Special Triangles & Applications

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Introduction​

Triangles are the foundation of geometry. Among all polygons, triangles are unique because they are the simplest closed shape — and yet they appear everywhere in architecture, engineering, art, and nature.
This guide covers every type of special triangle you need to know, complete with formulas, properties, diagrams explained in text, and practice problems to test your understanding.

1. Equilateral Triangle​

Properties​

• All three sides are equal
• All three angles = 60°
• All four triangle centers (I, O, G, H) coincide at the same point
• AD is simultaneously the median, altitude, angle bisector, and perpendicular bisector

Key Formulas​

Measurement	Formula
Height	h = (√3 ÷ 2) × a
Area	A = (√3 ÷ 4) × a²
Circumradius	R = a ÷ √3 = (2 ÷ 3) × h
Inradius	r = a ÷ (2√3) = h ÷ 3

Special Ratios​

• R : r = 2 : 1
• Area of circumcircle : Area of incircle = 4 : 1

Equilateral Triangle Relations​

Side	Height	Area
a	(√3 ÷ 2) × a	(√3 ÷ 4) × a²

If side = 2k → height = √3k → area = √3k²

Example​

An equilateral triangle has side = 6 cm

• Height = (√3 ÷ 2) × 6 = 3√3 cm ≈ 5.196 cm
• Area = (√3 ÷ 4) × 36 = 9√3 cm² ≈ 15.59 cm²
• Circumradius = 6 ÷ √3 = 2√3 cm ≈ 3.46 cm
• Inradius = 6 ÷ (2√3) = √3 cm ≈ 1.73 cm

2. Isosceles Triangle​

Properties​

• Two sides are equal: AB = AC
• All four triangle centers lie on AD (the axis of symmetry)

Key Formulas​

Measurement	Formula
Height	AD = √(a² − b²÷4)
Area	A = (b ÷ 4) × √(4a² − b²)

Where:

• a = equal sides
• b = base

Example​

Isosceles triangle with equal sides = 5 cm, base = 6 cm

• Height = √(25 − 9) = √16 = 4 cm
• Area = (6 ÷ 4) × √(100 − 36) = 1.5 × √64 = 1.5 × 8 = 12 cm²

3. Isosceles Right Triangle​

Properties​

• Angles: 45°, 45°, 90°
• Two legs are equal

Key Formulas​

Measurement	Formula
Legs	Leg = H ÷ √2
Area	A = H² ÷ 4
Perimeter	P = H(√2 + 1)

Where H = hypotenuse

Example​

Isosceles right triangle with hypotenuse = 10 cm

• Legs = 10 ÷ √2 = 5√2 cm ≈ 7.07 cm
• Area = 100 ÷ 4 = 25 cm²
• Perimeter = 10(√2 + 1) = 10(2.414) ≈ 24.14 cm

4. Right Angle Triangle​

Properties​

• One angle = 90°
• Inscribed in a semicircle (hypotenuse = diameter)

Key Formulas​

Measurement	Formula
Inradius	r = (P + B − H) ÷ (r + R) = (P + B) ÷ 2
Circumradius	R = BO = shortest median = H ÷ 2
Centroid distances	BG = H ÷ 3, GO = H ÷ 6
Area	A = r × S = S(S − 2R) = r² + 2rR

Where:

• P = perpendicular side
• B = base
• H = hypotenuse
• S = semi-perimeter

Example​

Right triangle with legs 3 cm and 4 cm

• Hypotenuse = √(9 + 16) = 5 cm
• Circumradius = 5 ÷ 2 = 2.5 cm
• Inradius = (3 + 4 − 5) ÷ 2 = 1 cm
• Area = 1 × (3+4+5÷2) = 6 cm²

5. Pythagorean Triplets​

Pythagorean triplets are sets of three integers that satisfy: a² + b² = c²

How to Generate Them​

Odd number method:

• Start with any odd number n
• Triplet: (n, (n²−1)÷2, (n²+1)÷2)
• Examples: (3,4,5) and (5,12,13)

Even number method:

