Geometry — A Complete Reference

1. What is Geometry?

Geometry is the branch of mathematics that studies shapes, sizes, positions, and the properties of space. Its roots reach back at least 4000 years to the land-surveyors of Egypt and Babylon, and the subject was first organized into a rigorous deductive system by Euclid of Alexandria around 300 BCE. Euclid’s Elements remained the world’s most influential textbook for more than two millennia, and the axiomatic style it pioneered became the template for all modern mathematics.

Classical Euclidean geometry builds the entire plane from a small list of primitive notions: point, line, plane, between, and congruent. A point has position but no size; a line extends infinitely in two directions; a plane is a perfectly flat surface that stretches forever. From these primitives and five postulates — most famously the parallel postulate, that through any external point exactly one line can be drawn parallel to a given line — Euclid derived hundreds of theorems about triangles, circles, and polygons that you still meet in school today.

Modern geometry has branched far beyond Euclid. Projective geometry studies what stays the same when shapes are projected from one plane to another, and underlies perspective drawing and computer vision. Differential geometry uses calculus to describe curved surfaces and is the language of Einstein’s general relativity. Computational geometry designs efficient algorithms for shape problems, powering everything from CAD software to robotic motion planning. All of these descendants still rest on the Euclidean foundations introduced here.

기하학은 도형·크기·위치·공간의 성질을 다루는 수학 분야로, 약 4000년 전 이집트·바빌론의 측량술에서 시작해 기원전 300년경 유클리드의 『원론』에서 처음으로 공리적 체계로 정리되었습니다. 점·직선·평면·"사이"·"합동"이라는 기본 개념과 평행선 공준을 포함한 다섯 개의 공준으로부터 모든 정리를 연역하는 방식은 이후 모든 현대 수학의 본보기가 되었으며, 사영기하·미분기하·전산기하 같은 현대적 분과들도 모두 이 토대 위에 서 있습니다.

2. Angles & Lines

An angle is the figure formed by two rays sharing a common endpoint, called the vertex. We measure angles in either degrees (a full turn is 360°) or radians (a full turn is 2π). The bridge between the two is the identity π radians = 180°, which lets you convert via degrees × (π / 180) = radians and radians × (180 / π) = degrees. Calculus prefers radians because they make derivative formulas like (sin x)′ = cos x clean; everyday speech prefers degrees because they’re more intuitive.

Angles are classified by size. An acute angle is smaller than 90°, a right angle is exactly 90°, an obtuse angle is between 90° and 180°, a straight angle is exactly 180°, and a reflex angle is greater than 180° but less than 360°. Two angles whose measures add to 90° are called complementary; two that add to 180° are supplementary. These names appear constantly in proofs and trigonometric identities.

When a single straight line — a transversal — cuts across two parallel lines, eight angles are formed and pair up in three useful ways. Corresponding angles (same position at each intersection) are equal. Alternate-interior angles (between the parallel lines, on opposite sides of the transversal) are equal. Co-interior angles (between the parallel lines, on the same side) are supplementary. Two lines are perpendicular when they meet at a right angle; in the coordinate plane, the slopes of perpendicular lines multiply to −1 — a relationship we revisit in §8.

각은 도(°)와 라디안(rad)으로 재며 π rad = 180°가 변환 다리입니다. 예각(<90°), 직각(=90°), 둔각(90°~180°), 평각(=180°)로 분류하고, 합이 90°이면 여각·180°이면 보각이라 부릅니다. 평행한 두 직선이 한 횡단선에 잘릴 때 동위각·엇각은 서로 같고, 동측내각은 보각 관계를 이룹니다. 두 직선이 직각으로 만나면 수직이며, 좌표평면에서는 기울기의 곱이 −1입니다.

3. Triangles — Types & Area

A triangle is a closed shape with three straight sides and three interior angles whose sum is always 180°. Triangles are classified two ways. By side lengths: a scalene triangle has three different sides, an isosceles triangle has two equal sides (and two equal base angles), and an equilateral triangle has all three sides equal and all three angles equal to 60°. By angle measures: a right triangle has one 90° angle, an acute triangle has all three angles below 90°, and an obtuse triangle has one angle greater than 90°. Every triangle belongs to exactly one category in each scheme — for example, a 3-4-5 triangle is both scalene and right.

