Triangle Area Calculator: Complete Triangle Solver
A comprehensive triangle calculator computes area, perimeter, missing sides, angles, height, and all triangle properties using geometric formulas including basic area (A = ½bh), Heron's formula for three sides, Pythagorean theorem for right triangles, trigonometric ratios, and the law of cosines. This all-in-one geometric tool performs calculations for right triangles, isosceles triangles, equilateral triangles, scalene triangles, solving for missing dimensions, finding hypotenuse, calculating heights, determining angles, computing perimeter, and analyzing triangular prisms including volume and surface area essential for students, engineers, architects, construction professionals, designers, mathematicians, and anyone requiring accurate triangle calculations for mathematics education, geometry problems, construction planning, structural analysis, roof design, engineering applications, or problem-solving in education, architecture, surveying, and technical fields.
▲ Comprehensive Triangle Calculator
Calculate all triangle properties
Calculate Triangle Area
Formula: A = ½bh
Area from Three Sides (Heron's Formula)
When all three sides are known
Right Triangle Calculator
Calculate missing sides and angles
Enter any two sides to find the third
Find Missing Side of Triangle
Using Law of Cosines
Triangle Perimeter Calculator
Sum of all three sides
Triangular Prism Calculator
Volume and surface area
Complete Triangle Analysis
All properties from three sides
Understanding Triangles
A triangle is a three-sided polygon with three vertices and three angles that sum to 180°. Types include: right triangles (one 90° angle), isosceles (two equal sides), equilateral (all sides equal), and scalene (all sides different). Triangles are fundamental shapes in geometry, architecture, engineering, and structural design due to their inherent stability and strength.
Triangle Area Formulas
Basic Area Formula
Area from Base and Height:
\[ A = \frac{1}{2}bh \]
Where:
\( A \) = area
\( b \) = base
\( h \) = height (perpendicular to base)
Heron's Formula (Three Sides)
Area from Three Sides:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
Where \( s \) is the semi-perimeter:
\[ s = \frac{a+b+c}{2} \]
\( a, b, c \) = lengths of three sides
Right Triangle Formulas
Pythagorean Theorem:
\[ a^2 + b^2 = c^2 \]
Area of Right Triangle:
\[ A = \frac{1}{2}ab \]
Where \( a, b \) = legs, \( c \) = hypotenuse
Perimeter and Other Formulas
Perimeter:
\[ P = a + b + c \]
Law of Cosines (Missing Side):
\[ c^2 = a^2 + b^2 - 2ab\cos(C) \]
Height from Area:
\[ h = \frac{2A}{b} \]
Triangular Prism Formulas
Volume:
\[ V = A_{base} \times L = \frac{1}{2}bh \times L \]
Surface Area:
\[ SA = 2A_{triangle} + P_{triangle} \times L \]
Where \( L \) = length of prism, \( P \) = perimeter of triangle
Step-by-Step Examples
Example 1: Calculate Area from Base and Height
Problem: Find the area of a triangle with base 10 cm and height 6 cm.
Formula: \( A = \frac{1}{2}bh \)
Calculation: \( A = \frac{1}{2} \times 10 \times 6 = 30 \text{ cm}^2 \)
Answer: 30 square centimeters
Example 2: Area Using Heron's Formula
Problem: Triangle with sides 5, 6, and 7 units.
Step 1: Calculate semi-perimeter
\( s = \frac{5+6+7}{2} = 9 \)
Step 2: Apply Heron's formula
\( A = \sqrt{9(9-5)(9-6)(9-7)} = \sqrt{9 \times 4 \times 3 \times 2} = \sqrt{216} \approx 14.70 \)
Example 3: Right Triangle (Find Hypotenuse)
Problem: Legs are 3 and 4 units. Find hypotenuse.
Pythagorean Theorem: \( c = \sqrt{a^2 + b^2} \)
\( c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)
Answer: Hypotenuse = 5 units (3-4-5 triangle)
Triangle Types and Properties
| Triangle Type | Properties | Area Formula | Special Notes |
|---|---|---|---|
| Right Triangle | One 90° angle | A = ½ab (legs) | Pythagorean theorem applies |
| Isosceles | Two equal sides | A = ½bh | Two equal angles |
| Equilateral | All sides equal | A = (√3/4)a² | All angles 60° |
| Scalene | All sides different | Heron's formula | All angles different |
Common Right Triangle Ratios
| Triangle Name | Side Ratios | Example | Application |
|---|---|---|---|
| 3-4-5 Triangle | 3:4:5 | 6-8-10, 9-12-15 | Construction squaring |
| 5-12-13 Triangle | 5:12:13 | 10-24-26 | Engineering |
| 45-45-90 Triangle | 1:1:√2 | 5-5-7.07 | Architecture |
| 30-60-90 Triangle | 1:√3:2 | 5-8.66-10 | Roof design |
Real-World Applications
Construction & Architecture
- Roof trusses: Calculate triangular roof support structures
- Foundation squaring: Use 3-4-5 triangle for right angles
- Gable ends: Determine triangular wall sections
- Structural bracing: Analyze triangular support strength
Surveying & Land Measurement
- Triangulation: Determine distances using triangle properties
- Plot division: Calculate triangular land parcels
- Elevation measurement: Use right triangle ratios
- Slope calculations: Determine grade using triangles
Engineering & Design
- Bridge trusses: Design triangular support structures
- Force analysis: Resolve forces using vector triangles
- Machine components: Calculate triangular parts
- Antenna towers: Design triangular support frames
Tips for Triangle Calculations
Best Practices:
- Check triangle inequality: Sum of any two sides > third side
- Verify angle sum: All angles should sum to 180°
- Use correct height: Height must be perpendicular to base
- Identify right triangles: Use Pythagorean theorem when applicable
- Choose correct formula: Match formula to known information
- Include units: Area in square units, perimeter in linear units
- Round appropriately: Maintain precision for engineering
Common Mistakes to Avoid
⚠️ Calculation Errors
- Forgetting ½ in area formula: Area = ½bh, not bh
- Using slant height: Must use perpendicular height
- Wrong Pythagorean application: Only for right triangles
- Triangle inequality violation: Impossible triangle sides
- Angle sum error: Angles must total 180°
- Confusing perimeter and area: Different formulas and units
- Missing square root: Remember √ in Pythagorean theorem
- Wrong semi-perimeter: s = (a+b+c)/2, not a+b+c
Frequently Asked Questions
How do you find the area of a triangle?
