Definitions Two coplanar lines m and n are parallel lines, written m || n , if and only if they have no points in common or they are identical. The segment (or line segment) with endpoints A and B, denoted AB , is the set consisting of the distinct points A and B and all points between A and B. The ray with endpoint A and containing a second point B, denoted AB

! "!!, consists of the

points on AB and all points for which B is between each of them and A. AB! "!!

and AC! "!!

are opposite rays if and only if A is between B and C. A convex set is a set in which every segment that connects points of the set lies entirely in the set. An instance of a conditional is a specific case in which both the antecedent (if part) and the consequent (then part) of the conditional are true. A counterexample of a conditional is a specific case in which the antecedent (if part) is true and the consequent (then part) of the conditional is false. The converse of p! q is q! p The midpoint of a segment AB is the point M on AB with AM=MB A circle is the set of all points in a plane at a certain distance, its radius, from a certain point, its center. The union of two sets A and B, written A! B is the set of elements which are in A, in B, or in both A and B. The intersection of two sets A and B, written A! B is the set of elements which are in both A and B. A polygon is the union of segments in the same plane such that each segment intersects exactly two others, one at each of its endpoints. An angle is the union of two rays that have the same endpoint. VR! "!

is the bisector of !PVQ if and only if VR! "!

(except for point V) is in the interior of !PVQ and m!PVR = m!RVQ If m is the measure of an angle, then the angle is

a. zero if and only if m=0

b. acute if and only if 0<m<90 c. right if and only if m=90 d. obtuse if and only if 90<m<180 e. straight if and only if m=180

If the measures of two angles are m1 and m2, then the angles are

a. complementary if and only if m1+m2=90 b. supplementary if and only if m1+m2=180

Two non-straight and non-zero angles are adjacent angles if and only if a common side is interior to the angle formed by the non-common sides. Two adjacent angles form a linear pair if and only if their non-common sides are opposite rays. Two non-straight angles are vertical angles if and only if the union of their sides is two lines. Two segments, rays, or lines are perpendicular if and only if the lines containing them form a right angle. For a point P not on a line, m, the reflection image of P over line m is the point Q if and only if m is the perpendicular bisector of PQ . For a point P on m, the reflection image of P over line m is P itself. A transformation is a correspondence between two sets of points such that 1. each point in the preimage set has a unique image 2. each point in the image has exactly one preimage The composite of a transformation S and a second transformation T, denoted T ! S , is the transformation that maps each point P onto T(S(P)) A translation, or slide, is the composite of two reflections over parallel lines A rotation is the composite of two reflections over intersecting lines. A vector is a quantity that can be characterized by its direction and magnitude. Let rm be a reflection and T be a translation with positive magnitude and direction parallel to m. Then G = T ! rm is a glide reflection. Two figures F and G are congruent figures, written F ! G , if and only if G is the image of F under an isometry. A plane figure F is a reflection-symmetric figure if and only if there is a line m such that rm (F) = F . The line m is a symmetry line for the figure.

A quadrilateral is a parallelogram if and only if both pairs of its opposite sides are parallel. A quadrilateral is a rhombus if and only if its four sides are equal in length. A quadrilateral is a rectangle if and only if it has four right angles. A quadrilateral is a square if and only if it has four equal sides and four right angles. A quadrilateral is a kite if and only if it has two distinct pairs of consecutive sides of the same length. A quadrilateral is a trapezoid if and only if it has at least one pair of parallel sides. A trapezoid is an isosceles trapezoid if and only if it has a pair of base angles equal in measure. A plane figure F is a rotation-symmetric figure if and only if there is a rotation R with magnitude between 0° and 360° such that R(F)=F. The center of R is the center of symmetry for F A regular polygon is a convex polygon whose angles are all congruent and whose sides are all congruent. An angle is an exterior angle of a polygon if and only if it forms a linear pair with one of the angles of the polygon. The perimeter of a polygon is the sum of the lengths of its sides.

! =Cd

where C is the circumference and d is the diameter of a circle

If a line, n, intersects a plane X at point P, then line n is perpendicular to plane X if and only if n is perpendicular to every line in X that contains P A cylindric solid is the set of points between a region and its translation image in space, including the region and its image. A cylinder is the surface of a cylindric solid whose base is a circle. A prism is the surface of a cylindric solid whose base is a polygon. Given a region (the base) and a point (the vertex) not in the plane of the base, a conic solid is the set of all points on segments joining the vertex and any point on the base.

