gre-math-review

Part 2: Geometry

Geometry involves the study of shapes, sizes, and the properties of space. Key concepts in geometry include:

1. Basic Shapes: Common geometric shapes include triangles, rectangles, squares, circles, and polygons. Each shape has unique properties and formulas related to its dimensions.

2. Angles: An angle is formed by two rays with a common endpoint. Angles are measured in degrees and can be classified as acute (< 90°), right (= 90°), obtuse (> 90° and < 180°), straight (= 180°), and reflex (> 180°).

3. Perimeter: The perimeter of a shape is the total distance around it. For example:

   • A rectangle's perimeter is calculated as $P = 2(length + width)$.

   • A circle's perimeter (circumference) is calculated as $C = 2 \text{π}r$, where $r$ is the radius.

4. Area: The area of a shape is a measure of the space contained within it. For example:

   • The area of a rectangle is given by $A = length \times width$.

   • The area of a triangle is given by $A = \frac{1}{2} \times base \times height$.

   • The area of a circle is given by $A = \text{π}r^2$.

5. Volume: Volume measures the amount of space an object occupies. Examples of volume formulas include:

   • The volume of a rectangular prism (box) is calculated as $V = length \times width \times height$.

   • The volume of a cylinder is given by $V = \text{π}r^2h$, where $h$ is the height.

6. Surface Area: The surface area of a three-dimensional object is the total area of its surface. Examples include:

   • The surface area of a rectangular prism is given by $SA = 2lw + 2lh + 2wh$, where $l$ is length, $w$ is width, and $h$ is height.

   • The surface area of a sphere is given by $SA = 4 \text{π}r^2$, where $r$ is the radius.

   • The surface area of a cylinder is given by $SA = 2 \text{π}r^2 + 2 \text{π}rh$, where $r$ is the radius and $h$ is the height.

7. The Pythagorean Theorem: In a right triangle, the relationship between the lengths of the legs and the hypotenuse is described by the formula: $a^2 + b^2 = c^2$, where $c$ is the length of the hypotenuse.

8. Similar and Congruent Shapes: Two shapes are similar if their corresponding angles are equal and their sides are proportional. Two shapes are congruent if they are identical in shape and size.

Understanding these fundamental geometry concepts provides a strong basis for solving geometric problems and proofs.