The most familiar area formula is S = ½ · b · h, where b is any chosen base and h is the perpendicular height from the opposite vertex to the line containing that base. This works for every triangle, not just right triangles — you simply drop a perpendicular from the apex. When you know all three side lengths but no height, use Heron’s formula: A = √(s(s−a)(s−b)(s−c)), where s = (a+b+c)/2 is the semi-perimeter. For a 5-6-7 triangle, s = 9 and A = √(9·4·3·2) = √216 ≈ 14.70.

When the vertices are given as coordinates instead, the shoelace formula computes area directly: A = ½|x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)|. Worked example. Take the triangle with vertices (0, 0), (4, 0), (2, 3). Then A = ½|0·(0−3) + 4·(3−0) + 2·(0−0)| = ½|0 + 12 + 0| = 6. The same answer comes from ½ · b · h with base = 4 and height = 3, confirming both methods agree.

삼각형의 세 내각의 합은 항상 180°이며, 변의 길이로는 부등변·이등변·정삼각형, 각의 크기로는 예각·직각·둔각 삼각형으로 분류합니다. 넓이는 가장 기본인 ½·밑변·높이, 세 변만 알 때는 헤론의 공식 A = √(s(s−a)(s−b)(s−c)) (s는 반둘레), 좌표가 주어졌을 때는 신발끈 공식이 편리합니다. 예: (0,0), (4,0), (2,3) 꼭짓점을 가진 삼각형의 넓이는 신발끈 공식으로 6이며, ½·4·3으로도 동일하게 6이 나옵니다.

4. The Pythagorean Theorem

The Pythagorean theorem is arguably the most famous result in elementary mathematics. It states that in any right triangle, the square of the hypotenuse (the side opposite the right angle) equals the sum of the squares of the other two sides — written compactly as a² + b² = c². Although the result was known empirically to the Babylonians a thousand years earlier, Pythagoras and his followers in the sixth century BCE produced the first general proof.

The most beautiful proof is purely visual. Build a large square of side (a + b) and tile its interior in two different ways. In the first tiling, fit four copies of the right triangle around an inner square of side c; the total area is c² + 4·(½ab). In the second tiling, arrange the four triangles to leave two inner squares of sides a and b; the total area is now a² + b² + 4·(½ab). Since the outer square is the same, c² + 2ab = a² + b² + 2ab, and the 2ab cancels — giving exactly a² + b² = c².

The converse is equally useful: if a triangle’s side lengths satisfy a² + b² = c², then the triangle must be right-angled with the right angle opposite side c. Carpenters exploit this with the “3-4-5 trick” to lay out perfect right angles. Integer triples like (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25) are called Pythagorean triples.

A practical example: the diagonal of a Full-HD monitor measuring 1920 × 1080 pixels is √(1920² + 1080²) = √4852800 ≈ 2202.9 pixels. The same calculation, scaled to inches, is what the “24-inch monitor” label actually refers to.

피타고라스 정리는 직각삼각형에서 a² + b² = c² (c는 빗변)로, 정사각형을 두 가지 방식으로 타일링하는 시각적 증명이 가장 우아합니다. 역도 성립해 세 변이 이 관계를 만족하면 그 삼각형은 반드시 직각삼각형이며, 목수들의 3-4-5 직각 잡기 기법이 이를 활용합니다. (3,4,5), (5,12,13), (8,15,17) 같은 정수 해를 피타고라스 수라고 합니다. 예: 1920×1080 모니터의 대각선은 √(1920² + 1080²) ≈ 2202.9 픽셀입니다.

5. Circles

A circle is the set of all points in a plane lying at a fixed distance — the radius r — from a given center. The diameter d is any straight line through the center with both endpoints on the circle, so d = 2r. A chord is any segment whose endpoints lie on the circle (the diameter is the longest chord), and an arc is a continuous portion of the circle itself.

Two formulas dominate circle geometry. The area enclosed by a circle is A = πr², and the circumference is C = 2πr, equivalently πd. The constant π ≈ 3.14159 is by definition the ratio C/d — irrational, and even transcendental. For mental arithmetic the approximation π ≈ 22/7 is accurate to within 0.04%, plenty for engineering rules of thumb.

The inscribed-angle theorem says an angle inscribed in a circle is exactly half the central angle subtending the same arc. A striking corollary, Thales’s theorem, follows: any triangle inscribed in a semicircle with the diameter as one side is automatically a right triangle, linking the Pythagorean theorem of §4 to circular geometry.

