Home / Topics / Geometry

Geometry

The study of shapes, sizes, angles, and the properties of space. From ancient Greek constructions to modern computational geometry.

Foundations of Geometry

Geometry is one of the oldest branches of mathematics, dating back to ancient civilizations who needed to measure land, build structures, and navigate the seas. The word "geometry" literally means "earth measurement" (from Greek geo = earth, metron = measure).

Euclid, often called the "Father of Geometry," established geometry as a rigorous deductive system around 300 BCE. His famous work Elements remained the primary geometry textbook for over 2,000 years.

Basic Undefined Terms

Euclid's Five Postulates

  1. A straight line can be drawn between any two points.
  2. A straight line segment can be extended indefinitely.
  3. A circle can be drawn with any center and radius.
  4. All right angles are equal.
  5. If a line intersects two other lines such that the interior angles on one side sum to less than 180°, those two lines will eventually meet on that side (the Parallel Postulate).

Angles and Lines

An angle is formed by two rays sharing a common endpoint (vertex). Angles are measured in degrees (°) or radians.

Types of Angles

Angle Relationships

Parallel Lines and Transversals

When a transversal crosses two parallel lines, it creates eight angles with special relationships:

Triangles

A triangle is a polygon with three sides, three vertices, and three angles. The sum of the interior angles of any triangle is always 180°.

Classification by Sides

Classification by Angles

Triangle Congruence

Two triangles are congruent if they have exactly the same shape and size. The congruence criteria are:

Triangle Similarity

Two triangles are similar if they have the same shape but not necessarily the same size. The similarity criteria are:

Quadrilaterals

A quadrilateral is a polygon with four sides. The sum of interior angles is always 360°.

Types of Quadrilaterals

Circles

A circle is the set of all points in a plane that are equidistant from a fixed point called the center.

Key Terms

Circle Theorems

Area and Perimeter

Perimeter is the total distance around a shape. Area is the amount of space enclosed by a shape.

Common Area Formulas

Rectangle: A = l × w
Triangle: A = ½ × b × h
Circle: A = πr²
Parallelogram: A = b × h
Trapezoid: A = ½(a + b) × h
Rhombus: A = ½ × d₁ × d₂

Example: Find the area of a triangle with base 10 cm and height 6 cm

A = ½ × b × h = ½ × 10 × 6 = 30 cm²

Heron's Formula

For a triangle with sides a, b, and c:

s = (a + b + c) / 2 (semi-perimeter)
A = √(s(s-a)(s-b)(s-c))

Volume and Surface Area

Volume measures the space inside a 3D object. Surface area is the total area of all the faces.

Common 3D Formulas

Cube: V = s³, SA = 6s²
Rectangular Prism: V = lwh, SA = 2(lw + lh + wh)
Cylinder: V = πr²h, SA = 2πr² + 2πrh
Sphere: V = (4/3)πr³, SA = 4πr²
Cone: V = (1/3)πr²h, SA = πr² + πrl
Pyramid: V = (1/3) × Base Area × h

Transformations

A geometric transformation changes the position, size, or orientation of a figure.

Types of Transformations

Translations, reflections, and rotations are rigid transformations (isometries) — they preserve both size and shape. Dilations change the size but preserve the shape.

Coordinate Geometry

Coordinate geometry (analytic geometry) combines algebra and geometry using the coordinate plane.

Distance Formula

d = √((x₂ - x₁)² + (y₂ - y₁)²)

Midpoint Formula

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

Equation of a Circle

(x - h)² + (y - k)² = r²

Where (h, k) is the center and r is the radius.

Example: Find the distance between (1, 2) and (4, 6)

d = √((4-1)² + (6-2)²) = √(9 + 16) = √25 = 5

The Pythagorean Theorem

Perhaps the most famous theorem in all of mathematics:

a² + b² = c²

In a right triangle, the square of the hypotenuse (c) equals the sum of the squares of the other two sides (a and b).