• Start with any even number n
• Triplet: (n, (n²÷4)−1, (n²÷4)+1)
• Examples: (6,8,10) and (8,15,17)

Common Pythagorean Triplets to Memorize​

Triplet	Verification
3, 4, 5	9 + 16 = 25
5, 12, 13	25 + 144 = 169
6, 8, 10	36 + 64 = 100
8, 15, 17	64 + 225 = 289
7, 24, 25	49 + 576 = 625

6. Square Inscribed in a Triangle​

Key Formulas​

Situation	Formula
Base segments x and y	Side of square = xy ÷ (x + y)
Right triangle	Side = ab ÷ (a + b)
Largest square	y = abc ÷ (a² + b² + ab), where x > y

Example​

Triangle with base segments x = 4 cm and y = 6 cm

• Side of inscribed square = (4 × 6) ÷ (4 + 6) = 24 ÷ 10 = 2.4 cm

7. Scalene Triangle​

Properties​

• All three sides are unequal
• All three angles are different

Key Formulas​

Measurement	Formula
Perimeter	P = a + b + c
Semi-perimeter	s = (a + b + c) ÷ 2
Area (Heron's)	A = √(s(s−a)(s−b)(s−c))
Area (inradius)	A = r × s
Area (circumradius)	A = abc ÷ (4R)

Example​

Scalene triangle with sides 5 cm, 7 cm, 8 cm

• s = (5 + 7 + 8) ÷ 2 = 10 cm
• Area = √(10 × 5 × 3 × 2) = √300 = 17.32 cm²

Quick Reference — All Triangle Formulas​

Triangle Type	Area Formula	Special Property
Equilateral	(√3÷4) × a²	All sides equal, all angles 60°
Isosceles	(b÷4) × √(4a²−b²)	Two equal sides
Isosceles Right	H² ÷ 4	Angles 45-45-90
Right Angle	r × s	One angle 90°
Scalene	√(s(s−a)(s−b)(s−c))	All sides different

Practice Problems​

Problem 1: An equilateral triangle has side = 8 cm. Find the height, area, and circumradius.
Problem 2: An isosceles triangle has equal sides = 10 cm and base = 12 cm. Find the height and area.
Problem 3: A right triangle has legs 5 cm and 12 cm. Find the hypotenuse, circumradius, and inradius.
Problem 4: Find the area of a scalene triangle with sides 6 cm, 8 cm, and 10 cm using Heron's formula.
Problem 5: A square is inscribed in a triangle with base segments 3 cm and 5 cm. Find the side of the square.

Answers​

Problem 1:

• Height = (√3÷2) × 8 = 4√3 ≈ 6.93 cm
• Area = (√3÷4) × 64 = 16√3 ≈ 27.71 cm²
• Circumradius = 8÷√3 = 4√3÷3 ≈ 4.62 cm

Problem 2:

• Height = √(100 − 36) = √64 = 8 cm
• Area = (12÷4) × √(400−144) = 3 × 16 = 48 cm²

Problem 3:

• Hypotenuse = √(25+144) = 13 cm
• Circumradius = 13÷2 = 6.5 cm
• Inradius = (5+12−13)÷2 = 2 cm

Problem 4:

• s = (6+8+10)÷2 = 12
• Area = √(12×6×4×2) = √576 = 24 cm²
• (Note: this is actually a right triangle! 6²+8²=10²)

Problem 5:

• Side = (3×5)÷(3+5) = 15÷8 = 1.875 cm

Common Mistakes to Avoid​

• Using slant height instead of perpendicular height — always use the height that is 90° to the base
• Forgetting to square root in Heron's formula — the answer is under a square root sign
• Confusing circumradius and inradius — circumradius R goes outside, inradius r goes inside
• Mixing up legs and hypotenuse in right triangle formulas

Summary​

Special triangles each have their own unique set of properties and formulas. The key to mastering them is:

1. Identify the triangle type first
2. Pick the right formula for what you need
3. Double-check your measurements — height vs slant, leg vs hypotenuse

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