Geometry

1. Basic Shapes and Properties

Geometry starts with the most fundamental building blocks: points, lines, and the shapes they form. Everything in geometry -- from simple triangles to complex 3D models -- is built from these primitives.

Points, Lines, Rays, and Line Segments

These are the atoms of geometry. Every shape is ultimately defined by points and the connections between them.

• Point: A single location in space. It has no size, no width, no length -- just a position. We label points with capital letters: A, B, C.
• Line: An infinite collection of points extending forever in both directions. A line through points A and B is written as AB with a double-headed arrow above it. Lines never end.
• Ray: Starts at a point and extends forever in one direction. Think of a flashlight beam -- it has a starting point but no end.
• Line Segment: A piece of a line with two endpoints. It has a definite, measurable length. The segment from A to B is written AB with a bar above it.

  Point         Line (infinite)       Ray (one direction)     Segment (finite)

    *            <----------->          *----------->           *-----------*
    A            A           B          A                       A           B

Triangles

The triangle is the most important polygon in geometry and in computer science. Every complex shape can be broken down into triangles -- this is why GPUs are optimized for triangle rendering.

    Equilateral        Isosceles          Scalene            Right

        /\                /\               /\                 |\
       /  \              /  \             /  \                | \
      /    \            /    \           /    \               |  \
     /______\          /______\         /______\              |___\
   60  60  60         70  70  40      50   60   70            90

A crucial fact: the angles in any triangle always sum to exactly 180 degrees. This is true regardless of the type of triangle. We will use this constantly.

Quadrilaterals

Quadrilaterals are four-sided polygons. Each type has specific properties that make them useful for different calculations.

  Square       Rectangle     Parallelogram    Trapezoid       Rhombus

  +------+    +----------+     ________       __________       /\
  |      |    |          |    /       /       /          \     /  \
  |      |    |          |   /       /       /            \   /    \
  +------+    +----------+  /-------/       /--------------\ \    /
                                                              \  /
                                                               \/

Circles

A circle is the set of all points that are the same distance from a center point. That distance is the radius.

• Radius (r): Distance from center to any point on the circle.
• Diameter (d): Distance across the circle through the center. Always d = 2r.
• Circumference (C): The distance around the circle (its perimeter). C = 2(pi)r.
• Chord: A line segment connecting any two points on the circle. The diameter is the longest possible chord.
• Arc: A portion of the circumference. Measured by angle or length.
• Sector: The "pie slice" region between two radii and an arc.
• Tangent: A line that touches the circle at exactly one point, perpendicular to the radius at that point.

Polygons

A polygon is a closed figure made of straight line segments. Triangles (3 sides) and quadrilaterals (4 sides) are the simplest, but polygons can have any number of sides.

A regular polygon has all sides equal and all angles equal. A convex polygon has all interior angles less than 180 degrees (no "dents"). These properties matter for CS algorithms like convex hull computation.

2. Angles

An angle is formed when two rays share a common endpoint (called the vertex). Angles are measured in degrees (from 0 to 360) or radians (from 0 to 2(pi)). Understanding angle relationships is essential for everything from triangle solving to rotation in graphics.

Types of Angles

   Acute (45)     Right (90)     Obtuse (120)    Straight (180)   Reflex (270)

       /              |               \              ___________      ___________
      /               |                \                                   |
     /____            |_____            \____                              |
                                                                           |

Complementary and Supplementary Angles

These are pairs of angles with special relationships:

Vertical Angles

When two lines cross, they form two pairs of vertical angles (also called "vertically opposite angles"). Vertical angles are always equal.

           Two lines crossing form 4 angles:

              \   a   /                a = c  (vertical angles)
               \     /                 b = d  (vertical angles)
            b   \   /   d             a + b = 180  (supplementary)
         --------\ /--------
                 / \
            d   /   \   b
               /     \
              /   c   \

Angles in a Triangle

Interior Angles of Polygons

This formula works because any polygon can be divided into (n - 2) triangles by drawing diagonals from one vertex. Since each triangle has angles summing to 180 degrees, the total is (n - 2) x 180.

3. Area and Perimeter

Perimeter is the total distance around a shape (sum of all sides). Area is the amount of surface the shape covers, measured in square units (cm squared, m squared, pixels squared, etc.).