A pyramid is the surface of a conic solid whose base is a polygon. A cone is the surface of a conic solid whose base is a circle. A sphere is the set of points in space at a certain distance (its radius) from a point (its center). A plane section of a three-dimensional figure is the intersection of that figure with a plane. For a point P which is not on a plane M, the reflection image of P over M is the point Q if and only if M is the perpendicular bisector of segment PQ. For a point P on plane M, the reflection image of P over M is P itself. Two three-dimensional figures F and G are congruent figures if and only if G is the image of F under a reflection or composition of reflections. A three-dimensional figure F is a reflection-symmetric figure if and only if there is a plane M (the symmetry plane) such that rM (F) = F A regular polyhedron is a convex polyhedron in which all faces are congruent regular polygons and the same number of edges intersect at each of its vertices. Two statements p and q are contradictory if and only if they cannot be true at the same time. Let O be a point and k be a positive real number. For any point, P, let S(P) = !P be the point on OP

! "!! with O !P = k "OP . Then S is the size change or size transformation with

center O and magnitude or size-change factor k. Two figures F and G are similar, written F ! G , if and only if there is a composite of size changes and reflections mapping one onto the other. A transformation is a similarity transformation if and only if it is the composite of size changes and reflections. Let a, b, and g be positive numbers. g is the geometric mean of a and b if and only if ag=gb

In right triangle ABC with right angle C, the tangent of !A , written tan(A) , is leg!opposite!!A

leg!adjacent !to!!A

In right triangle ABC with right angle C,

the sine of !A , written sin(A) , is leg!opposite!!Ahypotenuse

the cosine of !A , written cos(A) , is leg!adjacent !to!!A

hypotenuse

The degree measure of a minor arc or semicircle, AB! , of circle O, written mAB! , is the measure of its central angle, !AOB . The degree measure of a major arc, ACB! , of circle O, written mACB! , is 360° ! mAB! . An angle is an inscribed angle in a circle if and only if

a. the vertex of the angle is on the circle, and b. each side of the angle intersects the circle at a point other than at the vertex.

A secant to a circle is a line that intersects the circle in two points. A tangent to a circle is a line in the plane of the circle which intersects the circle in exactly one point (the point of tangency). Formulas Equilateral Polygon Perimeter: p = ns

Where p = perimeter, n = number of sides, s = length of side Rectangle Formula – The area A of a rectangle with dimensions b and h has area

A = bh

Triangle Area Formula: A =12bh

Where b = base, h = height

Trapezoid Area Formula: A =12b1 + b2( )h

Where b1 = one base, b2 = other base, h = height Parallelogram Area Formula: A = bh Where b = base, h = height Circle Circumference Formula: C = !d Where C = circumference, d = diameter

Circle Area Formula: A = !r2 Where A = area, r = radius Lateral Area of a Cylinder: LA = 2!rh Where LA=lateral area, r=radius, h=height Lateral Area of a Cone: LA = !rl Where LA=lateral area, r=radius, l=slant height Surface Area of a Sphere: A = 4!r2 Where A=surface area, r=radius Volume of a Rectangular Solid: V = lwh Where V=volume, l=length, w=width, h=height Volume of a Prism: V = Bh Where V=volume, B=area of base, h=height

Volume of a Pyramid: V =13Bh

Where V=volume, B=area of base, h=height Volume of a Cylinder: V = Bh = !r2h Where V=volume, B=area of base, h=height, r=radius

