Geometry Essentials for Beginning-Intermediate Algebra

Roots and Radicals

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. Square roots are only defined for non-negative numbers in real numbers.

• Key Point 1: $\sqrt{x}$ is not defined for $x < 0$ in real numbers.

• Key Point 2: $\sqrt{x}$ is defined for $x \geq 0$.

• Example: $\sqrt{25} = 5$ because $5 \times 5 = 25$.

Lines and Angles

Parallel and Perpendicular Lines

Parallel lines never intersect and have the same slope. Perpendicular lines intersect at a right angle (90 degrees).

• Key Point 1: Parallel lines: $l_1 \parallel l_2$

• Key Point 2: Perpendicular lines: $l_1 \perp l_2$

• Example: Angle $X$ is $90^{\circ}$ for perpendicular lines.

Shapes and Properties

Similar and Congruent Shapes

Similar shapes have the same shape, angles, and proportions, but may differ in size. Congruent shapes are identical in shape, angles, proportions, and size.

• Key Point 1: Similar shapes: same shape, angles, proportions.

• Key Point 2: Congruent shapes: same shape, angles, proportions, and size.

• Example: Two triangles with equal corresponding angles and proportional sides are similar; if their sides are also equal, they are congruent.

Circles

Area and Circumference

The area of a circle measures the space inside the circle, while the circumference is the distance around the circle.

• Key Point 1: Area formula: $A = \pi r^2$

• Key Point 2: Circumference formula: $C = 2\pi r$

• Example: For a circle with radius $r = 3$, $A = \pi \times 3^2 = 9\pi$ and $C = 2\pi \times 3 = 6\pi$.

Annulus (Ring-Shaped Region)

An annulus is the region between two concentric circles. Its area is found by subtracting the area of the smaller circle from the area of the larger circle.

• Key Point 1: Area of annulus: $A = \pi R^2 - \pi r^2$

• Example: If $R = 5$ and $r = 3$, $A = \pi (25 - 9) = 16\pi$.

Sectors and Arcs

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

• Key Point 1: Area of sector: $A = \frac{\theta}{360^{\circ}} \pi r^2$ where $\theta$ is the central angle in degrees.

• Key Point 2: Arc length: $S = \frac{\theta}{360^{\circ}} 2\pi r$

• Example: For $r = 4$ and $\theta = 60^{\circ}$, $A = \frac{60}{360} \pi \times 16 = \frac{1}{6} \pi \times 16 = \frac{16\pi}{6}$

Inscribed and Exterior Angles

Inscribed angles are formed at the circumference of a circle, while exterior angles are formed outside the circle.

• Key Point 1: Inscribed angle: $\theta = \frac{1}{2}$ arc angle.

• Key Point 2: Exterior angle: angle formed outside the circle.

• Example: If the arc is $80^{\circ}$, the inscribed angle is $40^{\circ}$.

Triangles

Area and Perimeter of Triangles

The area of a triangle is calculated using its base and height. The perimeter is the sum of its sides.

• Key Point 1: Area formula: $A = \frac{1}{2} b h$

• Key Point 2: Perimeter formula: $P = a + b + c$

• Example: For a triangle with $b = 4$, $h = 3$, $A = \frac{1}{2} \times 4 \times 3 = 6$

Equilateral Triangles

An equilateral triangle has all sides of equal length and all angles equal to $60^{\circ}$.

• Key Point 1: Area formula: $A = \frac{\sqrt{3}}{4} s^2$

• Key Point 2: Perimeter formula: $P = 3s$

• Example: For $s = 5$, $A = \frac{\sqrt{3}}{4} \times 25$

Rectangles and Squares

Area and Perimeter of Rectangles

Rectangles have opposite sides equal and four right angles. The area is the product of length and width.

• Key Point 1: Area formula: $A = l \times w$

• Key Point 2: Perimeter formula: $P = 2l + 2w$

• Example: For $l = 6$, $w = 3$, $A = 18$, $P = 18$

Area and Perimeter of Squares

Squares have all sides equal and four right angles. The area is the side squared.

• Key Point 1: Area formula: $A = s^2$

• Key Point 2: Perimeter formula: $P = 4s$

• Example: For $s = 4$, $A = 16$, $P = 16$

Prisms and Cylinders

Surface Area and Volume of Rectangular Prisms

A rectangular prism is a three-dimensional shape with six rectangular faces. Surface area is the sum of the areas of all faces, and volume is the product of length, width, and height.

• Key Point 1: Surface area formula: $SA = 2lw + 2lh + 2wh$

• Key Point 2: Volume formula: $V = lwh$

• Example: For $l = 2$, $w = 3$, $h = 4$, $SA = 2(2 \times 3) + 2(2 \times 4) + 2(3 \times 4) = 52$, $V = 24$

Surface Area and Volume of Cylinders

A cylinder has two parallel circular bases and a curved surface. Surface area includes the bases and the lateral area; volume is the area of the base times the height.

• Key Point 1: Surface area formula: $SA = 2\pi r^2 + 2\pi r h$

• Key Point 2: Volume formula: $V = \pi r^2 h$

• Example: For $r = 3$, $h = 5$, $SA = 2\pi \times 9 + 2\pi \times 3 \times 5 = 18\pi + 30\pi = 48\pi$, $V = 45\pi$

Spheres

Surface Area and Volume of Spheres

A sphere is a perfectly round three-dimensional shape. Surface area and volume are calculated using the radius.

• Key Point 1: Surface area formula: $SA = 4\pi r^2$

• Key Point 2: Volume formula: $V = \frac{4}{3}\pi r^3$

• Example: For $r = 3$, $SA = 36\pi$, $V = 36\pi$

Right Triangles and the Pythagorean Theorem

Pythagorean Theorem

The Pythagorean Theorem relates the sides of a right triangle. The square of the hypotenuse equals the sum of the squares of the other two sides.

• Key Point 1: Formula: $a^2 + b^2 = c^2$

• Key Point 2: Used to find the length of a side in a right triangle.

• Example: If $a = 3$, $b = 4$, $c = \sqrt{9 + 16} = 5$

Surface Area and Volume

Definitions

Surface area is the total area covering the surface of a 3D shape. Volume is the space inside a 3D shape, measured in cubic units.

• Key Point 1: Surface area is measured in square units.

• Key Point 2: Volume is measured in cubic units.

Summary Table: Area and Volume Formulas

The following table summarizes the main area and volume formulas for common shapes:

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	--	$SA = 2\pi r^2 + 2\pi r h$	$V = \pi r^2 h$
Sphere	--	$SA = 4\pi r^2$	$V = \frac{4}{3}\pi r^3$

Additional info: These notes cover essential geometry concepts relevant to beginning-intermediate algebra, including area, perimeter, surface area, and volume formulas, as well as properties of shapes and the Pythagorean theorem.