Triangle Calculator

    Solve triangle properties

    Triangle Calculator

    Calculate area, perimeter, and all angles/sides of any triangle

    Triangle Properties

    Area
    6
    Perimeter
    12
    Height (to side A)
    4

    Complete Triangle

    Side Lengths
    Side A
    3
    Side B
    4
    Side C
    5
    Angles
    Angle A
    36.9°
    Angle B
    53.1°
    Angle C
    90°

    What This Calculator Does

    The Triangle Calculator is your comprehensive digital tool for solving a wide range of triangle-related problems quickly and accurately. Whether you know two sides and an included angle, all three sides, or a combination of sides and angles, this calculator determines the missing triangle properties instantly. It provides precise values for area, perimeter, side lengths, angle measures, and heights, making it invaluable for students, professionals, and anyone needing fast triangle solutions.

    How to Use This Calculator

    1. Select what information you know: Start by choosing the option in the “What Do You Know?” dropdown that matches your given values. For example, you might know all three sides (SSS), two sides and the included angle (SAS), two angles and a side (ASA or AAS), or other combinations.
    2. Enter your known values: Fill in the relevant input fields for your triangle. These may include Side A Length, Side B Length, Side C Length, Angle A, Angle B, Angle C, or the Included Angle C, depending on your selected option.
    3. Review your inputs: Double-check your entries to ensure all required values are accurate and appropriate. The calculator will highlight missing or invalid entries if necessary.
    4. View your results: After entering your values, the calculator will display the computed properties, including area, perimeter, missing side lengths, missing angles, and the height to side A.
    5. Interpret the outputs: Use the detailed output fields to assist with homework, design projects, construction planning, or any scenario requiring triangle calculations.

    Definitions of Key Terms

    Side A Length
    The length of the side of the triangle labeled as A. This is typically used as a reference side for height calculations and may be any side of the triangle depending on your naming convention.
    Side B Length
    The length of the side of the triangle labeled as B. Used in conjunction with other sides or angles to solve the triangle.
    Side C Length
    The length of the side labeled as C, which is often considered the base or the side opposite a specific angle, such as Angle C.
    Angle A
    The internal angle of the triangle at vertex A, measured in degrees. Angle A is opposite Side A.
    Angle B
    The internal angle at vertex B, measured in degrees. Angle B is opposite Side B.
    Angle C
    The internal angle at vertex C, measured in degrees. Angle C is opposite Side C.
    Included Angle
    The angle formed between two known sides. For example, if you know Side A and Side B, the included angle is typically Angle C.
    Area
    The amount of space contained within the triangle, measured in square units. Calculating area requires knowing either the base and height, two sides and the included angle, or all three sides.
    Perimeter
    The total distance around the triangle, calculated as the sum of its three sides.
    Height (to side A)
    The perpendicular distance from vertex A’s opposite side to Side A. This represents the triangle’s height when Side A is used as the base.

    Calculation Methodology

    The Triangle Calculator uses classical trigonometric formulas and geometric principles to solve for missing sides, angles, area, perimeter, and height. The specific formulas depend on the known values you provide. Here are the main calculation methods:

    Given three sides (SSS):
    
    s = (a + b + c) / 2
    
    Area = sqrt[s * (s - a) * (s - b) * (s - c)]
    
    Perimeter = a + b + c
    
    Angle A = arccos[(b² + c² - a²) / (2bc)]
    Angle B = arccos[(a² + c² - b²) / (2ac)]
    Angle C = arccos[(a² + b² - c²) / (2ab)]
    
    Height (to side A) = (2 * Area) / a
    
    Given two sides and included angle (SAS):
    
    Area = (1/2) * a * b * sin(C)
    
    Side c = sqrt[a² + b² - 2ab*cos(C)]
    
    Use Law of Sines to find remaining angles:
    sin(A)/a = sin(B)/b = sin(C)/c
    
    Given two angles and a side (AAS or ASA):
    
    Angle C = 180° - (A + B)
    
    Use Law of Sines to find missing sides:
    a/sin(A) = b/sin(B) = c/sin(C)
    
    Area = (1/2) * base * height
    

    In these formulas, a, b, and c represent the lengths of Sides A, B, and C. Angles A, B, and C are their corresponding opposite angles, measured in degrees. The variable s is the semi-perimeter of the triangle, used in Heron's formula for area. The Law of Sines and Law of Cosines are used to solve for unknown sides and angles when different combinations of values are provided.

    Practical Scenarios

    • Home renovation and construction: If you need to determine the area of a triangular plot or the dimensions of a roof truss, input the side lengths or angle measurements you have to instantly calculate the missing properties and ensure precise materials planning.
    • Classroom learning and homework: Students can use the Triangle Calculator to check their work on geometry assignments involving triangle properties, making it easier to visualize and verify solutions for triangle-related problems.
    • Design and engineering projects: Architects and engineers often encounter triangular components in structural designs. By inputting known sides and angles, they can quickly derive all needed dimensions for accurate drafting and modeling.
    • Outdoor and DIY projects: Whether you’re building a triangular garden bed, laying out a deck, or setting up a tent, use the calculator to find exact measurements and angles, ensuring your project comes together smoothly.

    Advanced Tips & Best Practices

    • Check for triangle validity: Ensure the combination of sides and angles you enter can form a valid triangle. The sum of any two sides must be greater than the third, and the sum of all angles must equal 180 degrees.
    • Use the correct units consistently: Always enter all side lengths in the same unit (such as meters, feet, or centimeters) to guarantee accurate area and perimeter results.
    • Leverage included angles for precision: For the most accurate results when using two sides and an angle, make sure the angle entered is the included angle between those sides. This allows the calculator to apply the Law of Cosines directly.
    • Apply outputs to real-world measurements: Use the calculated height to side A for tasks like determining the altitude needed for structural supports, or the area for estimating paint, flooring, or fencing requirements.
    • Double-check ambiguous cases: When entering two sides and a non-included angle (SSA configuration), be aware that two different triangles may be possible, or sometimes no triangle at all. The calculator will indicate if the input leads to an ambiguous case.

    Frequently Asked Questions (Optional)

    Can I use this calculator for right triangles?
    Yes, you can use the Triangle Calculator for right triangles as well as acute and obtuse triangles. For right triangles, entering a 90-degree angle will allow the calculator to apply the appropriate formulas automatically.
    What if my triangle has all angles given but no sides?
    A triangle cannot be uniquely determined if only the angles are known. At least one side must be provided to calculate the triangle’s dimensions and area. Enter at least one side length along with your angles for complete results.
    Why am I getting an error or no result?
    Errors typically occur if the input values cannot form a valid triangle. Check that your side lengths and angles satisfy the triangle inequality and that all angles together sum to 180 degrees. The calculator will alert you if the input is invalid or ambiguous.

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    Frequently Asked Questions

    Is this calculator free to use?

    Yes, all calculators on Calculator Galaxy are completely free to use.

    How accurate are the results?

    Our calculators use standard mathematical formulas to provide accurate results.

    Can I save my calculations?

    Currently, results are not saved between sessions. We recommend taking a screenshot if you need to save your results.