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Complex Polygons: Concave, Convex, and Irregular Shapes

TX.G.10.A

Surface area of 3-dimensional shapes

Surface area and volume of prisms

Surface area and volume of pyramids

Surface area and volume of cylinders

Surface area and volume of cones

Surface area and volume of spheres

TX.G.10.B

Upper and lower bound

Upper and Lower Bound

auto-gen

TX.G.11.B

Areas and sectors of circles

TX.G.11.C

Introduction to surface area of 3-dimensional shapes

Nets of 3-dimensional shapes

TX.G.11.D

Introduction to volume

Calculate the Volume

TX.G.12.A

Angles in a circle

Chord properties

Tangent properties

Circles and circumference

Circle properties

Central angles and proofs

Inscribed angles and proofs

Central and inscribed angles in circles

TX.G.12.B

Arcs of a circle

Circle chord, tangent, and inscribed angles proofs

TX.G.12.D

Converting between degrees and radians

Trigonometric ratios of angles in radians

Radian measure and arc length

TX.G.12.E

Conics - Circle

TX.G.13.A

Permutations

Combinations

Problems involving both permutations and combinations

TX.G.13.B

Probability

TX.G.13.C

Probability of independent events

Addition rule for 

Addition Rule for "OR"

Addition rule for "OR"

Multiplication rule for 

Multiplication rule for "AND"

Multiplication Rule for "AND"

TX.G.13.D

Conditional probability

Probability with Venn diagrams

TX.G.13.E

Determining probabilities using tree diagrams and tables

Probability involving permutations and combinations

TX.G.2.A

Midpoint formula: \(M = ( \frac{x_1+x_2}2  ,\frac{y_1+y_2}2)\)

Midpoint formula

Midpoint formula: <katex>M = ( \frac{x_1+x_2}2  ,\frac{y_1+y_2}2)</katex>

Midpoint formula: <katex>M = ( \frac{x_1+x_2}2 ,\frac{y_1+y_2}2)</katex>

TX.G.2.B

Distance formula: \(d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

Distance formula: <katex>d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}</katex>

Distance formula

Slope equation: \(m = \frac{y_2-y_1}{x_2- x_1}\)

Slope equation: <katex>m = \frac{y_2-y_1}{x_2- x_1}</katex>

Parallel and perpendicular lines in linear functions

TX.G.2.C

Parallel line equation

Perpendicular line equation

Combination of both parallel and perpendicular line equations

TX.G.3.A

Rotational symmetry and transformations

Transformations of functions: Horizontal translations

Transformations of functions: Vertical translations

TX.G.3.B

Combining transformations of functions

Enlargements and reductions with scale factors

TX.G.3.C

Scale diagrams

TX.G.3.D

Line symmetry

Reflection across the y-axis: y = f(-x)

Reflection across the x-axis: y = -f(x)

TX.G.4.A

Inductive reasoning

Deductive reasoning

TX.G.4.B

Conditionals

Inverses, converses, and contrapositives

Biconditionals

TX.G.4.D

Algebraic proofs

TX.G.5.A

Congruence and congruent triangles

Pairs of lines and angles

Parallel lines and transversals

Perpendicular line proofs

TX.G.5.D

Classifying triangles

TX.G.6.A

Isosceles and equilateral triangles

Parallel line proofs

TX.G.6.B

Triangles congruent by SSS proofs

Triangles congruent by SAS and HL proofs

Triangles congruent by ASA and AAS proofs

Triangles

TX.G.6.C

Rectangles

TX.G.6.D

Combination of SohCahToa questions

Pythagorean theorem

Pythagorean Theorem

Using the pythagorean relationship

Applications of pythagorean theorem

TX.G.6.E

Classifying quadrilaterals

Classifying Quadrilaterals

Word Problem: Classifying Quadrilaterals

Parallelograms

Trapezoids, rhombi, and kites

Polygons

Complex polygons

TX.G.7.A

Similar triangles

Similar polygons

TX.G.8.B

Estimating square roots

TX.G.9.A

Use sine ratio to calculate angles and sides (Sin = o / h)

Use cosine ratio to calculate angles and sides (Cos = a / h)

Use tangent ratio to calculate angles and sides (Tan = o / a)

TX.G.9.B

Solving expressions using 45-45-90 special right triangles

Solving expressions using 30-60-90 special right triangles

Assignments

Challenge Zone

Foundation Review

Circles

Coordinate and transformational geometry

Logical argument and constructions

Proof and congruence

Two-dimensional and three-dimensional figures

math

This lesson explains complex polygons, focusing on the difference between convex and concave shapes such as concave pentagons, hexagons, and octagons, and walks through finding the area of an irregular polygon using triangle decomposition.

Area and perimeter of complex polygons

Complex Polygons | StudyPug

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Overview

Complex Polygons: Concave vs Convex Polygons

What You'll Learn

Identify a convex polygon by checking that every interior angle is less than 180 degrees

Recognize a concave polygon by spotting at least one reflex angle greater than 180 degrees

Compare concave pentagons, hexagons, and octagons using the same reflex-angle test regardless of side count

Apply triangle decomposition to calculate the area of an irregular or concave polygon

Subtract the area of a reflex notch when a diagonal of a concave shape falls outside the figure

Calculate total area by summing or subtracting the areas of simple shapes that make up a complex polygon

This Unit Includes

Learning resources