Euclidean geometry facts for kids

"Plane geometry" redirects here. For other uses, see Plane geometry (disambiguation).

Detail from Raphael's The School of Athens showing a Greek mathematician, possibly Euclid, drawing with a compass.

Euclidean geometry is a way of understanding shapes, sizes, and positions of objects in space. It's a mathematical system created by Euclid, an amazing ancient Greek mathematician. He wrote it all down in his famous textbook called Elements. Euclid's method starts with a few simple ideas, called axioms or postulates, that seem obviously true. From these, he figured out many other important rules and facts, called propositions or theorems. One special rule is the parallel postulate, which talks about parallel lines on a flat surface. Even though some of Euclid's ideas were known before, he was the first to put them all together in a logical system. In this system, every new idea was proven using the basic rules and earlier proven facts.

Euclid's Elements begins with plane geometry, which is about flat shapes like squares and circles. You still learn this in secondary school (high school)! It's often the first time you see how math can be built from basic rules. The book then moves on to solid geometry, which deals with 3D shapes like cubes and spheres. Many parts of Elements also explain ideas that we now call algebra and number theory, but Euclid described them using geometric shapes.

For a very long time, people thought Euclid's rules were the only way geometry could work. They seemed so true! But later, mathematicians discovered other types of geometry, called non-Euclidean geometries, like hyperbolic and elliptic geometry. These new geometries show that space isn't always exactly "Euclidean." For example, Albert Einstein's theory of general relativity suggests that the space around us isn't perfectly flat like Euclid imagined, especially near very heavy objects. However, Euclidean geometry is still a great way to understand space over short distances or where gravity isn't super strong.

Euclidean geometry is like building a house from the ground up using logical steps. This is different from analytic geometry, which René Descartes invented much later. Analytic geometry uses coordinates (like on a graph) and algebraic formulas to describe shapes and their properties.

Euclid's Famous Book: The Elements
Important Discoveries in Geometry
Geometry in the Real World
- Using Geometry in Engineering
- Geometry in Everyday Life
How Geometry Changed Over Time
Images for kids
See also
- Classical theorems
See also

Euclid's Famous Book: The Elements

Euclid's book, The Elements, brought together a lot of geometry knowledge that existed before him. His way of organizing it was so good that most older geometry books were forgotten.

The Elements has 13 books, each focusing on different parts of geometry:

Books I–IV and VI: These books are all about plane geometry. They cover shapes you can draw on a flat surface. For example, Euclid proved that "in any triangle, two angles added together are always less than two right angles." He also proved the famous Pythagorean theorem, which says that in a right-angled triangle, the square on the longest side (hypotenuse) is equal to the sum of the squares on the other two sides.
Books V and VII–X: These books dive into number theory. Euclid looked at numbers using geometry, like thinking of them as lengths of lines or areas of surfaces. He introduced ideas like prime numbers and rational and irrational numbers. He even proved that there are endlessly many prime numbers!
Books XI–XIII: These last books explore solid geometry. This is about 3D shapes. One cool result is that a cone has one-third the volume of a cylinder if they have the same height and base. He also showed how to build the platonic solids, which are special 3D shapes with identical faces.

Basic Rules: Axioms and Postulates

The parallel postulate (Postulate 5): If two lines cross a third line and the inner angles on one side add up to less than two right angles, then those two lines will eventually meet on that side if you extend them far enough.

Euclidean geometry is built on a few simple starting rules, called axioms or postulates. Think of them as the basic truths that everyone agrees on, without needing a proof. From these few rules, all other geometry facts (theorems) are figured out. For a long time, people thought these rules were obviously true in the real world.

Near the beginning of his first book, Euclid listed five postulates for plane geometry. These are like instructions for drawing:

You can draw a straight line from any point to any other point.
You can extend any straight line segment continuously in a straight line.
You can draw a circle with any center point and any distance (radius).
All right angles are equal to one another.
The parallel postulate: If a straight line crosses two other straight lines, and the inner angles on one side add up to less than two right angles (less than 180 degrees), then those two lines will eventually meet on that side if you extend them far enough.

Euclid also had five "common notions," which are general truths not just for geometry:

Things that are equal to the same thing are also equal to each other.
If you add equal things to equal things, the totals are equal.
If you subtract equal things from equal things, the differences are equal.
Things that perfectly match each other are equal.
The whole is always greater than any of its parts.

The Special Parallel Postulate

The parallel postulate always seemed a bit different from the others. It wasn't as "obvious" to ancient mathematicians. They wondered if it could be proven from the other simpler rules. Today, we know it cannot be proven. This is because you can create other types of geometry (called non-Euclidean geometries) where the other axioms are true, but the parallel postulate is false! Euclid himself seemed to know it was different, as he proved his first 28 ideas without using it.

Many other statements are actually the same as the parallel postulate. For example, Playfair's axiom says:

In a flat surface, through a point not on a given straight line, you can draw at most one line that will never meet the given line.

The "at most one" part is key, because we can prove that at least one parallel line always exists using the other axioms.

