Area of Triangle Calculator
Find the area of any triangle. This versatile tool supports calculation via Base & Height, Three Sides (Heron's formula), or Two Sides with the Included Angle (SAS). This free online triangle area calculator is perfect for students, surveyors, and geometry enthusiasts.
How to Use the Area of Triangle Calculator
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Mathematical Formula & Logic
Step-by-Step Worked Calculation
Scenario: Calculating Area with Three Sides 5, 6, and 7
A surveyor needs to find the acreage of a triangular plot of land with measured boundaries of 50 meters, 60 meters, and 70 meters.
Step 1: Set three sides a = 50, b = 60, c = 70.
Step 2: Calculate semiperimeter s = (50+60+70)/2 = 90.
Step 3: Apply Heron's formula: √[90 * (90-50) * (90-60) * (90-70)] = √[90 * 40 * 30 * 20].
Step 4: Compute Area ≈ 1469.69 square meters.
How to Use the Area of Triangle Calculator
- 1. Select your calculation method: Base & Height, Three Sides (Heron's), or Two Sides & Angle (SAS).
- 2. Input the required dimensions in the fields.
- 3. Check the computed surface area, perimeter, and geometric details.
What Is a Area of Triangle Calculator?
Area of Triangle Calculator is a mathematical computation tool that helps you calculate the area of any triangle using multiple formulas: Base & Height, Heron's formula (three sides), or SAS (two sides and one angle). Our free triangle area calculator provides accurate results with step-by-step explanations. It applies established mathematical principles to deliver accurate results, often showing the underlying formula and step-by-step working so you can understand the computation process.
Why This Calculation Matters
Mathematical calculations form the foundation of science, engineering, finance, and everyday problem-solving. Area of Triangle Calculator helps you work through calculations accurately and efficiently, reducing the risk of manual arithmetic errors. Whether you are a student learning concepts, a professional verifying work, or anyone needing quick and reliable math results, this tool ensures precision and saves time.
Historical Background
Mathematics has evolved over thousands of years, from ancient Babylonian clay tablets and Egyptian papyri to Greek formal proofs by Euclid and Archimedes. The development of algebra by Persian mathematician al-Khwarizmi in the 9th century and the invention of calculus by Newton and Leibniz in the 17th century laid the groundwork for modern computation. Area of Triangle Calculator continues this tradition by making mathematical operations accessible through digital technology.
Frequently Asked Questions
Complete indexable directory of answers (100 questions)
How do I calculate the area of a triangle?
Use this free triangle area calculator by selecting your calculation method (Base & Height, Three Sides, or SAS) and entering the required dimensions. The calculator instantly computes the area with step-by-step explanation.
What is the triangle area formula?
The triangle area formula depends on what you know: Base & Height: A = ½ × b × h, Three Sides (Heron's): A = √[s(s-a)(s-b)(s-c)], or SAS: A = ½ × a × b × sin(C). This calculator handles all three methods.
What is Heron's formula used for?
Heron's formula allows you to calculate the area of a triangle when you only know the lengths of all three sides, without needing to know any angles or altitude heights.
Can a triangle have sides of any length?
No. The Triangle Inequality Theorem states that the sum of the lengths of any two sides must always be strictly greater than the length of the remaining side (a + b > c).
Is this triangle area calculator free?
Yes, this free online triangle area calculator is 100% free with no signup required. Calculate triangle area instantly.
What is the area of a 3-4-5 right triangle?
Area = ½ × 3 × 4 = 6 square units. The hypotenuse (5) is not used in area calculation.
What is the area of a 5-12-13 right triangle?
Area = ½ × 5 × 12 = 30 square units. This is a famous Pythagorean triple.
What is the area of an equilateral triangle with side 6?
Area = (√3/4) × 6² = 9√3 ≈ 15.59 square units.
How do I calculate area using coordinates?
Use A = ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)| with the three vertex coordinates.
What is Heron's formula for a triangle with sides 3, 4, 5?
s = (3+4+5)/2 = 6. A = √(6×3×2×1) = √36 = 6 square units.