Use formula A = ½bh (half base times height). Example: base 10 cm, height 6 cm gives A = ½ × 10 × 6 = 30 cm². Height must be perpendicular to base. If three sides known, use Heron's formula: A = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2. For right triangle, A = ½ab where a and b are legs. Choose formula based on information available. Result always in square units (cm², m², ft²). Most fundamental geometry calculation.
How do you find the missing side of a triangle?
Method depends on triangle type. Right triangle: use Pythagorean theorem c² = a² + b². Example: legs 3 and 4 give hypotenuse √(9+16) = 5. Any triangle with two sides and included angle: Law of Cosines c² = a² + b² - 2ab·cos(C). If two angles and one side known: use Law of Sines. If all angles known but no sides: need at least one side length. Triangle inequality: sum of any two sides must exceed third side. Verify answer makes geometric sense.
What is Heron's formula and when do you use it?
Heron's formula calculates triangle area from three side lengths: A = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2 is semi-perimeter. Use when all three sides known but height unknown. Example: sides 5, 6, 7 give s = 9, then A = √[9×4×3×2] = √216 ≈ 14.70. Useful for irregular triangles, land surveying, navigation. Works for any triangle type. Alternative to base-height method. Named after Hero of Alexandria (1st century AD). Essential for real-world measurements where height difficult to determine.
How do you find the height of a triangle?
If area and base known: h = 2A/b. Rearrangement of area formula. Example: area 30 cm², base 10 cm gives h = 60/10 = 6 cm. For right triangle: each leg is height to other leg. For equilateral triangle with side a: h = (√3/2)a. Using Heron's formula: first find area, then calculate h = 2A/b. In practical applications: measure perpendicular distance from vertex to opposite side. Height is altitude—line segment from vertex perpendicular to opposite side (or extension).
What is a right triangle and how do you solve it?
Right triangle has one 90° angle. Sides: two legs (a, b) and hypotenuse (c, longest side opposite right angle). Pythagorean theorem: a² + b² = c². Example: legs 3, 4 give c = √(9+16) = 5. To solve: need two pieces of information (sides or angles). Trigonometric ratios: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. Common ratios: 3-4-5, 5-12-13, 8-15-17. Area = ½ab. Used extensively in construction, navigation, physics. Foundation of trigonometry.
How do you calculate the volume of a triangular prism?
Volume = area of triangular base × length (height) of prism. Formula: V = (½bh)L where b = triangle base, h = triangle height, L = prism length. Example: triangle base 10 cm, height 6 cm, prism length 15 cm gives V = (½×10×6)×15 = 30×15 = 450 cm³. Surface area = 2(triangle area) + perimeter×length. Used for tent volume, roof spaces, structural members. Ensure consistent units throughout. Triangular prism has two parallel triangular ends connected by three rectangular faces.
Key Takeaways
Understanding triangle calculations is fundamental for geometry, engineering, construction, and countless practical applications. From basic area formulas to complex right triangle solving, triangles provide the structural foundation for mathematics and real-world design.
Essential principles to remember:
- Triangle area: A = ½bh (base × height ÷ 2)
- Heron's formula: A = √[s(s-a)(s-b)(s-c)]
- Pythagorean theorem: a² + b² = c²
- Perimeter: P = a + b + c
- Angles sum to 180°
- Triangle inequality: sum of any two sides > third side
- Height must be perpendicular to base
- Right triangles: special formulas apply
- Law of Cosines for missing sides
- Area in square units, perimeter in linear units
Getting Started: Use the comprehensive calculator at the top of this page to calculate triangle area, perimeter, missing sides, angles, and all properties. Choose your calculation type—basic area, Heron's formula, right triangle solver, missing side finder, perimeter, or triangular prism—enter your known values, and receive instant results with step-by-step solutions. Perfect for students, engineers, architects, and anyone needing accurate triangle calculations.