PSAT 10 Math — Geometry & Right-Triangle Trigonometry (Learn-from-Scratch Notes)

Area

The amount of 2D space inside a boundary, measured in square units like $\text{cm}^2$ or $\text{ft}^2$.

Volume

The amount of 3D space an object occupies, measured in cubic units like $\text{cm}^3$.

Circumference

The perimeter of a circle, calculated as $C = 2\theta\text{r}$ where $\theta$ is approximated as $3.14$.

Pythagorean theorem

In a right triangle, relates the lengths of the sides: $a^2 + b^2 = c^2$.

Composite area

The area of a shape that can be divided into simpler shapes whose areas can be easily calculated.

Scaling factor

A multiplier that changes the size of an object; affects length, area, and volume differently.

Rectangle area formula

$A = l \times w$ where $l$ is length and $w$ is width.

Triangle area formula

$A = \frac{1}{2} b h$ where $b$ is the base and $h$ is the height.

Trapezoid area formula

$A = \frac{1}{2} (b_1 + b_2) h$ where $b_1$ and $b_2$ are the parallel bases.

Circle area formula

$A = \pi r^2$ where $r$ is the radius.

Rectangular prism volume formula

$V = lwh$, where $l$ is length, $w$ is width, and $h$ is height.

Cylinder volume formula

$V = \pi r^2 h$ where $r$ is radius and $h$ is height.

Cone volume formula

$V = \frac{1}{3} \pi r^2 h$ where $r$ is radius and $h$ is height.

Sphere volume formula

$V = \frac{4}{3} \pi r^3$ where $r$ is radius.

Surface area of a rectangular prism

$SA = 2(lw + lh + wh)$, calculating total area of all outer faces.

Surface area of a cylinder

$SA = 2\pi r^2 + 2\pi rh$, includes areas of two circles and the side rectangle.

Angle of elevation

The angle formed by a horizontal line and a line pointing upward to an object.

Angle of depression

The angle formed by a horizontal line and a line pointing downward to an object.

Vertical angles

Angles opposite each other when two lines intersect; they are always equal.

Complementary angles

Two angles that add up to $90^{\text{o}}$.

Supplementary angles

Two angles that add up to $180^{\text{o}}$.

Similar triangles

Triangles that have the same shape but may differ in size; corresponding angles are equal.

Congruent triangles

Triangles that are exactly the same in size and shape; corresponding sides and angles are equal.

Sine ratio

$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$.

Cosine ratio

$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$.

Tangent ratio

$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$.

Trigonometric ratios

Ratios in right triangles that define relationships between the angles and lengths of the sides.