Formula Sheet

Sum of interior angles of an $n$-gon:

In a convex polygon, every interior angle is less than $180°$.

Every interior angle of an $n$-gon is less than $(n-2)(180°)$.

Sum of angles around a point is $360°$.

A straight angle measures $180°$.

Vertical angles are congruent — when two lines intersect, opposite angles are equal.

If two parallel lines are cut by a transversal, corresponding angles are congruent.

If two parallel lines are cut by a transversal, alternate interior and alternate exterior angles are congruent.

Same-side interior/exterior angles formed by parallel lines and a transversal are supplementary (sum to $180°$).

The exterior angle of a triangle equals the sum of the two remote interior angles.

Equal sides imply equal opposite angles.

If $AB = AC$, then $\angle B = \angle C$.

Equal angles imply equal opposite sides.

If $\angle B = \angle C$, then $AB = AC$.

All sides and all angles are equal (each condition implies the other).

All angles $= 60°$.

The segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

When $\triangle ABC \sim \triangle DEF$, corresponding vertices match in order:

Corresponding sides are proportional.

When $\triangle ABC \cong \triangle DEF$, corresponding sides and angles are exactly equal.

Two sides and the included angle are equal ⟹ triangles are congruent.

All three corresponding sides are equal ⟹ triangles are congruent.

Two angles and the included side are equal ⟹ triangles are congruent.

Two pairs of corresponding angles are equal ⟹ triangles are similar.

In a right triangle:

where $c$ is the hypotenuse.

Side ratios (short leg : long leg : hypotenuse):

Side ratios (leg : leg : hypotenuse):

The circumcenter of a right triangle is the midpoint of the hypotenuse.

In a right triangle, the circumradius equals half the hypotenuse:

Similar figures have corresponding angles equal and corresponding sides proportional.

If corresponding side lengths scale by a factor $k$, areas scale by a factor $k^2$.

Properties of trapezoids, parallelograms, rhombi, rectangles, squares, and kites — each has unique angle, side, and diagonal properties.

What conditions imply each type of quadrilateral (e.g., equal diagonals + bisect each other ⟹ rectangle).

The midsegment of a trapezoid is parallel to both bases and has length:

Sector with central angle $x$:

The measure of an arc equals the measure of its corresponding central angle.

An inscribed angle measures half its intercepted arc:

Inscribed angles that intercept the same arc are congruent.

A radius drawn to the midpoint of a chord is perpendicular to the chord.

Tangent segments drawn from the same external point are equal in length.

Tangent lengths from a vertex to the incircle are equal.

For shaded-area problems involving circles: subtract the inner region from the outer, or add/subtract sector areas and triangle areas.

$\pi \approx \frac{22}{7} \approx 3.14$

where $s$ is the side length.

Parallelogram: $A = b \times h$

Trapezoid: $A = \frac{1}{2}(a + b) \times h$

Rhombus: $A = \frac{1}{2} d_1 \cdot d_2$

Complex shapes can be computed by adding or subtracting areas of simpler shapes.

Adding and subtracting segment lengths to find unknown distances.

Rectangle: $P = 2(l + w)$

Square: $P = 4s$

Circle (circumference): $C = 2\pi r$

Triangle: $P = a + b + c$

where $s$ is the side length.

All sides equal, all angles equal.

Interior angle: $\frac{(n-2) \times 180°}{n}$

Exterior angle: $\frac{360°}{n}$

Two figures are similar if they have the same shape but not necessarily the same size — corresponding angles are equal and sides are proportional.

Volume is the amount of 3D space enclosed by a solid figure, measured in cubic units.

Surface area is the total area of all outer faces/surfaces of a 3D figure.

Counterclockwise = positive rotation direction.

Clockwise = negative rotation direction.

$90°, 180°, 270°$ (CCW) vs $-90°, -180°$ (CW)

In congruence and similarity, the order of vertex labels defines which parts correspond:

A circle is the set of all points in a plane that are equidistant from a fixed center point. The distance is the radius.

The incircle is the largest circle that fits inside a triangle, tangent to all three sides. Its center (incenter) is at the intersection of the angle bisectors.

The circumcircle passes through all three vertices of a triangle. Its center (circumcenter) is at the intersection of the perpendicular bisectors.

The arc angle equals the central angle that subtends it.

The sector angle is the central angle that forms the sector — used to compute sector area and arc length.

Points on a perpendicular bisector are equidistant from the segment's endpoints.

If $P$ is on the perp. bisector of $\overline{AB}$, then $PA = PB$.

Points on an angle bisector are equidistant from the sides of the angle (distance measured perpendicular to each side).

The distance from a point to a line is measured along the perpendicular segment — this is always the shortest distance.

Two lines are parallel if and only if the distance between them is constant everywhere.

One radian is the angle subtended by an arc whose length equals the radius.

Degrees to radians:

Radians to degrees:

Degrees use the $°$ symbol: $45°$

Radians typically have no symbol: $\frac{\pi}{4}$

Occasionally written as "rad" in science/engineering.