How Euclid Proved Things

A proof from Euclid's Elements showing how to build an equilateral triangle (ΑΒΓ) using a compass and straightedge. You draw circles around points Α and Β, and where they cross is the third point of the triangle.

Euclidean geometry is very much about building things. Euclid's rules (postulates) don't just say that certain shapes exist; they also give you ways to create them using only a compass and an unmarked straightedge. This means you can actually draw and construct the shapes he talks about.

Euclid often used a clever method called proof by contradiction. This is where you pretend the opposite of what you want to prove is true. If that assumption leads to something impossible or silly, then you know your original idea must have been correct all along!

Understanding Geometric Language

Naming Points and Shapes

In geometry, we usually name points with capital letters, like A, B, or C. Shapes like lines, triangles, or circles are named by listing enough points to identify them clearly. For example, a triangle with corners at points A, B, and C would be called triangle ABC.

Complementary and Supplementary Angles

Complementary angles are two angles that add up to a right angle (90 degrees). Imagine a corner of a square; if you draw a line from that corner, it splits the 90 degrees into two complementary angles.
Supplementary angles are two angles that add up to a straight angle (180 degrees). Think of a straight line; if you draw another line from a point on it, it creates two supplementary angles.

Modern Ways to Talk About Lines

Today, we often measure angles in degrees (like 90 degrees for a right angle) or radians.

In school, you learn about different types of lines:

A line goes on forever in both directions.
A ray starts at one point and goes on forever in one direction.
A line segment has a definite start and end point.

Euclid didn't always use these exact terms. He might say "if the line is extended to a sufficient length" instead of talking about a ray. For Euclid, a "line" could even be curved, so he would say "straight line" when he meant a truly straight one.

Important Discoveries in Geometry

The pons asinorum (or bridge of asses) shows that in an isosceles triangle, the angles opposite the equal sides are also equal.
The triangle angle sum theorem states that the three angles inside any triangle always add up to 180 degrees.
The Pythagorean theorem says that in a right triangle, the square of the longest side (c) is equal to the sum of the squares of the other two sides (a and b).
Thales' theorem states that if you draw a triangle inside a circle, and one side of the triangle is the circle's diameter, then the angle opposite the diameter is always a right angle (90 degrees).

Pons Asinorum: The Bridge of Asses

The pons asinorum is a fancy name for a theorem that says: if a triangle has two sides of equal length (an isosceles triangle), then the angles opposite those sides are also equal. It's sometimes called the "bridge of asses" because it was often the first real challenge for students learning geometry. If you could understand this, you were ready for harder problems!

Congruent Triangles

Triangles are congruent if they have matching sides and angles. For example, if two sides and the angle between them (SAS) are the same, the triangles are congruent.

Two triangles are called congruent if they are exactly the same size and shape. You can tell if triangles are congruent if:

All three sides are equal (SSS).
Two sides and the angle between them are equal (SAS).
Two angles and the side between them are equal (ASA).
Two angles and a side that is not between them are equal (AAS).

If triangles only have three equal angles (AAA), they are similar (same shape, different size), but not necessarily congruent.

Triangle Angle Sum

A very useful rule is that the three angles inside any triangle always add up to 180 degrees. This means that an equilateral triangle (all sides equal) must have three 60-degree angles. It also means every triangle has at least two acute angles (less than 90 degrees).

The Pythagorean Theorem

The famous Pythagorean theorem is about right triangles (triangles with one 90-degree angle). It states that the area of the square built on the longest side (called the hypotenuse) is equal to the sum of the areas of the squares built on the other two sides (called the legs). If the legs are 'a' and 'b', and the hypotenuse is 'c', then $a^2 + b^2 = c^2$ .

Thales' Theorem

An example of congruence. The two shapes on the left are congruent (same size and shape). The third is similar (same shape, different size). The last one is different. Congruent shapes can be moved to perfectly match each other.

Thales' theorem, named after the ancient Greek mathematician Thales of Miletus, says that if you draw a triangle inside a circle, and one side of the triangle is the circle's diameter, then the angle opposite the diameter is always a right angle (90 degrees).

Measuring in Euclidean Geometry

Euclidean geometry uses two main types of measurements: angles and distances.

Angles are measured using a standard unit, like a right angle (90 degrees). So, a 45-degree angle would be half of a right angle.
Distances are measured by picking a certain length as a unit (like 1 centimeter). All other distances are then compared to this unit. You can "add" distances by putting line segments end-to-end.

Measurements for area (like for a square) and volume (like for a cube) come from these distances. For example, a rectangle that is 3 units wide and 4 units long has an area of 12 square units.

Euclid used the word "equal" (ἴσος) if two shapes had the same length, area, or volume. The word "congruent" means two figures are exactly the same size and shape. You could move one on top of the other, and they would match perfectly. For example, two squares with 5-inch sides are congruent.

Shapes that are the same shape but different sizes are called similar. For similar shapes, their corresponding angles are equal, and their corresponding sides are in proportion (meaning one shape is just a scaled-up or scaled-down version of the other).

Geometry in the Real World

Using Geometry in Engineering

Euclidean geometry is super important in many areas of engineering!

