Geometry

Slope of Perpendicular Lines: The −1 Rule

The slope of perpendicular lines follows one rule: the product of the two slopes is −1, so $m_1 \cdot m_2 = -1$. This article covers the negative-reciprocal rule, why it holds, how to find a perpendicular slope from any given slope, the vertical-horizontal exception, and six worked examples.

X and Y Axis in a Graph: Origin & Quadrants

In a graph, the x-axis is the horizontal number line and the y-axis is the vertical number line; they cross at the origin, the point (0, 0). This article explains the two axes, the four quadrants and their sign conventions, and how to plot any point from its ordered pair $(x, y)$, with six worked examples.

y = mx + b: Read Slope & Y-Intercept, Plot a Line

In the equation $y = mx + b$, the number $m$ is the slope (how steeply the line rises or falls) and the number $b$ is the y-intercept (where the line crosses the y-axis). This article shows how to read $m$ and $b$ straight off the equation, plot the line, write the equation from points or a graph, and rearrange any linear equation into this form.

Properties of a Rectangle: Sides, Angles & Diagonals

A rectangle is a quadrilateral with four right angles, opposite sides equal and parallel, and diagonals that are equal in length and bisect each other. This article covers every property of a rectangle by sides, angles, and diagonals, derives the area $l \times w$, perimeter $2(l+w)$, and diagonal $\sqrt{l^2 + w^2}$ formulas, and works through six examples.

Properties of a Kite: Sides, Angles & Diagonals

A kite is a quadrilateral with two pairs of adjacent equal sides, diagonals that cross at right angles, and one pair of equal opposite angles. This article covers every property of a kite by sides, angles, diagonals, and symmetry, derives the area formula $\tfrac{1}{2} \times d_1 \times d_2$, and works through six examples.

Parallel & Perpendicular Lines: Slope Rules

Parallel and perpendicular lines are two relationships you can read straight off slopes: parallel lines have equal slopes ($m_1 = m_2$), and perpendicular lines have slopes that multiply to −1 ($m_1 \cdot m_2 = -1$). This article covers both slope rules, why each works, how to tell lines apart from their equations, the special vertical-line case, and six worked examples.

Undefined Slope: Definition, Equation & Examples

An undefined slope is the slope of a vertical line, where every point shares the same x-coordinate, so the slope formula divides by zero. This article covers why a vertical line's slope is undefined, the equation x = a, the graph, how undefined slope differs from zero slope, and six worked examples.

Skew Lines: Definition, Distance & Examples

Skew lines are two straight lines in three-dimensional space that never intersect and are never parallel, because they lie in different planes. This article covers the definition, why skew lines exist only in 3D, how to spot them in a cube, the conditions that classify two lines, the distance formula, and six worked examples.

Collinear Points: Definition, How to Prove & Examples

Collinear points are three or more points that all lie on the same straight line. This article covers the definition, collinear versus non-collinear points, the three methods to prove collinearity (slope equality, zero triangle area, and the distance test), how each one works, and six worked examples.

Coplanar: Definition, Points, Lines & Examples

Coplanar means lying on the same flat plane: a set of points is coplanar if a single plane can contain all of them, and lines are coplanar if one plane holds both. This article covers the definition, coplanar versus collinear, coplanar and non-coplanar points and lines, how to test for coplanarity, and six worked examples.

Foci of an Ellipse: Definition, Formula, Examples

The foci of an ellipse are two fixed points on its major axis such that, for any point on the ellipse, the sum of its distances to the two foci is constant (equal to 2a). This article covers the definition, where the foci sit, the formula c² = a² − b², six worked examples, and the common mistakes.

Angle Between Two Vectors: Formula & Examples

The angle between two vectors is found from their dot product: $\cos\theta = \dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|,|\mathbf{b}|}$, so $\theta = \cos^{-1}!\left(\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|,|\mathbf{b}|}\right)$. This article covers the formula, where it comes from, 2D and 3D worked examples, the cross-product alternative, six examples, and the common mistakes.

Isosceles Obtuse Triangle: Properties & Examples

An isosceles obtuse triangle has one obtuse angle (between 90° and 180°) and two equal acute angles, with the two sides forming the obtuse angle equal in length. This article covers the definition, why such a triangle is possible, its properties, the area and perimeter formulas, six worked examples, and the common mistakes.

