefficients:
and so, from the Polynomial Factor Theorem:
:$\ds \map {Q_{n + 1} } x = \prod_{k \mathop = 1}^{n + 1} \paren {x - z_k}$
where $\set {z_1, \ldots, z_n, z_{n + 1} }$ are the roots of $\map {Q_{n + 1} } x$.
Hence:
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
 
== Examples ==
== Also known as ==
== Also see ==
* Definition:Elementary Symmetric Function
* Elementary Symmetric Function/Examples/Monic Polynomial
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Polynomial Equations: $103$
* : Theorem for a field.
* : Theorem for distinct roots.
Category:Viète's Formulas
Category:Polynomial Theory
Category:Algebra
Category:Elementary Symmetric Functions
Category:Proofs by Induction
Viète's Formulas/Also known as

== Viète's Formulas: Also known as ==
Viète's Formulas are also known (collectively) as Viète's Theorem or (the) Viète Theorem.
The Latin form of his name (Vieta) is also often seen.
== Sources ==
* : Theorem for a field.
* : Theorem for distinct roots.
Category:Viète's Formulas
Viète's Formulas/Examples

== Examples of Use of Viète's Formulas ==
=== Sum of Roots of Quadratic Equation ===
=== Product of Roots of Quadratic Equation ===
=== Coefficients of Cubic ===
=== Coefficients of Quartic ===
=== Two Numbers whose Sum is $4$ and whose Product is $8$ ===
=== Cubic with all Equal Roots ===
=== Monic Polynomial Formulas ===
Category:Examples of Use of Viète's Formulas
Viète's Formulas/Examples/Cubic

== Example of Use of Viète's Formulas ==
Consider the cubic equation:
:$x^3 + a_1 x^2 + a_2 x + a_3 = 0$
Let its roots be denoted $x_1$, $x_2$ and $x_3$.
Then:
== Proof ==
A specific instance of Viète's Formulas for $n = 3$.
== Sources ==
* : $3$: Elementary Analytic Methods: $3.8$ Algebraic Equations: Solution of Cubic Equations: $3.8.2$
* : $\S 9$: Solutions of Algebraic Equations: Cubic Equation: $x^3 + a_1 x^2 + a_2 x + a_3 = 0$: $9.5$
* : $\S 5$: Solutions of Algebraic Equations: Cubic Equation: $x^3 + a_1 x^2 + a_2 x + a_3 = 0$: $5.5.$
Category:Examples of Use of Viète's Formulas
Category:Cubic Equations
Viète's Formulas/Examples/Cubic with Equal Roots

== Example of Use of Viète's Formulas ==
The coefficient of $x$ in the expansion of cubic $\paren {x - 1}^3$ is $3$.
== Proof ==
The expansion:
:$\paren {x - 1}^3 = x^3 + a_2 x^2 + a_1 x + a_0$
Viète's Formulas for $z_1 = z_2 = z_3 = 1$ are: 
Category:Examples of Use of Viète's Formulas
Viète's Formulas/Examples/Monic Polynomial

== Example of Use of Viète's Formulas ==
Let:
:$\ds \map P x = x^N + \sum_{k \mathop = 0}^{N - 1} b_k x^k$
be a monic polynomial of degree $N$.
Let $U$ be the set of $N$ roots of equation $\map P x = 0$.
Then:
:$b_k = \paren {-1}^{N - k} \map {e_{N - k} } U, \quad 0 \le k \le N - 1$
where $\map {e_m} U$ denotes an elementary symmetric function.
== Proof ==
Let:
:$U = \set {x_1, \ldots, x_N}$
Translate Viète's Formulas from notation $a_0$ to $a_N$:
Let $N-k = j$ define a change of index.
Then $k = N-j$.
Apply the change of index:
 
Category:Examples of Use of Viète's Formulas
Viète's Formulas/Examples/Quartic

== Example of Use of Viète's Formulas ==
Consider the quartic equation:
:$x^4 + a_1 x^3 + a_2 x^2 + a_3 x + a_4 = 0$
Let its roots be denoted $x_1$, $x_2$, $x_3$ and $x_4$.
Then:
== Proof ==
A specific instance of Viète's Formulas for $n = 4$.
== Sources ==
* : $3$: Elementary Analytic Methods: $3.8$ Algebraic Equations: Solution of Quartic Equations: $3.8.3$
* : $\S 9$: Solutions of Algebraic Equations: Quartic Equation: $x^4 + a_1 x^3 + a_2 x^2 + a_3 x + a_4 = 0$: $9.8$
* : $\S 5$: Solutions of Algebraic Equations: Quartic Equation: $x^4 + a_1 x^3 + a_2 x^2 + a_3 x + a_4 = 0$: $5.8.$
Category:Examples of Use of Viète's Formulas
Category:Quartic Equations
Viète's Formulas/Examples/Sum 4, Product 8