Volume of a Cone: V =13Bh = 1

3!r2h

Where V=volume, B=area of base, h=height, r=radius

Volume of a Sphere: V =43!r3

Where V=volume, r=radius Distance Formula: Given two points, (x1, y1) and (x2 , y2 ) , the distance between the

points is d = x2 ! x1( )2 + y2 ! y1( )2 Equation for a Circle – The circle with center at (h,k) and radius r is the set of points (x,y) satisfying x ! h( )2 + y ! k( )2 = r2 Number Line Midpoint Formula – On a number line, the coordinate of the midpoint of

the segment with endpoints a and b is a + b2

Coordinate Plane Midpoint Formula – In the coordinate plane, the midpoint of the

segment with endpoints (x1, y1) and (x2 , y2 ) is x1 + x22

, y1 + y22

!"#

$%&

Three Dimension Distance Formula - Given two points, (x1, y1, z1) and (x2 , y2 , z2 ) , the

distance between the points is d = x2 ! x1( )2 + y2 ! y1( )2 + z2 ! z1( )2 Box Diagonal Formula – In a box with dimensions l, w, and h, the length, d, of the diagonal is given by d = l2 + w2 + h2 Equation for a Sphere – The sphere with center at (h,k,j) and radius r is the set of points (x,y,z) satisfying x ! h( )2 + y ! k( )2 + z ! j( )2 = r2

SAS Triangle Area Formula – In any triangle ABC, A =12absin(C)

Postulates Point-Line-Plane Postulate

a. Unique Line Assumption - Through any two points, there is exactly one line. b. Number Line Assumption - Every line is a set of points that can be put into a

one-to-one correspondence with the real numbers, with any point on it corresponding to 0 and any other point corresponding to 1.

c. Dimension Assumption - Given a line in a plane, there is at least one point in the plane that is not on the line. Given a plane in space, there is at least one point in space that is not in the plane.

d. Flat Plane Assumption – If two points lie in a plane, the line containing them lies in the plane.

e. Unique Plane Assumption – Through three non-collinear points, there is exactly one plane.

f. Intersecting Planes Assumption – If two different planes have a point in common, then their intersection is a line.

Distance Postulate

a. Uniqueness Property - On a line, there is a unique distance between two points.

b. Distance Formula - If two points on the line have have coordinates x and y, then the distance between them is x ! y .

c. Additive Property - If B is on AC , then AB + BC = AC. Triangle Inequality Postulate The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Angle Measure Postulate a. Every angle has a unique measure from 0° to 180° b. Given any ray VA

! "! and any real number, r, between 0 and 180, there is an

unique angle BVA in each half-plane of VA! "#

such that m!BVA = r c. If VA

! "!and VB! "!

are the same ray, then m!AVB = 0 d. If VA

! "!and VB! "!

are opposite rays, then m!AVB = 180 e. If VC

! "!!(except for point V) is in the interior of !AVB , then

m!AVC + m!CVB = m!AVB Corresponding Angles Postulate – Suppose two coplanar lines are cut by a transversal

a. If two corresponding angles have the same measure, then the lines are parallel b. If the lines are parallel, then the corresponding angles have the same measure

Reflection Postulate – Under a reflection

a. There is a 1-1 correspondence between points and their images b. Collinearity is preserved c. Betweeness is preserved d. Distance is preserved e. Angle measure is preserved f. Orientation is reversed

Area Postulate

a. Uniqueness Property – Given a unit region, every polygonal region has a unique area

b. Rectangle Formula – The area A of a rectangle with dimensions b and h has area A=bh

c. Congruence Property – Congruent figures have the same area d. Additive Property – The area of the union of two non-overlapping regions is the

sum of the areas of the regions. Volume Postulate

a. Uniqueness Property – Given a unit cube, every polyhedral region has a unique volume.

b. Box Volume Formula – The volume of a box is V = l !w !h c. Congruence Property – Congruent figures have the same volume. d. Additive Property – The volume of the union of two nonoverlapping solids is the

sum of the volumes of the solids. e. Cavalieri’s Principle – Let I and II be two solids included between parallel

planes. If every plnae, P, parallel to the given planes intersects I and II in sections with the same area, then Volume(I)=Volume(II)

Law of Detachment – From a true conditional p! q and a statement or given information, p, you may conclude q. Law of Transitivity – If p! q and q! r , then p! r is true.

Law of the Contrapositive – A conditional, p! q , and its contrapositive, ! q!!! p are either both true or both false. Law of Ruling Out Possibilities – When a statement p or statement q is true, and q is not true, then p is true. Law of Indirect Reasoning – If valid reasoning from a statement p leads to a false conclusion, then p is false.