Angle Angle Side (AAS) Congruence: Proof, Examples

Angle Angle Side (AAS) is a triangle congruence rule: if two angles and a non-included side of one triangle equal the corresponding two angles and side of another, the triangles are congruent. This article covers the statement, why it works, the proof from ASA, the difference between AAS and ASA, six worked examples, and the common mistakes.

Exterior Angle Theorem: Statement, Proof, Examples

The exterior angle theorem states that an exterior angle of a triangle equals the sum of the two interior angles not next to it: exterior angle = sum of the two remote interior angles. This article covers the statement, a clean proof, the exterior angle inequality version, six worked examples, the common mistakes, and where the theorem leads.

Consecutive Interior Angles: Theorem & Examples

Consecutive interior angles are the two non-adjacent interior angles that sit on the same side of a transversal, and the Consecutive Interior Angles Theorem says they are supplementary (sum to 180°) exactly when the two lines are parallel. This article covers the definition, the theorem, its converse for proving lines parallel, consecutive angles in a parallelogram, and six worked examples.

Same Side Interior Angles: Theorem & Examples

Same side interior angles are the two angles that lie between two lines and on the same side of the transversal, and when the two lines are parallel they are supplementary (their sum is 180°). This article covers the definition, the same side interior angles theorem and its converse, why they are supplementary, and six worked examples that solve for the supplementary angle.

Linear Pair of Angles: Definition & Axiom

A linear pair of angles is two adjacent angles whose non-common sides form a straight line, so they always add to 180°. This article covers the definition, the linear pair axiom and its converse, how a linear pair differs from supplementary and vertical angles, and six worked examples.

90 Degree Angle: Definition & Construction

A 90 degree angle is a right angle: the exact quarter turn formed when two lines meet perpendicularly, marked with a small square instead of an arc. This article covers the definition, how to construct one with a compass and verify it with the 3-4-5 rule, where right angles hold up buildings, and six worked examples.

45 Degree Angle: Definition & Construction

A 45 degree angle is an acute angle that measures exactly half of a right angle (90° ÷ 2 = 45°), and it is the angle each leg makes with the hypotenuse in an isosceles right triangle. This article covers the definition, how to construct one with a compass and by paper folding, its trigonometric values, where it shows up, and six worked examples.

Area of a Circle: Formula, Derivation & Examples

The area of a circle is the flat space enclosed inside its boundary, given by the formula $A = \pi r^2$, where $r$ is the radius. This article defines the area, derives πr² by unrolling the circle into a triangle, covers area from the diameter and circumference, and works through six examples.

Angle Bisector: Properties & Construction

An angle bisector is a ray, line, or segment that divides an angle into two equal smaller angles. This article covers the definition, the key properties, the compass-and-straightedge construction, the angle bisector of a triangle and the incenter, and six worked examples.

Median of a Triangle: Properties & Formula

The median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, so it always cuts that side into two equal halves. This article covers its definition, properties, the centroid and its 2:1 ratio, the length formula from Apollonius's theorem, six worked examples, and the mistakes students make most.

Interior Angles: Sum Formula & Examples

Interior angles are the angles inside a polygon, one at each vertex. The sum of all of them is $(n-2) \times 180°$ for an $n$-sided polygon, and in a regular polygon each one equals that sum divided by $n$. This article covers the definition, the sum formula and where it comes from, regular versus irregular polygons, the interior angles between parallel lines, and six worked examples.

Scale Factor: Definition, Formula & Examples

The scale factor is the number you multiply every length of a figure by to get the matching length of a similar figure, equal to new length ÷ original length. This article covers the formula, scaling up versus down, dilation on the coordinate plane, the area ($k^2$) and volume ($k^3$) rules, and six worked examples.

Similar Triangles: Theorems & Properties

Similar triangles are triangles with the same shape but not necessarily the same size: their corresponding angles are equal and their corresponding sides are in the same ratio. This article covers the definition, the AA, SAS, and SSS similarity criteria, the properties, the area-ratio rule, the difference from congruent triangles, six worked examples, and the mistakes students make most.

