Mathematics Geometry Syllabus

State the equation of a circle with center $(h, k)$ and radius $r$.

$$(x - h)^{2} + (y - k)^{2} = r^{2}$$

What is the volume of a rectangular prism (box) with dimensions $l$, $w$, $h$?

$V = l w h$

What is the volume of a cylinder with radius $r$ and height $h$?

$V = \pi r^{2} h$

What is the volume of a cone with radius $r$ and height $h$?

$V = \frac{1}{3}\pi r^{2} h$

What is the volume of a sphere of radius $r$?

$V = \frac{4}{3}\pi r^{3}$

What is the surface area of a sphere of radius $r$?

$A = 4\pi r^{2}$

What is the volume of a pyramid with base area $B$ and height $h$?

$V = \frac{1}{3} B h$

Define complementary and supplementary angles.

Complementary angles sum to $90^{\circ}$; supplementary angles sum to $180^{\circ}$.

State the Law of Cosines for a triangle with sides $a$, $b$, $c$ and angle $C$ opposite side $c$.

$$c^{2} = a^{2} + b^{2} - 2ab\cos C$$

State the Law of Sines for a triangle with sides $a$, $b$, $c$ and opposite angles $A$, $B$, $C$.

$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$

What are the triangle congruence criteria?

SSS, SAS, ASA, AAS, and HL (hypotenuse-leg, for right triangles).

What are the triangle similarity criteria?

AA (angle-angle), SSS (proportional sides), and SAS (two proportional sides with equal included angle).

When two parallel lines are cut by a transversal, what is true of alternate interior angles?

They are congruent (equal in measure).

When two parallel lines are cut by a transversal, what is true of corresponding angles?

They are congruent (equal in measure).

What is the relationship between vertical angles?

Vertical (opposite) angles are always congruent.

State the Inscribed Angle Theorem.

An inscribed angle is half the central angle that subtends the same arc: inscribed angle $= \frac{1}{2}\times$ central angle.

What is the measure of an angle inscribed in a semicircle?

$90^{\circ}$ (Thales' theorem); the angle subtended by a diameter is a right angle.

What is the arc length of a sector with central angle $\theta$ (radians) in a circle of radius $r$?

$s = r\theta$

What is the area of a circular sector with central angle $\theta$ (radians) and radius $r$?

$A = \frac{1}{2} r^{2}\theta$

State the Triangle Inequality for sides $a$, $b$, $c$ of a triangle.

The sum of any two sides exceeds the third: $a + b > c$, $a + c > b$, and $b + c > a$.

What is the lateral (curved) surface area of a cylinder with radius $r$ and height $h$?

$A = 2\pi r h$

What is the total surface area of a closed cylinder with radius $r$ and height $h$?

$A = 2\pi r h + 2\pi r^{2} = 2\pi r(h + r)$

What is the lateral surface area of a cone with radius $r$ and slant height $l$?

$A = \pi r l$

Define a chord, a diameter, and a radius of a circle.

A chord is a segment joining two points on the circle; a diameter is a chord through the center (the longest chord); a radius is a segment from the center to the circle, equal to half the diameter.

What is the centroid of a triangle and how does it divide each median?

The centroid is the intersection of the three medians; it divides each median in a $2:1$ ratio from vertex to midpoint.

How many degrees are in one full rotation, and what is this in radians?

$360^{\circ} = 2\pi$ radians.

State the conversion between degrees and radians.

$\text{radians} = \text{degrees}\times\frac{\pi}{180^{\circ}}$

What is the relationship between the diagonals of a rhombus?

The diagonals are perpendicular bisectors of each other (they meet at right angles and bisect one another).

For similar figures with linear scale factor $k$, how do areas and volumes scale?

Areas scale by $k^{2}$ and volumes scale by $k^{3}$.

What is the perimeter of a regular polygon with $n$ sides each of length $s$?

$P = n s$