Geometry Essentials for College Algebra: Circles, Triangles, and Solids

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 $y^2 = x$.

• Examples: $\sqrt{4} = 2$, $\sqrt{9} = 3$, $\sqrt{25} = 5$

• Not Defined: $\sqrt{-1}$ is not defined in real numbers.

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

• Parallel: $l_1 \parallel l_2$

• Perpendicular: $l_1 \perp l_2$

• Angle: $\angle X$

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. $A = \pi r^2$

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

• Diameter (d): $d = 2r$

• Circumference: Distance around the circle. $C = 2\pi r$

Annulus (Ring-Shaped Region)

An annulus is the region between two concentric circles.

• Area of Annulus: $A = \pi (R^2 - r^2)$, 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: $A = \frac{1}{2} r^2 \theta$ (where $\theta$ is in radians)

• Arc Length: $S = r\theta$

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: $A = \frac{1}{2} \text{base} \times \text{height}$

• Perimeter: Sum of all sides.

• Equilateral Triangle: All sides equal; $A = \frac{\sqrt{3}}{4} s^2$

Right Triangles and Pythagorean Theorem

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

• Pythagorean Theorem: $a^2 + b^2 = c^2$ (where c is the hypotenuse)

Rectangles and Squares

Rectangle Properties

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

• Area: $A = l \times w$

• Perimeter: $P = 2l + 2w$

Square Properties

Squares are rectangles with all sides equal.

• Area: $A = s^2$

• Perimeter: $P = 4s$

Solids: Prisms, Cylinders, and Spheres

Rectangular Prisms

Prisms are three-dimensional shapes with rectangular faces.

• Surface Area: $SA = 2lw + 2lh + 2wh$

• Volume: $V = lwh$

Cylinders

Cylinders have circular bases and a curved surface.

• Surface Area: $SA = 2\pi r^2 + 2\pi rh$

• Lateral Area: $LA = 2\pi rh$

• Volume: $V = \pi r^2 h$

Spheres

Spheres are perfectly round three-dimensional objects.

• Surface Area: $SA = 4\pi r^2$

• Volume: $V = \frac{4}{3}\pi r^3$

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

Shape	Area Formula	Perimeter/Surface Area Formula	Volume Formula
Circle	$A = \pi r^2$	$C = 2\pi r$	—
Rectangle	$A = l \times w$	$P = 2l + 2w$	—
Square	$A = s^2$	$P = 4s$	—
Triangle	$A = \frac{1}{2} b h$	$P = a + b + c$	—
Rectangular Prism	—	$SA = 2lw + 2lh + 2wh$	$V = lwh$
Cylinder	—	$SA = 2\pi r^2 + 2\pi rh$	$V = \pi r^2 h$
Sphere	—	$SA = 4\pi r^2$	$V = \frac{4}{3}\pi r^3$