What is the area of a triangle with base 10 and height 6?
Area = ½ × 10 × 6 = 30 square units.
What is the area of a triangle with two sides 7 and 10 and included angle 30°?
Area = ½ × 7 × 10 × sin(30°) = 35 × 0.5 = 17.5 square units.
What is the area of a 30-60-90 triangle with hypotenuse 10?
Legs are 5 and 5√3. Area = ½ × 5 × 5√3 = 12.5√3 ≈ 21.65 square units.
What is the area of a 45-45-90 triangle with hypotenuse 10?
Legs are 5√2 each. Area = ½ × 5√2 × 5√2 = 25 square units.
How do I verify if three sides form a valid triangle?
Check the triangle inequality: sum of any two sides must exceed the third. For 3, 4, 5: 3+4>5 ✓, 3+5>4 ✓, 4+5>3 ✓.
What is the area of a triangle with vertices (0,0), (4,0), (0,3)?
Using coordinates: A = ½|0(0-3) + 4(3-0) + 0(0-0)| = ½|12| = 6 square units.
What is the area of a triangle with sides 7, 8, 9?
s = 12. A = √(12×5×4×3) = √720 ≈ 26.83 square units.
What is the area of a triangle with base 15 and height 8?
Area = ½ × 15 × 8 = 60 square units.
What is the area of an equilateral triangle with side 10?
Area = (√3/4) × 100 = 25√3 ≈ 43.30 square units.
What is the difference between height and slant height?
Height (altitude) is the perpendicular distance from a vertex to the opposite side. Slant height is the length of a side, not the perpendicular distance.
How do I find the height of a triangle given three sides?
For Heron's formula: h = 2A/b. First find area A using Heron's, then divide by the base b.
What is the area of an isosceles triangle with equal sides 5 and base 6?
Height = √(5² - 3²) = 4. Area = ½ × 6 × 4 = 12 square units.
What is the area of a triangle with base 20 and height 12?
Area = ½ × 20 × 12 = 120 square units.
What is the semiperimeter in Heron's formula?
The semiperimeter s = (a + b + c) / 2, where a, b, and c are the three side lengths. It is half the perimeter.
What is the SAS formula for triangle area?
SAS (Side-Angle-Side): Area = ½ × a × b × sin(C), where a and b are two sides and C is the included angle between them.
What is the area of a triangle with sides 5, 5, and base 8?
Height = √(5² - 4²) = 3. Area = ½ × 8 × 3 = 12 square units.
What is the area of a triangle with sides 13, 14, 15?
s = 21. A = √(21×8×7×6) = √7056 = 84 square units.
How do I find the area of a right triangle?
For a right triangle, the two legs serve as base and height. Area = ½ × leg₁ × leg₂. The hypotenuse is not needed.
What is the area of a triangle with base 8 and height 10?
Area = ½ × 8 × 10 = 40 square units.
What is the area of a triangle with sides 6, 8, and included angle 60°?
Area = ½ × 6 × 8 × sin(60°) = 24 × (√3/2) = 12√3 ≈ 20.78 square units.
What is the area of an equilateral triangle with side 12?
Area = (√3/4) × 144 = 36√3 ≈ 62.35 square units.
What is the area of a triangle with base 7 and height 5?
Area = ½ × 7 × 5 = 17.5 square units.
How do I find the base given area and height?
Base = 2 × Area / Height. For example, if area = 30 and height = 6: base = 2 × 30 / 6 = 10.
How do I find the height given area and base?
Height = 2 × Area / Base. For example, if area = 40 and base = 8: height = 2 × 40 / 8 = 10.
What is the area of a triangle with vertices (1,1), (5,1), (3,4)?
A = ½|1(1-4) + 5(4-1) + 3(1-1)| = ½|-3+15+0| = 6 square units.
What is the area of a triangle with sides 9, 12, 15?
This is a right triangle (3-4-5 scaled by 3). Area = ½ × 9 × 12 = 54 square units.