Means-Extremes Property – If ab=cd

, then ad = bc

Theorems Line Intersection Theorem – Two different lines intersect in at most one point. Linear Pair Theorem – If two angles form a linear pair, then they are supplementary. Vertical Angles Theorem – If two angles are vertical angles, then they have equal measures. Corollary: Vertical angles are also congruent. Two Perpendiculars Theorem – If two coplanar lines are each perpendicular to a third line, then the two lines are parallel to each other. Perpendicular to Parallels Theorem – In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other. Perpendicular Lines and Slopes Theorem – Two non-vertical lines are perpendicular if and only if the product of their slopes is -1. Figure Reflection Theorem – If a figure is determined by certain points, then its reflection image is the corresponding figure determined by the reflection images of those points. Two Reflection Theorem for Translations – If lines m and n are parallel, the translation rm ! rn has magnitude two times the distance between m and n, in the direction from m perpendicular to n. Two Reflection Theorem for Rotations– If line m intersects line n, the rotation rm ! rn has center at the point of intersection of m and n and has magnitude twice the measure of the non-obtuse angle formed by these lines, in the direction n to m. Corresponding Parts of Congruent Figures (CPCF) Theorem – If two figures are congruent, then any pair of corresponding parts are congruent. Corollary: The corresponding parts have equal measure.

A-B-C-D Theorem – Every isometry preserves Angle measure, Betweeness, Collinearity (lines), and Distance (lengths of segments). Equivalence Properties of Congruence Theorem – For any figures F, G and H a. F ! F (congruence is reflexive) b. If , F ! G then G ! F (congruence is symmetric) c. If F ! G and G ! H , then F ! H (congruence is transitive) Segment Congruence Theorem – Two segments are congruent if and only if they have the same length. Angle Congruence Theorem – Two angles are congruent if and only if they have the same measure. AIA Theorem –

a. If two parallel lines are cut by a transversal, then alternate interior angles are congruent (and have same measure).

b. If two lines are cut by a transversal and form congruent alternate interior angles (or angles with the same measure), then the lines are parallel.

Perpendicular Bisector Theorem – If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Uniqueness of Parallels Theorem – Through a point not on a line, there is exactly one line parallel to the given line. Triangle-Sum Theorem – The sum of the measures of a triangle is 180° Quadrilateral-Sum Theorem – The sum of the measures of a convex quadrilateral is 360° Polygon-Sum Theorem – The sum of the measures of a convex n-gon is (n-2)180° Flip-Flop Theorem –

1. If F and G are points and rm (F) = G then rm (G) = F 2. If F and G are figures and rm (F) = G then rm (G) = F

Segment Symmetry Theorem – Every segment has exactly two symmetry lines:

1. its perpendicular bisector 2. the line containing the segment

Side-Switching Theorem – If one side of an angle is reflected over the line containing the angle bisector, its image is the other side of the angle

Angle Symmetry Theorem – The line containing the bisector of an angle is a symmetry line of the angle. Circle Symmetry Theorem – A circle is reflection symmetric to any line through its center. Symmetric Figures Theorem – If a figure is symmetric, then any pair of corresponding parts under the symmetry are congruent (and have the same measure). Isosceles Triangle Symmetry Theorem – The line containing the bisector of the vertex angle of an isosceles triangle is a symmetry line for the triangle. Isosceles Triangle Coincidence Theorem – In an isosceles triangle, the bisector of the vertex angle, the perpendicular bisector of the base, and the median to the base determine the same line. Isosceles Triangle Base Angles Theorem – If a triangle has two congruent sides, then the angles opposite them are congruent (and have the same measure). Equilateral Triangle Symmetry Theorem – Every equilateral triangle has three symmetry lines, which are the bisectors of its angles and are also the perpendicular bisectors of its sides. Equilateral Triangle Angle Theorem – If a triangle is equilateral, then it is equiangular and each angle measures 60°. Quadrilateral Hierarchy Theorem – see diagram on page 319 Kite Symmetry Theorem – The line containing the ends of a kite is a symmetry line for the kite. Kite Diagonal Theorem – The symmetry diagonal of a kite is the perpendicular bisector of the other diagonal and bisects the two angles at the ends of the kite. Rhombus Diagonal Theorem – Each diagonal of a rhombus is the perpendicular bisector of the other diagonal. Theorem – If a quadrilateral is a rhombus, then it is a parallelogram. Trapezoid Angle Theorem – In a trapezoid, consecutive angles between a pair of parallel sides are supplementary. Isosceles Trapezoid Symmetry Theorem – The perpendicular bisector of one base of an isosceles trapezoid is the perpendicular bisector of the other base and a symmetry line for the trapezoid.