Altitude of a Triangle: Formulas & Properties

The altitude of a triangle is the perpendicular segment from a vertex straight down to the line containing the opposite side, and its length is the height used in the area formula. This article covers the definition, the formulas for scalene, isosceles, equilateral, and right triangles, the orthocentre, six worked examples, and the mistakes students make most.

Equation of a Circle: Standard & General Form

The equation of a circle in standard form is (x − h)² + (y − k)² = r², where (h, k) is the centre and r is the radius. This article covers the standard and general forms, the derivation straight from the Pythagorean theorem, how to read the centre and radius off either form, how to convert between them by completing the square, and six worked examples.

Isosceles Right Triangle: Formulas & Examples

An isosceles right triangle is a right triangle whose two legs are equal, giving angles of 45°, 45°, and 90° and a fixed side ratio of 1 : 1 : √2. This article covers its definition, the 45-45-90 angle structure, area ($\tfrac{x^2}{2}$) and hypotenuse ($x\sqrt{2}$) formulas, six worked examples, and the mistakes students make most.

Alternate Exterior Angles: Theorem & Examples

Alternate exterior angles are the pair of angles that lie outside two lines and on opposite sides of the transversal crossing them. When the two lines are parallel, each pair is equal. This article covers the definition, the theorem and its converse, how to spot the pairs, and six worked examples.

Sector of a Circle: Area, Arc Length, Perimeter

A sector of a circle is the pie-slice region enclosed by two radii and the arc between them, and its area is the fraction of the whole circle set by its central angle: Area = (θ/360°) × πr². This article covers the definition, minor and major sectors, the area, arc length, and perimeter formulas in both degrees and radians, six worked examples, and the common mistakes.

Congruent Angles: Definition, Theorems, Examples

Congruent angles are two or more angles that have the same measure, written $\angle A \cong \angle B$ using the congruence symbol ≅. This article covers the definition and symbol, the theorems that guarantee congruent angles, the compass-and-straightedge construction, and six worked examples.

Y Intercept: Definition, Formula & Examples

The y intercept is the point where a graph crosses the y-axis, found by setting $x = 0$ and solving for $y$. For a line written as $y = mx + b$, the y intercept is simply $b$. This article covers the definition, the method for every equation form, the parabola case, six worked examples, and the mistakes students make most.

Arc Length: Formula, How to Find It, Examples

Arc length is the distance measured along the curved edge of a circle, a fraction of the full circumference set by the central angle: L = (θ/360°) × 2πr in degrees, or L = rθ in radians. This article covers the definition, both formulas, the derivation from the circumference, how to find arc length with and without the angle, six worked examples, and the common mistakes.

Is a Square a Rectangle? Yes — Here's Why

Yes — every square is a rectangle, because a rectangle is defined as a quadrilateral with four right angles, and a square has those four right angles plus the extra condition that all its sides are equal. This article explains the definitions, the quadrilateral family tree, why a square is a special rectangle, why the reverse is not always true, six examples, and the common mistakes.

Perpendicular Bisector: Definition & Construction

A perpendicular bisector of a line segment is a line that cuts the segment into two equal halves and meets it at a right angle ($90°$). Every point on it is equidistant from the two endpoints. This article covers the definition, the properties, the equidistance theorem, the compass-and-straightedge construction, the circumcentre, and six worked examples.

Conic Sections: Types, Formulas & Equations

A conic section is the curve you get when a flat plane slices through a cone, and tilting the slice produces exactly four shapes: the circle, ellipse, parabola, and hyperbola. This article covers the definition, the four types, their eccentricity values, the focus-directrix idea, standard equations, six worked examples, and the mistakes students make most.

Parts of a Circle: Names, Definitions & Diagram

The main parts of a circle are the centre, radius, diameter, circumference, chord, arc, sector, segment, tangent, and secant, every one of them defined by its relationship to the single fixed centre point. This article names and explains each part with a labelled diagram, the formulas tied to them, six worked examples, and the common mistakes students make.

Chord of a Circle: Formula, Theorems, Examples

A chord of a circle is a straight line segment joining any two points on the circle's boundary, and the longest possible chord is the diameter. This article covers the definition, the two formulas for chord length (from the perpendicular distance and from the central angle), the main chord theorems with proof, six worked examples, and the common mistakes.

