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Build skills through subject overviews, illustrated lessons, definitions, formulas, worked examples, and practice exercises. Follow the geometry curriculum from foundations to proofs, or choose another subject to study.
Foundations
Geometry studies position, size, and shape. Its simplest objects are idealized: a point has a location but no size; a line has length without thickness; a plane is a flat surface extending without end.
Point, line, and segment
A line continues forever in both directions. A ray begins at one endpoint and continues forever in one direction. A segment has two endpoints, so its length can be measured. Points on the same line are collinear.
Distance and midpoint
Distance is the length of the shortest path between two points. The midpoint splits a segment into two equal segments. If AB is 18 units long and M is its midpoint, then AM = MB = 9 units.
Parallel lines stay the same distance apart and never meet. Perpendicular lines meet at a right angle of 90°.
Angles
An angle is formed by two rays sharing an endpoint, called the vertex. Degrees measure the amount of turn: one full turn is 360°, a straight turn is 180°, and a quarter turn is 90°.
| Type | Measure | Example |
|---|---|---|
| Acute | Between 0° and 90° | 35° |
| Right | Exactly 90° | Corner of a square |
| Obtuse | Between 90° and 180° | 125° |
| Straight | Exactly 180° | Half turn |
| Reflex | Between 180° and 360° | 240° |
Two angles are complementary when they add to 90° and supplementary when they add to 180°. Opposite angles where two lines cross are equal. Adjacent angles on a straight line add to 180°.
When a line crosses two parallel lines, corresponding angles are equal, alternate interior angles are equal, and interior angles on the same side add to 180°. These relationships let you solve many diagrams without measuring them.
Triangles
A triangle is a polygon with three sides. Its interior angles always total 180° in flat Euclidean geometry. If two angles are known, subtract them from 180° to find the third.
Classify by sides
Equilateral: all three sides equal; every angle is 60°. Isosceles: at least two sides equal; the angles opposite those sides are equal. Scalene: all three sides have different lengths.
Classify by angles
An acute triangle has three acute angles. A right triangle has one 90° angle. An obtuse triangle has one angle greater than 90°.
The triangle inequality says the sum of any two side lengths must exceed the third. Lengths 3, 4, and 8 cannot form a triangle, because 3 + 4 is less than 8.
Right triangles
The side opposite the right angle is the hypotenuse. For legs a and b and hypotenuse c, the Pythagorean theorem gives:
A right triangle with legs 3 and 4 has hypotenuse √(3² + 4²) = 5. This also works backward: side lengths satisfying the equation form a right triangle.
Congruence and similarity
Congruent triangles have the same shape and size. Common ways to establish congruence are SSS (three corresponding sides), SAS (two sides and their included angle), and ASA (two angles and their included side). Similar triangles have the same shape but may differ in size; corresponding angles match, and corresponding sides share one scale factor.
Polygons
A polygon is a closed figure made of straight segments. A regular polygon has equal side lengths and equal interior angles. Triangles, quadrilaterals, pentagons, and hexagons have 3, 4, 5, and 6 sides respectively.
For a hexagon, (6 − 2) × 180° = 720°. In a regular hexagon, each interior angle is 720° ÷ 6 = 120°. The exterior angles of any convex polygon add to 360°.
| Quadrilateral | Defining property | Useful fact |
|---|---|---|
| Parallelogram | Both pairs of opposite sides parallel | Opposite angles and sides equal |
| Rectangle | Parallelogram with four right angles | Diagonals are equal |
| Rhombus | Parallelogram with four equal sides | Diagonals meet at right angles |
| Square | Four equal sides and four right angles | Both rectangle and rhombus |
| Trapezoid | At least one pair of parallel sides | Area uses the average base length |
Circles
A circle is the set of points in a plane the same distance from a center. That distance is its radius r. The diameter d passes through the center and equals 2r. A chord joins two points on the circle; an arc is a portion of its edge.
For a circle with radius 5 cm, the circumference is 10π cm (about 31.4 cm), and the area is 25π cm² (about 78.5 cm²).
A sector is a slice of a circle. If its central angle is θ degrees, its area is (θ ÷ 360) × πr², and its arc length is (θ ÷ 360) × 2πr. A tangent touches the circle at exactly one point and is perpendicular to the radius there.
Perimeter and area
Perimeter measures distance around a flat shape, using linear units such as centimeters. Area measures the region inside it, using square units such as cm². Keep these units distinct: doubling every side doubles the perimeter but multiplies the area by four.
| Shape | Perimeter | Area |
|---|---|---|
| Rectangle, width w, height h | 2(w + h) | wh |
| Square, side s | 4s | s² |
| Triangle, sides a, b, c; base b, height h | a + b + c | ½bh |
| Parallelogram, base b, height h | Sum of all sides | bh |
| Trapezoid, bases a and b, height h | Sum of all sides | ½(a + b)h |
| Circle, radius r | 2πr | πr² |
The height in a triangle or parallelogram is the perpendicular distance to its base, not necessarily a slanted side. For an irregular figure, split it into familiar shapes, find each area, and add or subtract as needed.