Isosceles Trapezoid Theorem – In an isosceles trapezoid, the non-base sides are congruent. Rectangle Symmetry Theorem – The perpendicular bisectors of the sides of a rectangle are symmetry lines for the rectangle. Theorem – If a figure possesses two lines of symmetry intersecting at a point P, then it is rotation-symmetric with a center of symmetry at P. Center of a Regular Polygon Theorem – In any regular polygon, there is a point (its center) which is equidistant from all of its vertices. Regular Polygon Symmetry Theorem – Every regular n-gon possesses

a. n symmetry lines, which are perpendicular bisectors of each of its sides and the bisectors of each of its angles

b. n-fold rotational symmetry Exterior Angle Theorem – In a triangle, the measure of an exterior angle is equal to the sum of the measures of the interior angles at the other two vertices of the triangle. Exterior Angle Inequality – In a triangle, the measure of an exterior angle is greater than the measure of the interior angle at each of the other two vertices. Unequal Sides Theorem – If the two sides of a triangle are not congruent, then the angles opposite them are not congruent. The larger angle is opposite the longer side. Unequal Angles Theorem – If two angles of a triangle are not congruent, then the sides opposite them are not congruent. The longer side is opposite the larger angle. Theorem – If two angles in one triangle are congruent to two angles in another triangle, then the third angles are congruent. SSS Congruence Theorem – If, in two triangles, three sides of one are congruent to three sides of another, then the triangles are congruent. SAS Congruence Theorem – If, in two triangles, two sides and the included angle of one are congruent to two sides and the included angle of another, then the triangles are congruent. ASA Congruence Theorem – If, in two triangles, two angles and the included side of one are congruent to two angles and the included side of another, then the triangles are congruent. AAS Congruence Theorem – If, in two triangles, two angles and a non-included side of one are congruent respectively to two angles and the corresponding non-included side of another, then the triangles are congruent.

Isosceles Triangle Base Angles Converse Theorem – If two angles of a triangle are congruent, then the sides opposite them are congruent. HL Congruence Theorem – If, in two triangles, the hypotenuse and a leg of one are congruent to the hypotenuse and a leg of another, then the triangles are congruent. SsA Congruence Theorem – If two sides and the angle opposite the longer of the two sides in one triangle are congruent, respectively, to two sides and the corresponding angle in another triangle, then the triangles are congruent. Properties of a Parallelogram Theorem – In any parallelogram,

a) opposite sides are congruent b) opposite angles are congruent c) the diagonals intersect at their midpoints

Theorem – The distance between two given parallel lines is constant. Parallelogram Symmetry Theorem – Every parallelogram has 2-fold rotation symmetry about the intersection of its diagonals. Sufficient Conditions for a Parallelogram Theorem – If, in a quadrilateral,

a) one pair of sides is both parallel and congruent, or b) both pairs of opposite sides are congruent, or c) the diagonals bisect each other, or d) both pairs of opposite angles are congruent

then the quadrilateral is a parallelogram. Exterior Angle Theorem – In a triangle, the measure of an exterior angle is equal to the sum of the measures of the interior angles at the other two vertices of the triangle. Exterior Angle Inequality – In a triangle, the measure of an exterior angle is greater than the measure of the interior angle at each of the other two vertices. Unequal Sides Theorem – If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side. Unequal Angles Theorem – If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle. Pythagorean Theorem – In any right triangle with legs of lengths a and b, and hypotenuse of length c, a2 + b2 = c2