Triangular Pyramid: Volume, Surface Area & Faces

A triangular pyramid, also called a tetrahedron, is a 3D solid with 4 triangular faces, 6 edges, and 4 vertices — the simplest possible polyhedron. This article covers its properties, the volume formula $V = \tfrac{1}{3} \times \text{base area} \times \text{height}$, the surface area formulas, the net, six worked examples, and the mistakes students make most.

Right Angled Triangle: Properties & Formulas

A right angled triangle is a triangle with one angle of exactly 90°, and the side opposite that angle, the hypotenuse, is always the longest. This article covers its definition, properties, the Pythagorean theorem, area and perimeter formulas, the special 45-45-90 and 30-60-90 types, six worked examples, and the mistakes students make most.

Isosceles Trapezoid: Properties, Area & Examples

An isosceles trapezoid is a four-sided shape with one pair of parallel sides (the bases) and two non-parallel sides (the legs) of equal length, which gives it equal base angles, equal diagonals, and a line of symmetry. This article covers its definition, properties, the area formula $A = \tfrac{1}{2}(a+b)h$, perimeter, diagonals, six worked examples, and where students go wrong.

Types of Angles: Acute, Right, Obtuse, Reflex

The six types of angles by measure are acute (under 90°), right (exactly 90°), obtuse (90° to 180°), straight (180°), reflex (180° to 360°), and full (360°). This article defines each type with a diagram, then covers the angle-pair relationships — complementary, supplementary, adjacent, and vertical — and the common mistakes.

Alternate Interior Angles: Theorem & Examples

Alternate interior angles are the pair of angles that sit between two lines and on opposite sides of the line crossing them, called a transversal. When the two lines are parallel, each pair is equal. This article covers the definition, the theorem and its converse, how to spot the pairs, co-interior angles, and six worked examples.

Slope Intercept Form: y = mx + b & Examples

Slope intercept form is the equation of a straight line written as $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. This article covers how to read $m$ and $b$ straight off the equation, how to find them from points or a graph, how to convert other forms, and six worked examples.

Line Segment: Definition, Properties & Examples

A line segment is a part of a straight line bounded by two distinct endpoints, so it has a fixed, measurable length — unlike a line or a ray, which run on forever. This article covers the definition and notation, how a segment differs from a line and a ray, its properties, the distance formula for length, and six worked examples.

Geometric Shapes: Types, Properties & Examples

Geometric shapes are closed figures built from points, lines, and curves, and they split into two families: flat 2D shapes and solid 3D shapes. This article covers the full list of types, the properties that separate one shape from the next, the area and volume formulas, six worked examples, and where shapes show up around you.

Angle Bisector Theorem: Statement, Proof, Examples

The angle bisector theorem states that the bisector of an angle in a triangle divides the opposite side into two segments whose ratio equals the ratio of the two adjacent sides: BD/DC = AB/AC. This article covers the statement, the internal and external versions, a clean similar-triangles proof, the converse, the length-of-bisector formula, six worked examples, the common mistakes, and where the theorem leads next.

Shapes in Geometry — Complete 2D and 3D Taxonomy

Geometric shapes split into 2D (flat, with length and width) and 3D (solid, with length, width, and height). 2D shapes group into polygons (triangles, quadrilaterals, regular and irregular) and non-polygons (circle, ellipse). 3D shapes group into polyhedra (prisms, pyramids, Platonic solids) and curved solids (sphere, cylinder, cone).

Adjacent Angles — Definition, Properties, and Examples

Adjacent angles are two angles that share a common vertex, share a common side (arm), and do not overlap. They sit next to each other — the word adjacent comes from Latin adjacens, meaning "lying near". Adjacent angles can be any size; they don't have to add to a specific number.

Radians to Degrees — Conversion Table and Formula

To convert radians to degrees, multiply by $180°/\pi$. The formula: $\text{degrees} = \text{radians} \times \dfrac{180°}{\pi}$. This article gives a complete conversion table for every common angle (multiples of $\pi/12$, $\pi/6$, $\pi/4$, $\pi/3$, $\pi/2$ and beyond), three worked examples, the reverse direction, and the most common mistakes.