== Example of Use of Viète's Formulas ==
Let $z_1$ and $z_2$ be two numbers whose sum is $4$ and whose product is $8$.
Then:
== Proof ==
Let $z_1$ and $z_2$ be the roots of the quadratic equation:
:$z^2 + b z + c = 0$
From Viète's Formulas:
and so $z_1$ and $z_2$ are the roots of the quadratic equation:
== Sources ==
* : $1$: Complex Numbers: Supplementary Problems: Polynomial Equations: $104$
Category:Examples of Use of Viète's Formulas
Volume Containing a Solid Foot

== Puzzle ==
:Mathematicians affirm that of all bodies contained under the same superficies, a sphere is the most capacious;
:but they have never considered the amazing capaciousness of a body,
:the name of which is now required,
:of which it may be truly affirmed that supposing:
::its greatest length $9$ inches,
::greatest breadth $4$ inches,
::and greatest depth $3$ inches,
:yet under these dimensions it contains a solid foot.
== Solution ==
A shoe.
== Sources ==
* 
* : Rational Amusements for Winter Evenings: $155$
Category:Trick Puzzles
Volume in Common between Two Equal Cylinders with Perpendicular Intersecting Axes/Historical Note

== Historical Note on Volume in Common between Two Equal Cylinders with Perpendicular Intersecting Axes ==
The Volume in Common between Two Equal Cylinders with Perpendicular Intersecting Axes was found by .
He demonstrates how it was found in Proposition $15$ in his .
== Sources ==
* : Chapter $\text {A}.5$: Archimedes (ca. B.C.)
Category:Historical Notes
Volume of Circular Cylinder

== Theorem ==
=== By Height ===
=== By Slant Height ===
== Sources ==
* : $\S 4$: Geometric Formulas: Circular Cylinder of Radius $r$ and Slant Height $l$
* : $\S 7$: Geometric Formulas: Circular Cylinder of Radius $r$ and Slant Height $l$
Category:Volume of Circular Cylinder
Category:Circular Cylinders
Category:Volume Formulas
Volume of Circular Cylinder/Height

== Theorem ==
Let $\CC$ be a circular cylinder such that:
:the bases of $\CC$ are circles of radius $r$
:the height of $\CC$ is $h$.
The volume $\VV$ of $\CC$ is given by the formula:
:$\VV = \pi r^2 h$
== Proof ==
From Volume of Cylinder in terms of Height and Base Area:
:$\VV = \AA h$
where $\AA$ is the area of the base of $\CC$.
From Area of Circle:
:$\AA = \pi r^2$
The result follows.
== Sources ==
* : $\S 4$: Geometric Formulas: Circular Cylinder of Radius $r$ and Slant Height $l$: $4.33$
* : $\S 7$: Geometric Formulas: Circular Cylinder of Radius $r$ and Slant Height $l$: $7.33.$
Category:Volume of Circular Cylinder
Volume of Circular Cylinder/Slant Height

== Theorem ==
Let $\CC$ be a circular cylinder such that:
:the bases of $\CC$ are circles of radius $r$
:the slant height of $\CC$ is $l$
:the inclination of the generatrices of $\CC$ to the base of $\CC$ is $\theta$.
The volume $\VV$ of $\CC$ is given by the formula:
:$\VV = \pi r^2 l \sin \theta$
== Proof ==
Let $h$ denote the height of $\CC$.
From Relation between Slant Height and Height of Cylinder:
:$h = l \sin \theta$
From Volume of Circular Cylinder in terms of Height:
:$\VV = \pi r^2 h$
The result follows.
== Sources ==
* : $\S 4$: Geometric Formulas: Circular Cylinder of Radius $r$ and Slant Height $l$: $4.33$
* : $\S 7$: Geometric Formulas: Circular Cylinder of Radius $r$ and Slant Height $l$: $7.33.$
Category:Volume of Circular Cylinder
Volume of Circular Paraboloid

== Theorem ==
Let $\CC$ be the region of space between:
:a circular paraboloid $\PP$
:a plane $\SS$ intersecting $\PP$ perpendicular to the axis $\AA$ of the parabola generating $\PP$.
Let $b$ be the radius of the circle which is the intersection of $\PP$ and $\SS$.
Let $a$ be the length of the line segment of $\AA$ between $\SS$ and the vertex of the parabola generating $\PP$.
The volume $\VV$ of $\CC$ is given by:
:$\VV = \dfrac {\pi b^2 a} 2$
== Proof ==
== Sources ==
* : $\S 4$: Geometric Formulas: Paraboloid of Revolution: $4.48$
* : $\S 7$: Geometric Formulas: Paraboloid of Revolution: $7.48.$
Category:Circular Paraboloids
Category:Volume Formulas
Volume of Cone

