Geometry and Trigonometry Overview

Lines and Angles: You'll need to understand the relationships between lines and angles, including vertical angles, complementary angles, supplementary angles, and angles formed by parallel lines cut by a transversal.

Polygons: A polygon is a closed figure formed by three or more line segments. You'll learn about the sum of interior angles in polygons, exterior angles, and the properties of regular polygons.

Quadrilaterals: Special types of four-sided figures include parallelograms, rectangles, rhombuses, squares, and trapezoids. Each has unique properties that you'll need to understand.

Problems in this category often involve:

• Finding unknown angle measures using angle relationships
• Determining the number of sides in a polygon given the sum of its interior angles
• Identifying properties of quadrilaterals based on given information
• Solving problems involving similar polygons

Triangle Properties: You'll need to understand the sum of angles in a triangle (always 180°), the Pythagorean theorem for right triangles, and special right triangles (30-60-90 and 45-45-90).

Similar Triangles: Triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. This concept is crucial for solving many geometric problems.

Trigonometric Functions: The basic trigonometric functions (sine, cosine, tangent) relate the angles of a right triangle to the ratios of its sides. These functions are defined as:

$\sin \theta =\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}$

$\cos \theta =\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}$

$\tan \theta =\frac{\mathrm{opposite}}{\mathrm{adjacent}}$

Problems in this category often involve:

• Finding unknown sides or angles in right triangles using the Pythagorean theorem or trigonometric functions
• Using properties of special right triangles to find unknown measurements
• Solving problems involving similar triangles
• Applying trigonometric relationships to solve real-world problems

Basic Circle Properties: You'll need to understand the radius, diameter, circumference, and area of a circle, as well as the relationships between these measurements.

Angles in Circles: Central angles, inscribed angles, and angles formed by tangents and chords have special properties that you'll need to understand.

Arcs and Sectors: The length of an arc and the area of a sector are proportional to the central angle that defines them.

Tangents and Chords: A tangent to a circle is perpendicular to the radius at the point of tangency. Chords have special properties related to their distance from the center of the circle.

Problems in this category often involve:

• Finding the circumference or area of a circle
• Determining the length of an arc or the area of a sector
• Finding unknown angle measures in circles
• Solving problems involving tangents and chords

Equation of a Circle: The standard form of the equation of a circle with center $(h,k)$ and radius $r$ is:

${(}^x+{(}^y=r^2$

Distance Formula: The distance between two points $(x_1,y_1)$ and $(x_2,y_2)$ is:

$\sqrt{{(}^{x_2}+{(}^{y_2}}$

Problems in this category often involve:

• Writing the equation of a circle given its center and radius
• Finding the center and radius of a circle given its equation
• Determining whether a point lies inside, on, or outside a circle
• Finding the distance between points in the coordinate plane

Area: The area of a shape is the amount of space it occupies in a two-dimensional plane. Common formulas include:

• Rectangle: $\mathrm{Area}=\mathrm{length}\times \mathrm{width}$
• Triangle: $\mathrm{Area}=\frac{1}{2}\times \mathrm{base}\times \mathrm{height}$
• Circle: $\mathrm{Area}=\pi r^2$

Surface Area: The surface area of a three-dimensional shape is the total area of all its faces. Common formulas include:

• Rectangular Prism: $\mathrm{Surface\; Area}=2\times \mathrm{length}\times \mathrm{width}+2\times \mathrm{length}\times \mathrm{height}+2\times \mathrm{width}\times \mathrm{height}$
• Cube: $\mathrm{Surface\; Area}=6\times {\mathrm{side}}^2$

Volume: The volume of a three-dimensional shape is the amount of space it occupies. Common formulas include:

• Rectangular Prism: $\mathrm{Volume}=\mathrm{length}\times \mathrm{width}\times \mathrm{height}$
• Cube: $\mathrm{Volume}={\mathrm{side}}^3$
• Cylinder: $\mathrm{Volume}=\pi r^{2}h$
• Sphere: $\mathrm{Volume}=\frac{4}{3}\pi r^3$

Note: The formulas for area, surface area, and volume of various shapes are provided on the SAT reference sheet.

Problems in this category often involve:

• Finding the area of two-dimensional shapes
• Calculating the surface area of three-dimensional shapes
• Determining the volume of three-dimensional shapes
• Solving problems involving scaling (how changing dimensions affects area and volume)
• Finding the dimensions of a shape given its area, surface area, or volume

Create or Redraw Diagrams: Whenever possible, create or redraw diagrams on your scratch paper. This helps you visualize the problem and identify relationships between different elements.

Extend Lines: It often helps to draw imaginary lines extending from polygons. This can reveal hidden relationships and make it easier to apply geometric principles.

Create Triangles: When you can, try to create triangles within more complex shapes. Triangles have well-defined properties that can be used to solve problems.

The SAT provides a reference sheet with formulas for area, volume, and other geometric measurements. Make sure you're familiar with this sheet and know how to use it effectively.

Remember that you don't need to memorize these formulas, but you should understand what they mean and how to apply them.

Complex geometry problems can often be broken down into simpler parts. Look for:

• Right triangles, which allow you to use the Pythagorean theorem
• Special triangles (30-60-90, 45-45-90), which have known side ratios
• Similar figures, which have proportional sides
• Congruent figures, which have equal sides and angles

By identifying these simpler parts, you can often solve complex problems step by step.