Geometry Fundamentals

Square Roots and Irrational Numbers

Understanding square roots is essential for solving equations and working with geometric formulas. The square root of a negative number is not defined in the set of real numbers.

• Square Root Definition: The square root of a positive number x is a value y such that .

• Examples: , ,

• Not Defined: is not defined in real numbers.

Parallel and Perpendicular Lines: Parallel lines never intersect; perpendicular lines intersect at a right angle.

• Parallel:

• Perpendicular:

• Angle:

Similarity and Congruence

Shapes can be classified as similar or congruent based on their properties.

• Similar Shapes: Same shape, angles, and proportions.

• Congruent Shapes: Same shape, angles, proportions, and size.

Proofs: Reasoning and logic are used to establish geometric facts.

Circles

Area and Circumference

The area and circumference of a circle are fundamental concepts in geometry.

• Area: Measure of the space inside a circle.

• Radius (r): Distance from center to circumference.

• Diameter (d):

• Circumference: Distance around the circle.

Annulus (Ring-Shaped Region)

An annulus is the region between two concentric circles.

• Area of Annulus: , where R is the outer radius and r is the inner radius.

Sectors and Arcs

A sector is a region of a circle bounded by two radii and the arc between them. An arc is a portion of the circle's circumference.

• Area of Sector: (where is in radians)

• Arc Length:

Central and Inscribed Angles

Central angles are formed at the center of the circle, while inscribed angles are formed on the circumference.

• Central Angle: Angle whose vertex is at the center.

• Inscribed Angle: Angle whose vertex is on the circle; its measure is half the central angle.

Triangles

Triangle Properties

Triangles are classified by their sides and angles. The area and perimeter are key properties.

• Area:

• Perimeter: Sum of all sides.

• Equilateral Triangle: All sides equal;

Right Triangles and Pythagorean Theorem

Right triangles have one 90-degree angle. The Pythagorean theorem relates the sides.

• Pythagorean Theorem: (where c is the hypotenuse)

Rectangles and Squares

Rectangle Properties

Rectangles have opposite sides equal and all angles are right angles.

• Area:

• Perimeter:

Square Properties

Squares are rectangles with all sides equal.

• Area:

• Perimeter:

Solids: Prisms, Cylinders, and Spheres

Rectangular Prisms

Prisms are three-dimensional shapes with rectangular faces.

• Surface Area:

• Volume:

Cylinders

Cylinders have circular bases and a curved surface.

• Surface Area:

• Lateral Area:

• Volume:

Spheres

Spheres are perfectly round three-dimensional objects.

• Surface Area:

• Volume:

Defining Area and Volume

Area

Area is measured in square units and refers to the space inside a 2D shape.

Volume

Volume is measured in cubic units and refers to the space inside a 3D shape.

Summary Table: Area and Volume Formulas