What is the area of a triangle with base 25 and height 8?
Area = ½ × 25 × 8 = 100 square units.
What is the area of a triangle with sides 10, 10, 12?
s = 16. A = √(16×6×6×4) = √2304 = 48 square units.
What is the area of a triangle with base 6 and height 3?
Area = ½ × 6 × 3 = 9 square units.
What is the area of a triangle with sides 4, 13, 15?
s = 16. A = √(16×12×3×1) = √576 = 24 square units.
How do I calculate the area of a scalene triangle?
Use Heron's formula when you know all three sides, or the SAS formula when you know two sides and the included angle.
What is the area of a triangle with base 40 and height 20?
Area = ½ × 40 × 20 = 400 square units.
What is the area of a triangle with sides 20, 21, 29?
This is a right triangle (20-21-29 is a Pythagorean triple). Area = ½ × 20 × 21 = 210 square units.
What is the area of a triangle with base 18 and height 14?
Area = ½ × 18 × 14 = 126 square units.
What is the area of a triangle with vertices (0,0), (6,0), (3,5)?
A = ½|0(0-5) + 6(5-0) + 3(0-0)| = ½|30| = 15 square units.
What is the area of a triangle with sides 2, 3, 4?
s = 4.5. A = √(4.5×2.5×1.5×0.5) = √8.4375 ≈ 2.905 square units.
What is the area of a triangle with base 50 and height 30?
Area = ½ × 50 × 30 = 750 square units.
What is the area of a triangle with sides 6, 7, 8?
s = 10.5. A = √(10.5×4.5×3.5×2.5) = √413.4375 ≈ 20.333 square units.
How do I find the area of an obtuse triangle?
Use the same formulas. For base-height, the altitude falls outside the triangle. For Heron's or SAS, the formulas work regardless of triangle type.
What is the area of a triangle with base 30 and height 24?
Area = ½ × 30 × 24 = 360 square units.
What is the area of a triangle with sides 11, 13, 20?
s = 22. A = √(22×11×9×2) = √4356 = 66 square units.
What is the area of a triangle with base 14 and height 9?
Area = ½ × 14 × 9 = 63 square units.
What is the area of a triangle with sides 8, 15, 17?
This is a right triangle. Area = ½ × 8 × 15 = 60 square units.
What is the area of an equilateral triangle with side 15?
Area = (√3/4) × 225 = 56.25√3 ≈ 97.43 square units.
What is the area of a triangle with base 35 and height 16?
Area = ½ × 35 × 16 = 280 square units.
What is the area of a triangle with sides 5, 12, 13?
This is a right triangle. Area = ½ × 5 × 12 = 30 square units.
What is the area of a triangle with base 22 and height 11?
Area = ½ × 22 × 11 = 121 square units.
What is the area of a triangle with vertices (2,3), (8,3), (5,9)?
A = ½|2(3-9) + 8(9-3) + 5(3-3)| = ½|-12+48+0| = 18 square units.
What is the area of a triangle with base 45 and height 18?
Area = ½ × 45 × 18 = 405 square units.
What is the area of a triangle with sides 10, 24, 26?
This is a right triangle (5-12-13 scaled by 2). Area = ½ × 10 × 24 = 120 square units.
What is the area of a triangle with base 16 and height 7?
Area = ½ × 16 × 7 = 56 square units.
What is the area of a triangle with sides 3, 25, 26?
s = 27. A = √(27×24×2×1) = √1296 = 36 square units.
What is the area of a triangle with base 55 and height 22?
Area = ½ × 55 × 22 = 605 square units.
What is the area of a triangle with sides 7, 24, 25?
This is a right triangle. Area = ½ × 7 × 24 = 84 square units.
What is the area of a triangle with base 28 and height 15?
Area = ½ × 28 × 15 = 210 square units.
What is the area of a triangle with sides 9, 40, 41?
This is a right triangle. Area = ½ × 9 × 40 = 180 square units.
What is the area of a triangle with base 60 and height 25?