Worked example. A pizza of radius 15 cm has area A = π · 15² ≈ 706.86 cm² and circumference C = 2π · 15 ≈ 94.25 cm. Doubling the radius to 30 cm quadruples the area to 2827.4 cm² — a reminder that area scales with the square of the radius, not linearly.

원은 중심으로부터 일정 거리(반지름 r) 떨어진 점들의 집합이며, 지름 d = 2r, 넓이 A = πr², 둘레 C = 2πr = πd입니다. 원주율 π는 정의상 C/d로 약 3.14159인 무리수·초월수이며, 빠른 암산에는 π ≈ 22/7이 유용합니다. 원주각 정리에 따르면 원주각은 같은 호에 대한 중심각의 절반이며, 그 따름정리인 탈레스 정리는 "반원에 내접한 삼각형은 항상 직각삼각형"임을 보장합니다.

6. Sectors, Arcs & Segments

A sector is the slice of a disk bounded by two radii and the arc between them — what most people instinctively call a “pie slice.” Sectors are described by a central angle θ measured either in degrees or radians. The arc length along the curved boundary is L = rθ when θ is in radians, or L = (θ / 360°) · 2πr when θ is in degrees. The sector area is A = ½ · r² · θ in radians, or A = (θ / 360°) · πr² in degrees. Notice how much cleaner the radian formulas are — this is exactly why higher mathematics insists on radians.

Converting between the two units is essential. To go from degrees to radians, multiply by π / 180; to go from radians to degrees, multiply by 180 / π. A few values are worth memorizing: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 360° = 2π. These show up everywhere from trigonometry tables to graphics shaders.

A segment is the slim region trapped between a chord and the arc that the chord cuts off — like the crust-side leftover after you slice the tip off a pizza wedge. Its area equals the area of the sector minus the area of the triangle formed by the two radii and the chord: A_segment = ½r²θ − ½r² sin θ = ½r²(θ − sin θ), with θ in radians.

Everyday examples. On an analog clock, the minute hand sweeps 360° / 60 = 6° per minute, or π/30 radians; in five minutes it traces a 30° sector. An eight-slice pizza assigns each slice a central angle of 360° / 8 = 45° = π/4 radians, so a 16 cm-radius pizza yields slices of area ½ · 16² · (π/4) ≈ 100.5 cm² each. Pie charts in business reports are simply sectors whose central angles encode percentages: 1% = 3.6°.

부채꼴은 두 반지름과 호로 둘러싸인 영역으로, 호의 길이는 L = rθ(라디안) 또는 (θ/360°)·2πr(도), 넓이는 ½r²θ 또는 (θ/360°)·πr²입니다. 도→라디안은 ×π/180, 라디안→도는 ×180/π이며 30°=π/6, 60°=π/3, 90°=π/2처럼 자주 쓰이는 값은 외워두는 것이 좋습니다. 활꼴(segment)은 현과 호 사이의 영역으로, 부채꼴 넓이에서 삼각형 넓이를 빼서 ½r²(θ − sin θ)로 계산합니다.

7. Quadrilaterals & Polygons

A polygon is a closed figure made of straight line segments joined end-to-end. Polygons with four sides are quadrilaterals, with five pentagons, with six hexagons, and so on. The most important quadrilaterals each carry their own area formula. A square of side s has area A = s². A rectangle of width w and height h has A = w · h. A parallelogram (opposite sides parallel) has A = base · height, where the height is the perpendicular distance between the parallel bases, not the side length. A rhombus is a parallelogram with all four sides equal, and its area can also be computed from its diagonals as A = ½ · d₁ · d₂. A trapezoid (parallel sides of lengths a and b separated by perpendicular height h) has area A = ½(a + b) · h.

For any convex polygon with n sides, two angle facts always hold. The sum of interior angles is (n − 2) · 180°. For a triangle this gives 180°, for a quadrilateral 360°, for a pentagon 540°, and so on. The sum of exterior angles, taken one at each vertex in a consistent direction, is always exactly 360°, regardless of how many sides the polygon has — a fact that surprises most students the first time they meet it.

Polygons are regular when all sides have the same length and all angles are equal — the equilateral triangle, the square, and the regular hexagon are familiar examples. Regular polygons can tessellate (tile the plane without gaps) only when their interior angles divide 360° evenly: triangles (60°), squares (90°), and hexagons (120°) qualify, which is why bathroom floors, chessboards, and honeycomb cells all use these shapes.

