Geometry
takes shape.

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.

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°.

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.

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.

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.

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°.

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.

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 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.

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.

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.

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.

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.

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.

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.

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.

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°?

1. Draw a line through one vertex parallel to the opposite side.
2. The other two angles of the triangle match angles on that new line by alternate interior angles.
3. Those two angles and the angle at the vertex form a straight angle.
4. 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.

Try each question before opening its answer. A quick sketch and labeled units make most problems easier.