Introduction
Before confronting the infinite complexity of the circle, we master the two specific triangles that build it.
Past Knowledge
This comes directly from Geometry (Isosceles Triangles and Equilateral Triangles split in half).
Today's Goal
We memorize the side ratios: for 45-45-90 and for 30-60-90.
Future Success
In Calculus, you will evaluate limits like and compute definite integrals. You will not have a calculator. You must be able to visualize these triangles instantly to find exact values.
Key Concepts
45-45-90 (Isosceles Right)
From a square cut in half diagonally.
Legs: x, x
Hypotenuse:
"Legs are same, Hypotenuse gets root 2"
30-60-90 (Scalene Right)
From an equilateral triangle cut in half.
Short Leg (opp 30): x
Long Leg (opp 60):
Hypotenuse: 2x
"Hypotenuse is double the short leg"
Worked Examples
Example 1: Finding Hypotenuse
Given a 45-45-90 triangle with legs of length 5, find the hypotenuse.
Identify Pattern
Legs are . Hypotenuse is .
Substitute
.
Example 2: Working Backwards
The hypotenuse of a 30-60-90 triangle is 10. Find the legs.
Find Short Leg First
This is crucial. Always find the short leg (opposite 30) first.
.
.
Find Long Leg
Long Leg is .
Long Leg:
Example 3: Area Calculation
Find the area of an equilateral triangle with side length 8.
Split into Right Triangles
Drop a height altitude. This splits the base (8) into two segments of 4. Now we have a 30-60-90 triangle.
Find Height
Short leg = 4. Angle opposite height is 60.
Height = Long Leg = .
Calculate Area
.
Common Pitfalls
Mixing up the 60 and 30 sides
Remember: The Shortest side is always opposite the Smallest angle (30). The Longest side (hypotenuse) is opposite the Largest angle (90).
Real-World Application
Construction & Carpentry
Roof pitches often come in standard ratios. A "12/12 pitch" roof rises 12 inches for every 12 inches of run. This creates a perfect 45-45-90 triangle. Carpenters use this fact ('Hip Rafters' = run x 1.414, where 1.414 is ) to cut beams without measuring every single one.
Practice Quiz
Practice Quiz
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