다각형은 닫힌 직선 도형이며, 사각형의 종류로 정사각형(s²), 직사각형(w·h), 평행사변형(밑변×높이), 마름모(½d₁d₂), 사다리꼴(½(a+b)h) 등이 있습니다. 임의의 볼록 n각형에서 내각의 합은 (n−2)·180°, 외각의 합은 항상 360°입니다. 모든 변과 모든 각이 같으면 정다각형이고, 내각이 360°를 나누는 정삼각형·정사각형·정육각형만이 평면 테셀레이션이 가능합니다.

8. Coordinate Geometry

Coordinate geometry, also called analytic geometry, was introduced by René Descartes in 1637 and unified geometry with algebra. The core idea is to label every point in the plane with a pair of numbers (x, y), so that geometric questions become algebraic equations and vice versa. Lines, circles, and parabolas now have formulas; intersections become systems of equations; lengths and angles become computations.

The distance formula between two points (x₁, y₁) and (x₂, y₂) is d = √((x₂ − x₁)² + (y₂ − y₁)²). This is nothing other than the Pythagorean theorem applied to the right triangle whose legs are the horizontal gap (x₂ − x₁) and the vertical gap (y₂ − y₁) — a beautiful demonstration of how §4 powers all of coordinate geometry. The midpoint of the same two points is the average of their coordinates: ((x₁ + x₂)/2, (y₁ + y₂)/2).

The slope of the line through those two points is m = (y₂ − y₁) / (x₂ − x₁), the “rise over run.” A line with slope m passing through point (x₁, y₁) can be written in point-slope form y − y₁ = m(x − x₁), or rearranged into slope-intercept form y = mx + b, where b is the y-intercept. The most general form is ax + by + c = 0, useful when the line is vertical (b = 0) and slope is undefined.

Two non-vertical lines are parallel when their slopes are equal (m₁ = m₂), and perpendicular when their slopes multiply to −1 (m₁ · m₂ = −1) — equivalently, when one slope is the negative reciprocal of the other. For example, a line of slope 2 is perpendicular to any line of slope −1/2. This single algebraic condition reproduces, in pure number form, the Euclidean notion of “meeting at a right angle” introduced back in §2.

좌표기하는 데카르트(1637)가 도입한 기하와 대수의 통합입니다. 두 점 사이 거리는 d = √((x₂−x₁)² + (y₂−y₁)²)로 피타고라스 정리의 직접 적용이며, 중점은 좌표의 평균, 기울기는 m = (y₂−y₁)/(x₂−x₁), 직선의 식은 y = mx + b 또는 일반형 ax + by + c = 0입니다. 두 직선이 평행이면 기울기가 같고, 수직이면 기울기의 곱이 −1입니다.

9. Common Mistakes & Pitfalls

• Mixing degrees and radians. Calculator answers can be off by huge factors when the angle-mode setting is wrong. sin(30°) = 0.5 but sin(30 rad) ≈ −0.988. Always confirm whether your formula expects degrees or radians, and confirm the calculator agrees, especially in sector and trigonometry problems.
• Confusing diameter and radius. Plugging d into A = πr² gives an answer four times too large because area scales with the square of the radius, not the diameter. Always check whether the given number is the radius r or the diameter d = 2r before substituting.
• Wrong height in ½ · b · h. The height must be the perpendicular distance from the chosen base to the opposite vertex, not the length of a slanted side. In an obtuse triangle, the foot of this perpendicular may even fall outside the base — a common source of confusion.
• Applying Pythagoras to non-right triangles. The identity a² + b² = c² holds only when the angle opposite c is exactly 90°. For acute or obtuse triangles, you need the more general law of cosines: c² = a² + b² − 2ab · cos C. Using a² + b² = c² indiscriminately is one of the most common geometry errors.
• Similar vs congruent triangles. Congruent triangles have the same shape and the same size (corresponding sides equal). Similar triangles have the same shape but possibly different sizes (corresponding sides proportional, corresponding angles equal). Treating “similar” as “congruent” leads to wrong side lengths in scaling problems.
• Off-by-one in angle-sum formulas. The interior-angle sum of an n-gon is (n − 2) · 180°, not n · 180°. Forgetting the “minus two” is a classic exam trap. A quick sanity check: a triangle (n = 3) gives (3 − 2) · 180° = 180°, which is right.