Area = ½ × 60 × 25 = 750 square units.
What is the area of a triangle with sides 11, 60, 61?
This is a right triangle. Area = ½ × 11 × 60 = 330 square units.
What is the area of a triangle with base 32 and height 19?
Area = ½ × 32 × 19 = 304 square units.
What is the area of a triangle with sides 12, 35, 37?
This is a right triangle. Area = ½ × 12 × 35 = 210 square units.
What is the area of a triangle with base 70 and height 35?
Area = ½ × 70 × 35 = 1225 square units.
What is the area of a triangle with sides 13, 84, 85?
This is a right triangle. Area = ½ × 13 × 84 = 546 square units.
What is the area of a triangle with base 80 and height 40?
Area = ½ × 80 × 40 = 1600 square units.
What is the area of a triangle with sides 16, 63, 65?
This is a right triangle. Area = ½ × 16 × 63 = 504 square units.
What is the area of a triangle with base 90 and height 45?
Area = ½ × 90 × 45 = 2025 square units.
What is the area of a triangle with base 100 and height 50?
Area = ½ × 100 × 50 = 2500 square units.
What is the area of a triangle with sides 28, 45, 53?
This is a right triangle. Area = ½ × 28 × 45 = 630 square units.
What is the area of a triangle with base 110 and height 55?
Area = ½ × 110 × 55 = 3025 square units.
What is the area of a triangle with sides 33, 56, 65?
s = 77. A = √(77×44×21×12) = √850944 ≈ 922.5 square units.
What is the area of a triangle with base 120 and height 60?
Area = ½ × 120 × 60 = 3600 square units.
What is the area of a triangle with sides 36, 77, 85?
s = 99. A = √(99×63×22×14) = √1917468 ≈ 1384.7 square units.
What is the area of a triangle with base 150 and height 75?
Area = ½ × 150 × 75 = 5625 square units.
What is the area of a triangle with sides 39, 80, 89?
This is a right triangle. Area = ½ × 39 × 80 = 1560 square units.
What is the area of a triangle with base 200 and height 100?
Area = ½ × 200 × 100 = 10000 square units.
What is the area of a triangle with sides 48, 55, 73?
s = 88. A = √(88×40×33×15) = √1742400 ≈ 1319.9 square units.
What is the area of a triangle with base 250 and height 125?
Area = ½ × 250 × 125 = 15625 square units.
What is the area of a triangle with sides 65, 72, 97?
s = 117. A = √(117×52×45×20) = √5475600 ≈ 2340.0 square units.
What is the area of a triangle with base 300 and height 150?
Area = ½ × 300 × 150 = 22500 square units.
What is the area of a triangle with sides 20, 99, 101?
This is a right triangle. Area = ½ × 20 × 99 = 990 square units.
What is the area of a triangle with base 400 and height 200?
Area = ½ × 400 × 200 = 40000 square units.
What is the area of a triangle with base 500 and height 250?
Area = ½ × 500 × 250 = 62500 square units.
What is the area of a triangle with sides 3, 4, 5?
This is the classic right triangle. Area = ½ × 3 × 4 = 6 square units.
What is the area of a triangle with base 1000 and height 500?
Area = ½ × 1000 × 500 = 250000 square units.
What is the area of a triangle with sides 5, 5, 6?
s = 8. A = √(8×3×3×2) = √144 = 12 square units.
What is the area of a triangle with base 2000 and height 1000?
Area = ½ × 2000 × 1000 = 1000000 square units.
What is the area of a triangle with sides 8, 8, 12?
s = 14. A = √(14×6×6×2) = √1008 ≈ 31.75 square units.
What is the future of triangle area calculators?
Future calculators will integrate AI for automatic problem recognition from photos, AR for real-world measurement, and educational step-by-step solutions.
What mathematical formula does the Area of Triangle Calculator use?
The Area of Triangle Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.
How can I verify the Area of Triangle Calculator results manually?
Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.
What types of inputs does the Area of Triangle Calculator accept?
The Area of Triangle Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.