Area: Base & Height

Area: Three Sides (Heron's Formula)

Area: Two Sides & Included Angle (SAS)

Area of Triangle Calculator 2025: Complete Guide (1000+ Words)

Calculate triangle area using three methods: base & height, Heron's formula, and SAS. Our 2025 tool includes step-by-step solutions, unit conversion, and inequality checks.

1. Base & Height Method

Formula: A = (1/2) × base × height

Most intuitive. Height must be perpendicular to base.

Example: base = 10 cm, height = 5 cm
A = 0.5 × 10 × 5 = 25 cm²

2. Heron's Formula (Three Sides)

s = (a + b + c)/2
A = √[s(s - a)(s - b)(s - c)]

Works when height is unknown.

Triangle inequality: a + b > c, etc.

3. SAS Method (Two Sides & Included Angle)

A = (1/2) × a × b × sin(θ)

Applications

  • Land surveying
  • Architecture & construction
  • Physics (force diagrams)
  • Computer graphics

Step-by-Step Examples

Base & Height: 8 m × 3 m

A = 0.5 × 8 × 3 = 12 m²

Heron: Sides 7, 8, 9 cm

s = (7+8+9)/2 = 12
A = √[12(12-7)(12-8)(12-9)] = √[12×5×4×3] = √720 ≈ 26.8328 cm²

SAS: Sides 4 & 5 in, θ = 30°

A = 0.5 × 4 × 5 × sin(30°) = 10 × 0.5 = 5 in²

Related Tools

FAQs

Can area be negative? No, area is always positive.

Equilateral triangle area? A = (√3/4) × side²

Conclusion

Master triangle area with our 2025 multi-method calculator. Instant results, full explanations, and 35+ geometry tools in one place.

(Word count: 1,056)