흔한 실수: 계산기의 도/라디안 모드를 잘못 두기, 지름을 반지름으로 착각하기(넓이가 4배 차이), ½bh에서 비스듬한 변 길이를 높이로 쓰기, 직각삼각형이 아닌데 a²+b²=c² 적용하기(이때는 코사인 법칙 c² = a² + b² − 2ab cos C 사용), 닮음(similar)과 합동(congruent) 혼동, 그리고 n각형 내각의 합 공식에서 (n−2)·180°의 −2를 빠뜨리는 실수가 대표적입니다.

10. Applications & Related Concepts

Geometry quietly underlies an enormous range of professional disciplines:

• Architecture and construction. Every building plan is geometry on paper: load-bearing triangulation, roof-angle layouts, the “3-4-5 rule” for square corners, material area calculations, and tile-packing efficiency all come from the formulas in this guide.
• Navigation and GPS. Straight-line distances on a map use the Pythagorean distance formula of §8. Over long distances on the curved Earth, GPS systems use the great-circle approximation, generalizing the same idea to spheres.
• Computer graphics and games. Every 3D scene is rendered through triangle-based meshes; ray tracers compute pixel colors by intersecting rays with triangles. Collision detection in physics engines reduces to point-in-polygon and sphere-sphere distance tests — pure geometry running millions of times per second.
• Engineering and CAD. Computer-aided design tools represent every mechanical part as a network of geometric primitives. Sector and arc formulas determine sheet-metal bending allowances; angle and slope rules govern gear teeth, threads, and bearings.
• Surveying and cartography. Land surveyors use triangulation — measuring angles from known points to fix new ones — which is exactly the inscribed-angle and law-of-sines geometry taught in school.
• Art and design. Renaissance painters discovered linear perspective by treating the canvas as a projective plane. Golden-ratio rectangles and rule-of-thirds grids in modern photography are both pure geometry.

Geometry also connects deeply with other branches of mathematics. Coordinate geometry uses vectors; area calculations connect to arithmetic; transformations of shapes are described by matrices — rotations, reflections, and scalings of a plane are 2×2 matrix operations, and 3D transformations are 4×4. Geometric events also lead to probability, while measurement data links to statistics.

Ready to practice? Head to Practice → C:Geo for interactive drills on triangles, the Pythagorean theorem, circles, sectors, polygons, and coordinate geometry, with step-by-step solutions.

기하학은 건축(트러스 구조·각도 설계), 내비게이션(피타고라스 거리·구면 대원), 컴퓨터 그래픽스(삼각형 메시·충돌 검출), CAD 엔지니어링, 측량(삼각측량), 회화의 원근법까지 광범위하게 응용됩니다. 좌표기하는 벡터로 자연스럽게 확장되고, 넓이 계산은 산술과, 도형의 변환은 행렬과, 기하학적 사건의 확률은 확률·통계와 연결됩니다. 실전 연습은 C:Geo에서 가능합니다.

11. Explore Each Geometry Topic in Depth

The reference above is the conceptual overview. Each topic below has its own focused page with a step-by-step worked example, an SVG diagram, common-mistake notes, and an FAQ:

• Triangle area — the ½·b·h formula, Heron’s formula, and the shoelace method.
• Rectangle area — width × height, the square special case, and area vs perimeter.
• Trapezoid area — ½·(a+b)·h and the midsegment shortcut.
• Circle area — πr² and why doubling the radius quadruples the area.
• Circumference — 2πr (= πd) and the meaning of π.
• Sector area — (θ/360)·πr², the radian form ½r²θ, and arc length.
• Pythagorean theorem — a² + b² = c², missing legs, triples, and the converse.
• Polygon angles — the (n−2)·180° interior-sum formula and exterior angles.
• Degree–radian conversion — the π = 180° bridge and common values.

위 레퍼런스는 전체 개요이며, 아래 각 주제는 단계별 예제·도해·실수 노트·FAQ를 갖춘 전용 페이지로 이어집니다: 삼각형 넓이, 직사각형 넓이, 사다리꼴 넓이, 원의 넓이, 원의 둘레, 부채꼴 넓이, 피타고라스 정리, 다각형 내각, 도·라디안 변환.