Dynamics: How Things Move

Vibration Analysis: Geometry helps engineers understand and control vibrations in machines and buildings. This is key for designing things that are strong and work well.

Vibration - oscillations

Aircraft Wing Design: The shape of aircraft wings is carefully designed using geometry. This shape directly affects how much lift (the force that keeps a plane in the air) and drag (the force that slows it down) the wing creates.

Airfoil nomenclature

Satellite Orbits: Geometry is used to calculate and predict the paths, or orbits, of satellites around Earth. This is vital for successful space missions and keeping satellites working.

Animation of orbit by eccentricity

CAD Systems: Designing with Computers

3D Modeling: In CAD (computer-aided design) systems, Euclidean geometry is the basic tool for making accurate 3D models of parts. These models help engineers see and test designs before they are actually built.
Design and Manufacturing: Much of CAM (computer-aided manufacturing) also relies on Euclidean geometry. The shapes in CAD/CAM are often made of planes, cylinders, cones, and other Euclidean forms. Today, CAD/CAM is used to design everything from cars and airplanes to ships and smartphones.

3D CAD model

Circuit Design: Electronics Layouts

PCB Layouts: Printed circuit board (PCB) design uses Euclidean geometry to place and connect electronic parts efficiently. A good layout helps prevent signal interference and makes the circuit work better.

PCB of a DVD player

Antenna Design and Fields

Antenna Design: The geometry of antennas helps engineers design them so they can send and receive electromagnetic waves effectively.

NASA Cassegrain antenna.

Field Theory: In understanding how inviscid flow (like water without friction) and electromagnetic fields behave, geometry helps visualize and solve problems in 3D space.

Control Systems

Control System Analysis: Euclidean geometry helps in designing and analyzing control systems, which are used to manage and optimize how other systems behave, like in robots or automated factories.

Basic feedback loop.

Geometry in Everyday Life

A surveyor uses a level to measure land.
Sphere packing helps arrange objects like oranges efficiently.
A parabolic mirror focuses parallel light rays to a single point.

Surveying: One of the oldest uses of geometry is surveying, which is measuring and mapping land. Surveyors use tools like levels and theodolites to measure distances and angles.
Packing Problems: Geometry helps figure out the best ways to pack objects, like how to stack spheres (think of oranges in a box) to take up the least amount of space. This is useful in many areas, including fixing errors in data.
Art and Architecture: Geometry is used a lot in architecture and art to create beautiful and stable designs.

Geometry is used in art and architecture.
The water tower consists of a cone, a cylinder, and a hemisphere. Its volume can be calculated using solid geometry.
Geometry can be used to design origami.

Origami: Even origami (paper folding) uses geometry! Some classic geometry problems that are impossible with just a compass and straightedge can actually be solved using paper folding.

How Geometry Changed Over Time

Archimedes and Apollonius

A sphere has 2/3 the volume and surface area of its circumscribing cylinder. A sphere and cylinder were placed on the tomb of Archimedes at his request.

Archimedes (around 287–212 BCE) was another brilliant ancient mathematician, famous for his original work. He figured out formulas for the volumes and areas of many 2D and 3D shapes.

Apollonius of Perga (around 240–190 BCE) is best known for his studies of conic sections, which are shapes like circles, ellipses, parabolas, and hyperbolas.

Descartes and Analytic Geometry

René Descartes. Portrait after Frans Hals, 1648.

In the 17th century, René Descartes (1596–1650) created analytic geometry. This was a new way to do geometry by connecting it to algebra.

With analytic geometry, a point on a flat surface is described by its coordinates (like an (x, y) pair on a graph). A line is described by an equation. This allowed mathematicians to use algebraic formulas to solve geometric problems. For example, the distance between two points P=(p_x, p_y) and Q=(q_x, q_y) can be found using the formula:

|PQ|=\sqrt{(p_x-q_x)^2+(p_y-q_y)^2} \,

This formula is called the Euclidean metric.

Non-Euclidean Geometries and Einstein

A photograph from a 1919 solar eclipse showing how starlight (marked with lines) was bent by the Sun's gravity. This supported Einstein's idea that space is not perfectly Euclidean.

In the 19th century, mathematicians like János Bolyai and Nikolai Ivanovich Lobachevsky discovered non-Euclidean geometry. These were geometries where the parallel postulate was not true. This showed that Euclid's parallel postulate couldn't be proven from his other rules.

Later, in the 20th century, Albert Einstein's theory of general relativity showed that the space around us isn't perfectly Euclidean, especially near strong gravity. For example, light rays can bend near massive objects, meaning that a triangle made of light rays might not have angles that add up to exactly 180 degrees. This was proven by observations during a solar eclipse in 1919. These ideas are now part of the technology we use every day, like the GPS.

Images for kids

Squaring the circle: This problem, trying to make a square with the same area as a given circle using only a compass and straightedge, was proven impossible in 1882.
A comparison of different types of geometry: Euclidean (flat), Spherical (like a ball), and Hyperbolic (like a saddle).
A sphere has 2/3 the volume and surface area of its circumscribing cylinder. A sphere and cylinder were placed on the tomb of Archimedes at his request.
René Descartes. Portrait after Frans Hals, 1648.