== Theorem ==
Let $K$ be a cone whose base is of area $A$ and whose height is $h$.
Then the volume of $K$ is given by:
:$V_K = \dfrac {A h} 3$
== Proof ==
Let $V_K$ be the volume of $K$.
Let $V_C$ be the volume of a cylinder of base $A$ and of height $h$.
From Volume of Cylinder in terms of Height and Base Area:
:$V_C = A h$
From Volume of Cone is Third of Cylinder on Same Base and of Same Height:
== Sources ==
* 
* 
Category:Cones
Category:Volume Formulas
Volume of Cone is Third of Cylinder on Same Base and of Same Height/Historical Note

== Historical Note ==
This theorem is Proposition $10$ of Book $\text{XII}$ of .
The treatise of , discovered in $1906$, tells us that this result was discovered by .
== Sources ==
* : Chapter $\text {A}.3$: Democritus (ca. B.C.)
* : Chapter $\text {B}.6$: How Archimedes Discovered Integration
Category:Euclid Book XII
Category:Historical Notes
Volume of Cone is Third of Cylinder on Same Base and of Same Height/Proof 2

== Theorem ==
== Proof ==
Let the cone be of height $h$.
Let the area of the base of the cone be $A$.
From Volume of Cylinder in terms of Height and Base Area, the volume of a cylinder of base $A$ and height $h$ is $A h$.
Let the cone be divided by planes parallel to its base each positioned some small distance $d$ apart.
Let $d$ be sufficiently small that they can be approximated to cylinders in shape.
Let there be $n$ of these small cylinders in total.
Let the volumes of each of these small cylinders be:
:$v_1, v_2, v_3, \ldots, v_n$
starting from the base of the cone and working up.
We have that:
:$v_k = d a_k$
where $a_n$ is the 
Let the areas of the bases of each of these small cylinders be:
:$a_1, a_2, a_3, \ldots, a_n$
starting from the base of the cone and working up.
== Historical Note ==
== Sources ==
* : Chapter $\text {A}.10$: Kepler ()
Category:Volume of Cone is Third of Cylinder on Same Base and of Same Height
Volume of Cuboid

== Theorem ==
Let $\CC$ be a cuboid whose edges are of length $a$, $b$ and $c$.
The volume $V$ of $\CC$ is given by:
:$V = a b c$
== Proof ==
== Sources ==
* : $\S 4$: Geometric Formulas: Rectangular Parallelepiped of Length $a$, Height $l$, Width $c$: $4.26$
* 
* 
* 
* 
* : $\S 7$: Geometric Formulas: Rectangular Parallelepiped of Length $a$, Height $b$, Width $c$: $7.26.$
Category:Cuboids
Category:Volume Formulas
Volume of Cuboid/Mistake

== Source Work ==
:Chapter $5$: Trigonometrical Functions:
::Relationships between Sides and Angles of a Spherical Triangle
This mistake can be seen in the edition as published by Schaum: ISBN 0-07-060224-7 (unknown printing).
== Mistake ==
;Rectangular Parallelepiped of Length $a$, Height $l$, Width $c$
:$4.26 \quad$ Volume $= a b c$
== Correction ==
The title has the height given as $l$, but the formula has it as $b$.
This was been corrected by the time of the $3$rd edition.
== Sources ==
* : $\S 4$: Geometric Formulas: Rectangular Parallelepiped of Length $a$, Height $l$, Width $c$: $4.26$
Category:Mistakes/Mathematical Handbook of Formulas and Tables
Volume of Cylinder

== Theorem ==
=== By Height and Cross-Sectional Area ===
=== By Slant Height and Cross-Sectional Area ===
=== By Height and Base Area ===
=== By Slant Height and Base Area ===
== Sources ==
* : $\S 4$: Geometric Formulas: Cylinder of Cross-sectional Area $A$ and Slant Height $l$
* : $\S 7$: Geometric Formulas: Cylinder of Cross-sectional Area $A$ and Slant Height $l$
Category:Volume of Cylinder
Category:Cylinders
Category:Volume Formulas
Volume of Cylinder/Height and Base Area

== Theorem ==
Let $\CC$ be a cylinder such that:
:the bases of $\CC$ have area $A$
:the height of $\CC$ is $h$.
The volume $\VV$ of $\CC$ is given by the formula:
:$\VV = A h$
== Proof ==
:
Consider a cylinder $C$ whose base is of area $A$ and whose height is $h$.
Consider a cuboid $K$ whose height is $h$ and whose base is also $A$.
Let $C$ be positioned with its base in the same plane as the base of $K$.
By Cavalieri's Principle $C$ and $K$ have the same volume.
From Volume of Cuboid, $K$ has volume given by:
:$V_K = A h$
Hence the result.
== Sources ==
* 
* 
* : $\S 7$: Geometric Formulas: Cylinder of Cross-sectional Area $A$ and Slant Height $l$
Category:Volume of Cylinder
Volume of Cylinder/Height and Cross-Sectional Area