Pythagorean Triples

Sets of three positive integers that satisfy a² + b² = c²:

Converse of the Pythagorean Theorem

If a² + b² = c² for the sides of a triangle, then the triangle is a right triangle. We can also determine:

Example: A ladder is 13 feet long and leans against a wall. If the base is 5 feet from the wall, how high does the ladder reach?

5² + h² = 13²

25 + h² = 169

h² = 144

h = 12 feet

Geometric Proofs

A geometric proof is a logical argument that uses definitions, postulates, and previously proven theorems to demonstrate that a geometric statement is true. Proofs are at the heart of mathematical thinking and develop rigorous reasoning skills.

Two-Column Proofs

The most common proof format in school geometry. Each step consists of a statement and a reason.

Example: Prove that vertical angles are equal

Given: Two lines intersect, forming angles 1, 2, 3, 4.

Prove: ∠1 = ∠3 (vertical angles)

1. ∠1 + ∠2 = 180° — (Linear pair postulate)

2. ∠2 + ∠3 = 180° — (Linear pair postulate)

3. ∠1 + ∠2 = ∠2 + ∠3 — (Substitution from steps 1 & 2)

4. ∠1 = ∠3 — (Subtraction property of equality) ✓

Paragraph Proofs

A proof written as a flowing paragraph, common in higher mathematics.

Example: Prove the exterior angle of a triangle equals the sum of the two non-adjacent interior angles

Let triangle ABC have an exterior angle at vertex C, call it ∠ACD, formed by extending side BC to point D. Since ∠ACB and ∠ACD form a linear pair, we have ∠ACB + ∠ACD = 180°. We also know the angle sum property: ∠A + ∠B + ∠ACB = 180°. Setting these equal: ∠A + ∠B + ∠ACB = ∠ACB + ∠ACD. Subtracting ∠ACB from both sides gives ∠ACD = ∠A + ∠B. ✓

Proof by Contradiction (Indirect Proof)

Assume the opposite of what you want to prove, then show this assumption leads to a contradiction.

Example: Prove that a triangle cannot have two right angles

Suppose a triangle has two right angles: ∠A = 90° and ∠B = 90°.

Then ∠A + ∠B = 180°. But the angle sum of a triangle is 180°, so ∠C = 180° − 180° = 0°.

An angle of 0° is impossible in a triangle — contradiction.

Therefore, a triangle cannot have two right angles. ✓

When writing proofs, every step must be justified. The main types of justifications are: definitions, postulates (accepted without proof), previously proven theorems, and algebraic properties (equality, inequality).

Important Theorems to Know

Compass and Straightedge Constructions

Classical constructions use only two tools: a compass (for drawing circles and arcs) and a straightedge (for drawing straight lines, without markings). These constructions date back to the ancient Greeks.

Fundamental Constructions

Constructing Regular Polygons

Some regular polygons can be constructed with compass and straightedge:

Gauss proved in 1796 that a regular polygon with n sides is constructible if and only if n is a product of a power of 2 and distinct Fermat primes (primes of the form 2^(2^k) + 1). The known Fermat primes are 3, 5, 17, 257, and 65537.

Regular Polygons

A regular polygon has all sides equal and all interior angles equal.

Interior Angles

Sum of interior angles = (n − 2) × 180°
Each interior angle = (n − 2) × 180° / n

Exterior Angles

Sum of exterior angles = 360° (always, for any convex polygon)
Each exterior angle = 360° / n

Example: Find the interior angle of a regular octagon (8 sides)

Each interior angle = (8 − 2) × 180° / 8 = 6 × 180° / 8 = 1080° / 8 = 135°

Area of a Regular Polygon

A = (1/2) × perimeter × apothem = (1/2) × n × s × a

Where s is the side length, n is the number of sides, and a is the apothem (distance from center to the midpoint of a side).

Number of Diagonals

D = n(n − 3) / 2

Example: How many diagonals does a decagon (10 sides) have?

D = 10(10 − 3)/2 = 10 × 7/2 = 35 diagonals

Common Regular Polygons

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane at different angles. They are among the most important curves in mathematics and physics.

The Four Conic Sections

Circle

(x − h)² + (y − k)² = r²

Center (h, k), radius r. A special case of an ellipse where both axes are equal.