Coordinate geometry
The Cartesian plane locates points with ordered pairs (x, y). The x-axis runs horizontally; the y-axis runs vertically. Their intersection is the origin, (0, 0). Positive x moves right and positive y moves up.
Slope
Slope describes how a line rises or falls. Between points (x₁, y₁) and (x₂, y₂), slope is the change in y divided by the change in x.
A positive slope rises to the right; a negative slope falls. A horizontal line has slope 0. A vertical line has undefined slope because its change in x is 0.
Distance and midpoint
These formulas follow from the Pythagorean theorem and averaging. Between (1, 2) and (4, 6), the horizontal change is 3, the vertical change is 4, and the distance is 5.
Transformations
A transformation maps each point of a figure to a new point. Some transformations preserve lengths and angles; these are called rigid motions. Dilation changes size while keeping shape.
| Transformation | What happens | Coordinate example |
|---|---|---|
| Translation | Slides without turning | (x, y) → (x + 3, y − 2) |
| Reflection | Flips across a line | Across y-axis: (x, y) → (−x, y) |
| Rotation | Turns around a center | 90° counterclockwise: (x, y) → (−y, x) |
| Dilation | Scales from a center | Factor 2 from origin: (x, y) → (2x, 2y) |
Translations, reflections, and rotations preserve size, so the image is congruent to the original. A dilation preserves angles and proportional side lengths, so its image is similar. With scale factor k, lengths multiply by k, areas by k², and volumes by k³.
Solid geometry
Three-dimensional figures have length, width, and height. Surface area measures the area of all exterior faces in square units. Volume measures the space inside in cubic units.
| Solid | Volume | Surface area |
|---|---|---|
| Rectangular prism, l × w × h | lwh | 2(lw + lh + wh) |
| Cube, side s | s³ | 6s² |
| Cylinder, radius r, height h | πr²h | 2πr² + 2πrh |
| Cone, radius r, height h | ⅓πr²h | πr² + πrℓ, where ℓ is slant height |
| Sphere, radius r | ⁴⁄₃πr³ | 4πr² |
| Pyramid, base area B, height h | ⅓Bh | Depends on its faces |
A prism has two congruent parallel bases. Its volume is base area × height. A pyramid with the same base and height occupies one third of that volume. For a cylinder with r = 3 and h = 10, V = π × 3² × 10 = 90π cubic units.
Euler’s formula connects many convex polyhedra: vertices − edges + faces = 2. A cube has 8 vertices, 12 edges, and 6 faces, so 8 − 12 + 6 = 2.
Geometric reasoning
A drawing suggests relationships; a proof explains why they must hold. Start from definitions and known facts, state each step clearly, and show how the conclusion follows. Measurement is useful for checking an idea, but a measurement alone does not prove a general rule.
Why do triangle angles total 180°?
- Draw a line through one vertex parallel to the opposite side.
- The other two angles of the triangle match angles on that new line by alternate interior angles.
- Those two angles and the angle at the vertex form a straight angle.
- A straight angle measures 180°, so the triangle’s three angles total 180°.
This argument assumes a flat Euclidean plane. On a curved surface such as a sphere, a triangle’s angle sum can exceed 180°. Geometry begins with assumptions, so always notice which setting a rule belongs to.
Practice
Try each question before opening its answer. A quick sketch and labeled units make most problems easier.
Two supplementary angles are in the ratio 2:3. Find both angles.
Show solution
The ratio has five parts, so each part is 180° ÷ 5 = 36°. The angles are 72° and 108°.
A right triangle has legs 8 cm and 15 cm. Find its hypotenuse and area.
Show solution
Hypotenuse = √(8² + 15²) = √289 = 17 cm. Area = ½ × 8 × 15 = 60 cm².
What is each interior angle of a regular octagon?
Show solution
Angle sum = (8 − 2) × 180° = 1080°. Divide by 8 to get 135°.
A circle has diameter 12 m. Find its circumference and area in terms of π.
Show solution
Radius = 6 m. Circumference = 12π m. Area = 36π m².
Find the midpoint and distance between (−2, 3) and (4, 11).
Show solution
Midpoint = ((−2 + 4)/2, (3 + 11)/2) = (1, 7). Distance = √(6² + 8²) = 10.
A square’s side length triples. What happens to its perimeter and area?
Show solution
Perimeter triples. Area becomes 3² = 9 times as large.
A cylinder has radius 4 cm and height 9 cm. Find its volume.
Show solution
V = πr²h = π × 16 × 9 = 144π cm³.
Quick glossary
Bisector: a line or ray splitting an angle or segment into two equal parts.
Chord: a segment joining two points on a circle.
Congruent: identical in size and shape.
Diagonal: a segment connecting non-adjacent vertices.
Hypotenuse: the longest side of a right triangle.
Parallel: coplanar lines that never meet.
Perpendicular: intersecting at 90°.
Radius: distance from a circle’s center to its edge.
Similar: same shape with proportional lengths.
Vertex: the point where edges or rays meet.