Diameter of a Circle — Formula, Worked Examples, and Properties

The diameter of a circle is the longest chord, passing through the centre, equal to twice the radius. Formula: $d = 2r$, or $d = C/\pi$ from circumference, or $d = 2\sqrt{A/\pi}$ from area. This article gives the three diameter formulas, three worked examples (Quick, Standard, Stretch), and the historical thread from Archimedes to modern usage.

Vertical Angles — Definition, Theorem, Proof, and Examples

Vertical angles (also called vertically opposite angles) are the pair of non-adjacent angles formed when two straight lines cross at a single point. Sitting opposite each other across the intersection, they share only a vertex — never a side. The Vertical Angles Theorem states that vertical angles are always congruent (equal in measure), no matter how the two lines are oriented.

Transversal — All 8 Angles and Pair Relationships

A transversal is a line that crosses two or more other lines at distinct points. When the transversal crosses two parallel lines, exactly $8$ angles form — grouped into four named pair-relationships (corresponding, alternate interior, alternate exterior, co-interior).

Types of Triangles — Classification Matrix

Triangles are classified two ways — by **side lengths** (equilateral, isosceles, scalene) and by **angle measures** (acute, right, obtuse). Combining the two axes gives a $3 \times 3$ matrix with **seven** valid types and two impossible ones. This article gives the complete matrix, properties of each type, three worked examples, and the impossibilities that come from the triangle angle-sum theorem.

Hexagon Shape — Definition, Types, Properties, and Area Formula

Hexagon is a six-sided closed two-dimensional polygon with six vertices and six interior angles. In a regular hexagon, all six sides are equal, all six interior angles are $120°$, and the sum of interior angles is $720°$. The area of a regular hexagon with side $s$ is $\frac{3\sqrt{3}}{2}s^2$.

Corresponding Angles — Postulate, Pair Table, Examples

Corresponding angles are pairs of angles that occupy the same relative position at each of the two intersections formed when a transversal crosses two lines. When the two lines are parallel, corresponding angles are equal.

Complementary Angles — Definition, Properties, Examples

Complementary angles are any two angles whose measures sum to exactly 90°. The two angles can sit side by side (forming a right angle — a corner) or be drawn anywhere on the page — what matters is the sum. This article covers the definition, properties, the two types (adjacent vs non-adjacent), the right-triangle connection, three worked examples.

Supplementary Angles — Definition, Properties, Examples

Supplementary angles are any two angles whose measures sum to exactly 180°. The two angles can be next to each other (forming a straight line — a linear pair) or completely separate — what matters is the sum, not the position.

Supplementary vs Complementary Angles — A Side-by-Side Comparison

Supplementary angles add to $180°$; complementary angles add to $90°$. This article compares the two side by side, gives three worked examples (Quick, Standard, Stretch), explains the most common mistake, and offers the mnemonic that stops students mixing them up — Complementary forms a Corner, Supplementary forms a Straight line.

Types of Angles — Acute, Right, Obtuse, Reflex

An acute angle measures between $0°$ and $90°$ — but it is only one of five named angle types. This article gives a complete reference table for all five (acute, right, obtuse, straight, reflex), with definitions, diagrams, real-world examples, and the common mistakes students make when sorting them.

Tangent in Geometry — Definition, Formula, Examples

A tangent is a straight line that touches a curve at exactly one point and does not cross it there. This article covers the geometric tangent (focused on the circle), the formulas that describe a tangent line, the two foundational tangent theorems, three worked examples (Quick, Standard, Stretch), and the mistakes students make most often.

Triangular Prism - Volume, Surface Area, Formulas

A triangular prism is a 3D solid with 2 triangular bases and 3 rectangular lateral faces — total 5 faces, 9 edges, 6 vertices. The volume is $V = (\text{area of triangle}) \times L = \tfrac{1}{2}bh \times L$ where $b, h$ are the triangle's base and height, and $L$ is the prism's length. The surface area = sum of the two triangle areas + the three rectangle areas.

Quadrants of Coordinate Plane - I, II, III, IV

The coordinate plane is divided by the x-axis and y-axis into four quadrants, numbered I, II, III, IV counterclockwise starting from the upper right. Each quadrant has a specific sign convention for $(x, y)$: Quadrant I: both positive; II: $x$ negative, $y$ positive; III: both negative; IV: $x$ positive, $y$ negative.