Rectangle

        width (w)
       +----------+
       |          |
 length |          |  A = l x w
 (l)   |          |  P = 2(l + w)
       +----------+

Square

Triangle

The height (or altitude) must be perpendicular to the base. It is not necessarily one of the sides -- sometimes you drop a perpendicular from the top vertex down to the base.

         *
        /|\
       / | \     height (h)
      /  |  \
     /   |   \
    /____|____\
      base (b)

    A = (1/2) x b x h

Circle

Parallelogram

Notice that the area formula looks like a rectangle's. That is because if you "cut" one end of a parallelogram and move it to the other end, you get a rectangle with the same base and height.

      __________
     /          /    height (h) is perpendicular
    /     h    /     to the base, NOT the slanted side
   /__________/
      base (b)

Trapezoid (Trapezium)

       ________     <-- top base (a)
      /        \
     /    h     \    height is perpendicular
    /            \   distance between bases
   /______________\  <-- bottom base (b)

4. Volume and Surface Area

Volume measures the 3D space inside a solid shape (in cubic units). Surface area is the total area of all the outer faces. Think of volume as how much water a container holds, and surface area as how much wrapping paper you need to cover it.

Cube

Rectangular Prism (Box)

Cylinder

        _____
       /     \     <-- top circle: area = pi x r squared
      |       |
      |   h   |    <-- curved surface "unwraps" to a rectangle:
      |       |        width = 2(pi)r, height = h
       \_____/     <-- bottom circle

Sphere

Cone

Notice the cone volume is exactly 1/3 of a cylinder with the same base and height. This is not a coincidence -- it can be proven with calculus.

          *        <-- apex
         /|\
        / | \  l (slant height)
       /  |h \
      /   |   \
     /____|____\
         r

5. Pythagorean Theorem

What Is It and Why Does It Exist?

The Pythagorean theorem answers one of the most practical questions in all of math: if I know two sides of a right triangle, how do I find the third? That is it. It is a tool for finding missing distances.

It only works on right triangles (triangles with one 90-degree angle). The formula says: the two shorter sides, when squared and added together, equal the longest side squared.

Why Does It Work? (Visual Proof)

This is not just a formula someone invented -- there is a beautiful geometric reason it is true. If you build a square on each side of a right triangle, the area of the two smaller squares adds up exactly to the area of the big square:

When Would I Actually Use This?

Finding the Hypotenuse

If you know both legs, solve for the hypotenuse:

Finding a Leg

If you know the hypotenuse and one leg, rearrange to find the other leg:

Pythagorean Triples

Some right triangles have all integer side lengths. These are called Pythagorean triples. Memorizing a few common ones saves time:

The Distance Formula

The distance formula is the Pythagorean theorem in disguise. To find the distance between two points on a coordinate plane, you form a right triangle where the horizontal and vertical distances are the legs.

    (x2, y2) *
             |\
             | \   d = distance (hypotenuse)
    (y2 - y1)| d\
             |   \
             |____\
    (x1, y1) *     * (x2, y1)
           (x2 - x1)

6. Trigonometry Basics

What Problem Does Trig Solve?

The Pythagorean theorem connects sides to sides. But what if you know an angle and a side, and need to find another side? Or you know two sides and need the angle? That is what trigonometry does. It connects angles to side lengths.

Here is the real-world problem: you are standing 50 meters from a building. You look up at the top and measure the angle is 35 degrees. How tall is the building? You cannot use Pythagoras -- you only know one side and one angle, not two sides. You need trig.

SOH CAH TOA

SOHCAHTOA is a mnemonic for the three ratios. Each one divides two specific sides of a right triangle:

What Each Ratio Actually Tells You

• sin(angle) = "what fraction of the hypotenuse is the opposite side?" If sin(30) = 0.5, that means the opposite side is always half the hypotenuse in a 30-degree triangle.
• cos(angle) = "what fraction of the hypotenuse is the adjacent side?" If cos(60) = 0.5, the adjacent side is half the hypotenuse.
• tan(angle) = "how many times bigger is the opposite side than the adjacent side?" If tan(45) = 1, they are equal. If tan(60) = 1.73, the opposite side is 1.73 times the adjacent.