Ellipse

(x − h)²/a² + (y − k)²/b² = 1

Center (h, k), semi-major axis a, semi-minor axis b. The sum of distances from any point on the ellipse to the two foci is constant: 2a.

Foci: c² = a² − b² (where c is the focal distance)
Eccentricity: e = c/a (0 < e < 1 for an ellipse)

Example: Find the foci of x²/25 + y²/9 = 1

a² = 25, b² = 9, so c² = 25 − 9 = 16, c = 4

Since a² is under x², the major axis is horizontal.

Foci: (±4, 0)

Parabola

Standard forms (vertex at origin):

y² = 4px (opens right if p > 0, left if p < 0)
x² = 4py (opens up if p > 0, down if p < 0)

The focus is at distance p from the vertex, and the directrix is at distance p on the opposite side.

Parabolas have a remarkable reflective property: any signal coming parallel to the axis of symmetry reflects off the parabola and passes through the focus. This is why satellite dishes, telescope mirrors, and car headlights are parabolic.

Hyperbola

(x − h)²/a² − (y − k)²/b² = 1 (opens left/right)
(y − k)²/a² − (x − h)²/b² = 1 (opens up/down)

The difference of distances from any point on the hyperbola to the two foci is constant: 2a.

Foci: c² = a² + b²
Asymptotes: y − k = ±(b/a)(x − h) (for horizontal transverse axis)

Example: Find the asymptotes of x²/16 − y²/9 = 1

a² = 16 → a = 4, b² = 9 → b = 3

Asymptotes: y = ±(3/4)x

Identifying Conic Sections

The general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 can be classified (when B = 0):

Vectors in Geometry

Vectors provide a powerful bridge between geometry and algebra, enabling concise proofs and computations.

Position Vectors

Every point P = (x, y) in the plane corresponds to the position vector OP⃗ = (x, y) from the origin to P.

Vector Proofs

Many geometric theorems can be proved elegantly using vectors:

Example: Prove that the diagonals of a parallelogram bisect each other

Let parallelogram ABCD have A at origin, B at vector b, D at vector d.

Then C = b + d (since ABCD is a parallelogram).

Midpoint of AC = (A + C)/2 = (0 + b + d)/2 = (b + d)/2

Midpoint of BD = (B + D)/2 = (b + d)/2

Since both midpoints are equal, the diagonals bisect each other. ✓

Applications: Section Formula

The point dividing segment AB in the ratio m:n is:

P = (nA + mB) / (m + n)

Example: Find the point that divides A(2, 3) and B(8, 15) in the ratio 1:2

P = (2·(2,3) + 1·(8,15)) / (1 + 2) = ((4,6) + (8,15)) / 3 = (12, 21)/3 = (4, 7)

Non-Euclidean Geometry

For over 2,000 years, mathematicians attempted to prove Euclid's fifth postulate (the parallel postulate) from the other four axioms. In the 19th century, Nikolai Lobachevsky, János Bolyai, and Bernhard Riemann independently discovered that replacing the parallel postulate with different axioms produced perfectly consistent geometries — with startling properties.

Hyperbolic Geometry (Lobachevsky/Bolyai)

In hyperbolic geometry, through a point not on a given line, there are infinitely many lines parallel to the given line.

Elliptic (Spherical) Geometry (Riemann)

In elliptic geometry, there are no parallel lines — every pair of lines intersects.

Einstein's General Relativity showed that the geometry of spacetime is not Euclidean — massive objects curve spacetime, and the resulting geometry is Riemannian. Non-Euclidean geometry went from a curiosity to the mathematical foundation of our understanding of gravity.

Comparison Table

Explore Geometry Lessons

Dive deeper into specific geometry topics with our focused lessons:

📐 Triangles & Proofs Classification, congruence, similarity, the Pythagorean theorem, and proof techniques. ⭕ Circles & Measurement Arcs, sectors, inscribed angles, area formulas for all shapes, and 3D volume. 📍 Coordinate Geometry Distance, midpoint, lines, conic sections, and geometric transformations.