Pythagorean Converse Theorem – If a triangle has sides of lengths a, b, and c, and a2 + b2 = c2 , then the triangle is a right triangle. Line-Plane Perpendicularity Theorem – If a line is perpendicular to two different lines at their intersection, then it is perpendicular to the plane that contains those lines. Four Color Theorem – Suppose regions which share a border of some length must have different colors. Then any map of regions on a plane or a sphere can be colored in such a way that only four colors are needed. Midpoint Connector Theorem – The segment connecting the midpoints of two sides of a triangle is parallel to and half the length of the third side. Size Change Theorem – When k>0, the transformation Sk where Sk (x, y) = (kx,ky) is the size change with center (0,0) and magnitude k. Size-Change Distance Theorem – Under a size change with magnitude k>0, the distance between any two image points is k times the distance between their preimages. Size-Change Preservation Properties Theorem – Every size transformation preserves (1) angle measure, (2) betweenness, and (3) collinearity. Figure Size-Change Theorem – If a figure is determined by certain points, then its size-change image is the corresponding figure determined by the size-change images of those points. Similar Figures Theorem – If two figures are similar, then

1. corresponding angles are congruent, and 2. corresponding lengths are proportional

Fundamental Theorem of Similarity – If two figures are similar with ratio of similitude k, then

a. corresponding angle measures are equal b. corresponding lengths and perimeters are in the ratio k c. corresponding areas and surface areas are in the ratio k2 d. corresponding volumes are in the ratio k 3

SSS Similarity Theorem – If three sides of one triangle are proportional to three sides of a second triangle, then the triangles are similar. AA Similarity Theorem – If two angles of one triangle are congruent to two angles of another, then the triangle are similar. SAS Similarity Theorem – If, in two triangles, the ratios of two pairs of corresponding sides are equal and the included angles are congruent, then the triangles are similar.

Side-Splitting Theorem – If a line is parallel to a side of a triangle and intersects the other two sides in distinct points, it splits these sides into proportional segments. Side-Splitting Converse Theorem – If a line intersects two sides of a triangle in distinct points, and it splits these sides into proportional segments, then the line is parallel to the third side of the triangle. Geometric Mean Theorem – The positive geometric mean of the positive numbers a and b is ab Right Triangle Altitude Theorem – In a right triangle,

1. The altitude to the hypotenuse is the geometric mean of the segments into which it divides the hypotenuse.

2. Each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg.

Isosceles Right Triangle Theorem – In an isosceles right triangle, is a leg is x, then the hypotenuse is x 2 30-60-90 Triangle Theorem – In a 30-60-90 (right) triangle, if the shrter leg is x, then the longer leg is x 3 and the hypotenuse is 2x . Arc-Chord Congruence Theorem – In a circle or in congruent circles,

1. If two arcs have the same measure, they are congruent and their chords are congruent

2. If two chords have the same length, their minor arcs have the same measure. Chord-Center Theorem –

1. The line that contains the center of a circle and is perpendicular to a chord bisects the chord.

2. The line that contains the center of a circle and the midpoint of a chord bisects the central angle of the chord.

3. The bisector of the central angle of a chord is the perpendicular bisector of the chord.

4. The perpendicular bisector of a chord of a circle contains the center of the circle. Inscribed Angle Theorem – In a circle, the measure of an inscribed angle is one-half the measure of its intercepted arc. Theorem – An angle inscribed in a semicircle is a right angle. Theorem – In a circle, if two inscribed angles intercept the same arc, then they have the same measure. Angle-Chord Theorem – The measure of the angle formed by two intersecting chords is one-half the sum of the measures of the arcs intercepted by it and its vertical angle.

Angle-Secant Theorem – The measure of an angle formed by two secants intersecting outside a circle is half the difference of the arcs intercepted by the angle. Radius-Tangent Theorem – A line is tangent to a circle if and only if it is perpendicular to the radius at the radius’s endpoint on the circle. Tangent-Chord Theorem – The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. Tangent-Secant Theorem – The measure of the angle between two tangents, or between a tangent and a secant, is half the difference of the intercepted arcs. Secant-Length Theorem – Suppose one secant intersects a circle at A and B, and a second secant intersects the circle at C and D. If the secants intersect at P, then AP ! BP = CP !DP Tangent-Square Theorem – The power of point P for circle O is the square of the length of a tangent to circle O from P. Isoperimetric Theorem – Of all plane figures with the same perimeter, the circle has maximum area (alternatively, of all plane figures with same area, the circle has least perimeter). Isoperimetric Theorem (3 dimensions) – Of all solids with the same surface area, the sphere has maximum volume (alternatively, of all solids with same volume, the sphere has least surface area).