Shadow Arts Slides SHADOWS &
SHAPES
Trigonometry in Art and the Real World
SINE COSINE TANGENT
The Ratio Toolkit
SOH CAH TOA
Sine = Opposite / Hypotenuse
Cosine = Adjacent / Hypotenuse
Tangent = Opposite / Adjacent
Trigonometry allows us to find missing lengths and angles using the relationships between sides in right triangles.
Adjacent Opposite Hypotenuse θ
Part 1: Shadow Hunters
How do you measure a skyscraper or a flagpole without a ladder?
The Mission
1 Measure the length of a shadow cast by a stationary object.
2 Determine the angle of elevation of the Sun.
3 Apply the Tangent ratio to solve for the object's height.
Shadow Trigonometry
Part 2: Trig Tapestry
Mathematics is the skeleton of beauty. You will design a geometric masterpiece where every line is calculated.
Must include 5 different right triangles.
Must prove side lengths using ratios.
Must use symmetry and repetition.
Project Timeline
Day 1
Field Work: Data collection & shadow tracking.
Day 2
Math Lab: Height calculations & art sketching.
Day 3
Studio Time: Final art construction & portfolio prep.
Day 4
Exhibition: Gallery walk & peer peer reviews.
Ready to measure the world?
Shadow Hunter Worksheet Shadow Hunter Record
Field Operations: Trigonometric Surveying
Name:
Date:
The Tangent Method
To find the height of an object, we use the shadow it casts and the angle of the sun. In a right triangle formed by the object and its shadow:
Formula
tan(θ) = Height / Shadow
Solve For Height
Height = Shadow × tan(θ)
θ (Angle) Shadow Length Object Height
Sun Elevation Diagram
Field Measurements
Object Name Shadow (m) Angle (θ) Calculated Height Example: Flagpole 4.5 m 32° ≈ 2.81 m
Show Your Work
Select one object and show the full step-by-step calculation for its height.
Analysis & Reflection
1. If you measured the shadow of the same object two hours later, how would the Angle of Elevation change? How would this affect the shadow length?
2. What are some potential sources of error when measuring shadows or sun angles in the field?
Trig Tapestry Project Guide Trig Tapestry
Geometric Art Design Project
Artist:
Period:
The Creative Challenge
Your task is to create a complex geometric pattern or artistic composition using at least five unique right triangles . Every triangle in your design must be "math-legal"—meaning its side lengths and angles must be verified using SOH CAH TOA.
Success Criteria
Minimum of 5 right triangles of different sizes.
Verify at least one triangle with Sine , one with Cosine , and one with Tangent .
Final piece must use at least three distinct colors.
Incorporate a field-work element: One triangle's proportions should match a shadow-ratio from your field worksheet.
Calculation Proof
Show the math for your primary design triangle:
Angle A (Degrees)
Side b (cm)
Step-by-Step Proof
Protractors Used
Colored Pencils Ready
Ratios Calculated
Design Canvas
One Grid Square = 1 cm²
Art Construction Zone
Precision leads to beauty. Calculate twice, draw once.
Ratio Masters Teacher Guide Ratio Masters
Teacher Implementation & Rubric Guide
Target Level
MS / HS Geom
Learning Objectives
• Apply the Tangent ratio to solve indirect measurement problems in real-world contexts.
• Verify geometric constructions using Sine, Cosine, and Tangent identities.
• Analyze the relationship between sun elevation and shadow length.
Standards
CCSS.HSG.SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
CCSS.8.G.B.7
Apply the Pythagorean Theorem to determine unknown side lengths in right triangles.
Instructional Scaffolding
Middle School
Focus on Tangent as a slope relationship. Use protractors for elevation.
High School
Require inverse trig functions to find angles from measured heights.
Extension
Incorporate Law of Sines/Cosines for non-right artistic triangles.
Supplies Checklist
Meter sticks or tape measures
Clinometers (or protractors/straws)
Scientific calculators
Colored pencils / Art markers
Drafting compasses / Rulers
Final Project Rubric
50 Points Total
Criteria Proficient (10) Developing (6) Emerging (3) Field Data 3+ measurements; accurate use of Tangent. 2 measurements; minor calc errors. Missing data or incorrect formula. Math Proof Proof uses SOH CAH TOA for 5 triangles. Proof for 3 triangles; minor errors. Missing proofs or incorrect IDs. Artistic Quality Design is clean, colorful, and symmetric. Design is complete but lacks neatness. Unfinished or messy construction. Geometric Logic Creative use of right triangles throughout. Uses triangles but design is disjointed. Right triangles not identifiable. Reflection Insightful analysis of variables/errors. Basic answers to reflection questions. Incomplete or missing reflection.
