Geometry Basics

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• A point is an exact location. It has no size, just position.
• A line goes straight forever in both directions (two arrowheads).
• A segment is part of a line with two endpoints.
• A ray starts at one endpoint and goes forever in one direction.
• An angle is formed by two rays that share an endpoint, called the vertex. We measure angles in degrees ($^\circ$).

An angle measures a turn.

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• Acute: greater than $0^\circ$ and less than $90^\circ$.
• Right: exactly $90^\circ$ (marked with a small square).
• Obtuse: greater than $90^\circ$ and less than $180^\circ$.
• Straight: exactly $180^\circ$ (a straight line).

Two angles are complementary if their measures add to $90^\circ$. Two angles are supplementary if their measures add to $180^\circ$.

By sides:

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• Equilateral: all three sides equal.
• Isosceles: exactly two sides equal.
• Scalene: no sides equal.

By angles:

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• Acute: all three angles less than $90^\circ$.
• Right: one angle exactly $90^\circ$.
• Obtuse: one angle greater than $90^\circ$.

An angle measures a turn.

The three angles of any triangle always add up to $180^\circ$:

The perimeter is the total distance around a shape: just add up the lengths of all the sides. Perimeter is a length, so it uses plain units (cm, m, in).

An angle measures a turn.

Area is the amount of surface a flat shape covers, measured in square units.

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• Rectangle: $A = l \times w$
• Square: $A = s^2$
• Parallelogram: $A = b \times h$
• Triangle: $A = \dfrac{1}{2} b h$
• Trapezoid: $A = \dfrac{1}{2}(b_1 + b_2)\,h$

Here $h$ is always the perpendicular height (straight up from the base), not a slanted side.

An angle measures a turn.

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• The radius $r$ is the distance from the center to the edge.
• The diameter $d$ goes all the way across through the center: $d = 2r$, so $r = \dfrac{d}{2}$.
• Circumference (distance around): $C = \pi d$, which is the same as $C = 2\pi r$.
• Area: $A = \pi r^2$.

We use $\pi \approx 3.14$.

In a right triangle, the two shorter sides (the legs, $a$ and $b$) and the longest side (the hypotenuse, $c$, opposite the right angle) are related by

The hypotenuse $c$ is always the longest side and sits across from the $90^\circ$ angle.

An angle measures a turn.

For a box with length $l$, width $w$, and height $h$:

Volume is measured in cubic units (cm$^3$); surface area in square units (cm$^2$).

For a cylinder with radius $r$ and height $h$:

(The base is a circle of area $\pi r^2$; multiply by the height.)

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• Decide what the question is really asking: distance around (perimeter/circumference), surface covered (area), or space filled (volume).
• Pick the matching formula and substitute the numbers.
• Keep the units, and label the answer with the correct kind of unit.

Take four copies of a right triangle with legs $a$, $b$ and hypotenuse $c$, and arrange them inside a big square whose side is $a + b$. The four hypotenuses form a tilted square of side $c$ in the middle.

Compute the big square's area two ways:

Expand the left side and simplify:

The $2ab$ cancels from both sides, leaving the theorem. This shows the rule is not a coincidence: it follows from just adding up areas.

An angle measures a turn.

You can find a triangle's area from its side lengths alone---no height needed. First compute the semiperimeter (half the perimeter):

Then

This is the tool to reach for when you know all three sides but cannot easily measure a perpendicular height.

Two triangles are similar if they have the same shape but not necessarily the same size: their corresponding angles are equal, and their corresponding sides are in the same ratio (scale factor). We write $\triangle ABC \sim \triangle DEF$.

The key fact is that matching sides form equal fractions:

This lets you find a distance you cannot measure directly (like the height of a tree) by comparing it to one you can.

Because a triangle's angles sum to $180^\circ$ and a straight line also measures $180^\circ$, an exterior angle (formed by extending one side) equals the sum of the two remote interior angles:

Here the exterior angle at $B$ equals $x + y$. Why: the interior angle at $B$ is $180^\circ - x - y$ (angle sum), and the exterior angle is its supplement, $180^\circ - (180^\circ - x - y) = x + y$.

An inscribed angle has its vertex on the circle; a central angle has its vertex at the center. If both open onto the same arc, then

A famous special case (Thales' theorem): any angle inscribed in a semicircle is a right angle, because it opens onto a $180^\circ$ arc, and $\tfrac{1}{2}(180^\circ) = 90^\circ$. So if one side of an inscribed triangle is a diameter, the triangle is a right triangle.

Imagine unwrapping a can. You get two circles (the top and bottom) plus one rectangle (the curved side, or “label”). Each circle has area $\pi r^2$. When you unroll the side, its width is the circumference $2\pi r$ and its height is $h$, so the rectangle's area is $2\pi r \cdot h$. Add the pieces:

This is why the formula looks the way it does---it is just areas of shapes you already know, glued together.