Common Angle Values

Memorize these. They come up constantly in math and CS.

Solving for a Side

Solving for an Angle

To find an angle when you know the sides, use the inverse trig functions (arcsin, arccos, arctan):

The Unit Circle (Introduction)

The unit circle is a circle with radius 1 centered at the origin. It extends trigonometry beyond right triangles to any angle (including negative angles and angles greater than 360 degrees).

                    90 (pi/2)
                    (0, 1)
                      |
                      |
  180 (pi)  (-1, 0)---+---(1, 0)  0 (0 radians)
                      |
                      |
                    (0, -1)
                   270 (3pi/2)

On the unit circle, for any angle theta measured from the positive x-axis:

• The x-coordinate of the point = cos(theta)
• The y-coordinate of the point = sin(theta)

This means any point on a circle of radius r can be described as:

Radians Explained

Radians are another way to measure angles. One radian is the angle created when the arc length equals the radius of the circle.

Fundamental Trigonometric Identities

These identities are relationships that are always true for any angle. They're essential for simplifying expressions and solving equations.

Reciprocal Identities

Pythagorean Identities

Quotient Identity

Law of Sines

For any triangle (not just right triangles), the Law of Sines relates sides and angles:

Law of Cosines

This generalizes the Pythagorean theorem to all triangles:

When to Use Each Law

SOHCAHTOA Practice Problems

7. Coordinate Geometry

Coordinate geometry (analytic geometry) bridges algebra and geometry by placing shapes on a number grid. Instead of abstract shapes, everything gets precise coordinates, which is exactly how computers work with geometry.

The Coordinate Plane

The coordinate plane has two perpendicular number lines: the x-axis (horizontal) and y-axis (vertical). Every point is described by an ordered pair (x, y).

              y
              |
         Q2   |   Q1
       (-,+)  |  (+,+)
              |
   -----------+-----------> x
              |
         Q3   |   Q4
       (-,-)  |  (+,-)
              |

The four quadrants are numbered counterclockwise starting from the upper right (Q1). The origin is the point (0, 0).

Distance Formula

(Derived from the Pythagorean theorem -- see Section 5 above.)

This is the Euclidean distance -- the "straight line" or "as the crow flies" distance. But it is not the only way to measure distance, and in programming you will use different distance formulas for different situations.

Euclidean Distance vs Manhattan Distance

Imagine you are in a city with a grid of streets. You want to get from point A to point B. You can't walk through buildings -- you have to walk along the streets. The distance you actually walk (along the grid) is the Manhattan distance (named after Manhattan's grid layout). The straight-line distance if you could fly is the Euclidean distance.

When to Use Which Distance

Chebyshev Distance (Bonus)

There is a third distance you'll see in programming: Chebyshev distance (also called "chessboard distance"). It is the maximum of the horizontal and vertical distances. Think of how a king moves in chess -- it can move diagonally, so the distance is just the larger of the two axis differences.

3D Distance

All three formulas extend naturally to 3D (and beyond) by adding the z dimension:

Midpoint Formula

The midpoint is the point exactly halfway between two points. You just average the x-coordinates and average the y-coordinates.

Slope of a Line

The slope measures how steep a line is -- how much y changes for each unit change in x. It is the "rise over run."

Equation of a Line

There are several forms for writing the equation of a line. Each has its uses.

8. Transformations

A transformation changes a shape's position, size, or orientation. There are four fundamental types, and they are the backbone of computer graphics, animation, and game development.

Translation (Sliding)

A translation moves every point of a shape the same distance in the same direction. The shape does not rotate, flip, or change size -- it just slides.

  Before                After (translated by (4, 2))

    *                         *
   / \         --->          / \
  /___\                     /___\
 (1,1)                     (5,3)

Rotation

A rotation turns a shape around a fixed point (center of rotation) by a given angle. Counterclockwise is the positive direction.

Common rotations around the origin:

Reflection

A reflection flips a shape across a line (the line of reflection), creating a mirror image.

Scaling (Dilation)

Scaling changes the size of a shape by multiplying all coordinates by a scale factor.

9. Practice Quiz

Test your understanding. Click the answer you think is correct.