Trapezium - Definition, Properties, Area and Examples

A trapezium (US: trapezoid) is a quadrilateral with one pair of parallel sides. The parallel sides are called bases; the non-parallel sides are legs. The area formula is $A = \tfrac{1}{2}(a + b) \cdot h$ — the average of the parallel sides times the height.

Circumference of a Circle - Formula, Examples

The circumference of a circle is the distance around it — its perimeter. Given the radius $r$, the formula is $C = 2\pi r$. Given the diameter $d = 2r$, equivalently $C = \pi d$. The constant $\pi \approx 3.14159$ is the ratio of any circle's circumference to its diameter — a universal property of all circles.

Pentagon Shape - Properties, Area, and Perimeter

A pentagon is a polygon with 5 sides and 5 interior angles summing to $540°$. A regular pentagon has all sides equal and all angles equal to $108°$ each. Its area formula is $A = \tfrac{1}{4}\sqrt{5(5 + 2\sqrt{5})} \cdot s^2 \approx 1.72 s^2$, and its perimeter is $P = 5s$.

Rectangular Prism - Volume, Surface Area, Formulas

A rectangular prism (also called a cuboid) is a 3D solid with 6 rectangular faces, 12 edges, and 8 vertices. Its volume is $V = l \times w \times h$ (length × width × height), and its surface area is $S = 2(lw + lh + wh)$.

What is Adjacent? Meaning, Adjacent Angles, Solved Examples

In geometry, adjacent means "next to each other" — sharing a common side, edge, or vertex. Adjacent angles share a vertex and a side but don't overlap. Adjacent sides in a polygon share a common vertex. Adjacent in a triangle (with respect to an angle) is the side touching the angle that isn't the hypotenuse.

Congruent (Congruence) - Meaning, Definition, Examples

Congruent means identical in shape AND size. Two figures are congruent if one can be transformed into the other by rigid motions — translation, rotation, reflection — without stretching or shrinking. The symbol is $\cong$. For triangles, the five congruence theorems (SSS, SAS, ASA, AAS, RHS) let you prove congruence without measuring every side and angle.

Horizontal Line - Definition, Equation, and Slope

A horizontal line is a straight line that runs parallel to the x-axis. Its equation has the form $y = b$ (where $b$ is a constant), its slope is exactly $\mathbf{0}$, and it intersects the y-axis at the single point $(0, b)$.

Parabola - Definition, Formula, Graph, Examples

A parabola is the set of all points in a plane that are equidistant from a fixed point (the focus) and a fixed line (the directrix). It's one of four classical conic sections — created by slicing a cone with a plane parallel to its slant side. The standard equation is $y^2 = 4ax$ (horizontal opening) or $(x - h)^2 = 4p(y - k)$ (vertex form).

Ellipse - Equation, Formula, Properties, Graphing

An ellipse is the set of all points in a plane whose distances to two fixed points (called foci) sum to a constant. Its standard equation is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a$ is the semi-major axis and $b$ is the semi-minor axis.

Symmetry in Geometry - Types, Definition, Examples

Symmetry in geometry means a shape looks identical after being transformed — moved, rotated, or flipped. There are three core types: reflection symmetry (mirror image across a line), rotational symmetry (looks the same after rotation by a fixed angle), and point symmetry (every point has a matching point through a central point)

Isosceles Triangle - Definition, Types, Formulas

An isosceles triangle is a triangle with two sides of equal length — called the legs — and one side of different length called the base. The two angles opposite the equal sides (the base angles) are also equal.

Slope of a Line - Formula, Calculation, Examples

The slope of a line - sometimes called the gradient — measures the line's steepness as the ratio of vertical change to horizontal change between any two points: $m = \frac{y_2 - y_1}{x_2 - x_1}$, or "rise over run."

Perpendicular Lines - Definition, Slope, Examples

Trapezoid: Properties, Area, and Formula Guide

Geometric Transformations: Definition, Types and Examples

Rhombus: Properties, Area, and Perimeter

Cylinder — Shape, Formula, Examples

Parallelogram - Properties, Area, and Formulas

Pythagoras Theorem - Formula, Proof, Examples

Pythagoras theorem says a² + b² = c². Learn the formula, four proofs, common mistakes, the 4,000-year-old Babylonian tablet, and worked examples.