== Theorem ==
Let $\CC$ be a cylinder such that:
:the cross-sections of $\CC$ at right angles to the generatrices of $\CC$ have area $A$
:the height of $\CC$ is $h$
:the inclination of the generatrices of $\CC$ to the base of $\CC$ is $\theta$.
The volume $\VV$ of $\CC$ is given by the formula:
:$\VV = \dfrac {A h} {\sin \theta} = A h \csc \theta$
== Proof ==
== Sources ==
* : $\S 4$: Geometric Formulas: Cylinder of Cross-sectional Area $A$ and Slant Height $l$: $4.35$
Category:Volume of Cylinder
Volume of Cylinder/Mistake/First Edition

== Source Work ==
:
:Chapter $4$: Geometric Formulas
::Cylinder of Cross-sectional Area $A$ and Slant Height $l$
== Mistake ==
:$4.35. \quad$ Volume $= A l = \dfrac {A h} {\sin \theta} = A h \csc \theta$
:$4.36. \quad$ Lateral surface area $= p l = \dfrac {p h} {\sin \theta} = p h \csc \theta$
== Correction ==
The formulae quoted are indeed for the cylinder whose cross-section is of area $A$.
However, the accompanying diagram illustrates the area of the base, not the cross-section.
Also, $p$ needs to be explicitly specified as being the perimeter of the cross-section.
This has been left implicit from the accompanying diagram, which appears to illustrate $p$ as being the perimeter of the base.
Lack of clarity and confusion are likely outcomes.
== Also see ==
* Volume of Cylinder Mistake: Third Edition
== Sources ==
* : $\S 4$: Geometric Formulas: Cylinder of Cross-sectional Area $A$ and Slant Height $l$
Category:Mistakes/Mathematical Handbook of Formulas and Tables
Volume of Cylinder/Mistake/Third Edition

== Source Work ==
:
:Chapter $7$: Geometric Formulas
::Cylinder of Cross-sectional Area $A$ and Slant Height $l$
== Mistake ==
:$7.35. \quad$ Volume $= A h = A l \sin \theta$
:$7.36. \quad$ Lateral surface area $= p h = p l \sin \theta$
== Correction ==
The formulae quoted are for the cylinder whose bases are of area $A$.
However, the title states that they are for the cross-section.
Also, $p$ needs to be explicitly specified as being the perimeter of the base.
This has been left implicit from the accompanying diagram, which does in fact appear to illustrate $p$ as being the perimeter of the base.
Lack of clarity and confusion are likely outcomes.
== Also see ==
* Volume of Cylinder Mistake: First Edition
== Sources ==
* : $\S 7$: Geometric Formulas: Cylinder of Cross-sectional Area $A$ and Slant Height $l$
Category:Mistakes/Mathematical Handbook of Formulas and Tables
Volume of Cylinder/Slant Height and Base Area

== Theorem ==
Let $\CC$ be a cylinder such that:
:the bases of $\CC$ have area $A$
:the slant height of $\CC$ is $l$
:the inclination of the generatrices of $\CC$ to the base of $\CC$ is $\theta$.
The volume $\VV$ of $\CC$ is given by the formula:
:$\VV = A l \sin \theta$
== Proof ==
== Sources ==
* : $\S 7$: Geometric Formulas: Cylinder of Cross-sectional Area $A$ and Slant Height $l$: $7.35.$
Category:Volume of Cylinder
Volume of Cylinder/Slant Height and Cross-Sectional Area

== Theorem ==
Let $\CC$ be a cylinder such that:
:the cross-sections of $\CC$ at right angles to the generatrices of $\CC$ have area $A$
:the slant height of $\CC$ is $l$
The volume $\VV$ of $\CC$ is given by the formula:
:$\VV = A l$
== Proof ==
== Sources ==
* : $\S 4$: Geometric Formulas: Cylinder of Cross-sectional Area $A$ and Slant Height $l$: $4.35$
Category:Volume of Cylinder
Volume of Displaced Fluid equals Volume of Submerged Object

== Physical Law ==
Let a non-buoyant solid body $B$ be completely immersed in a container of fluid $F$.
Then the volume of $F$ displaced by $B$ equals the volume of $B$.
== Proof ==
== Also see ==
* Archimedes' Principle, with which this is often confused.
== Historical Note ==
Category:Hydrostatics