Angles & Triangles Calculator

Angles & Triangles Calculator

MathCrave's online tool simplifies triangle geometry by allowing you to calculate missing sides, angles, area, and perimeter. Just input the known values to solve any triangle, whether it's a right triangle or an oblique one.

Triangle Solver

Input the sides and angles you know. The solver will automatically apply the correct formulas and provide the full solution for the remaining values.

Triangle Formulas & Rules

Learn the fundamental principles behind solving triangles, including the Law of Sines, Law of Cosines, and the Angle Sum Theorem, with clear examples and explanations.

Angle Units & Conversion

Degrees ($^\circ$): A common unit where a full circle is $360^\circ$.

Minutes ($'$): $1^\circ = 60'$.

Seconds ($''$): $1' = 60''$. So, $1^\circ = 3600''$.

Radians (rad): The SI unit. $2\pi \text{ rad} = 360^\circ$. $1 \text{ rad} = \frac{180^\circ}{\pi} \approx 57.3^\circ$.

Degrees to DMS / Radians

DMS to Degrees / Radians

° ' ''

Radians to Degrees / DMS

Angle Arithmetic (DMS)

Add/Subtract Angles in DMS

Enter two angles in Degrees, Minutes, Seconds format.

° ' ''
° ' ''

Interactive Triangle Constructor & Analyzer

Construct & Analyze Triangle

Angle Definitions

Acute Angle

An angle less than $90^\circ$.

θ

Right Angle

An angle equal to $90^\circ$.

Obtuse Angle

An angle greater than $90^\circ$ but less than $180^\circ$.

θ

Straight Angle

An angle equal to $180^\circ$.

θ

Reflex Angle

An angle greater than $180^\circ$ but less than $360^\circ$.

θ

Full Angle

An angle equal to $360^\circ$.

θ

Complementary Angles

Two angles whose sum is $90^\circ$.

αβ

Supplementary Angles

Two angles whose sum is $180^\circ$.

αβ

Relevant Tools

To further enhance your learning and problem-solving skills, explore these additional resources

Parallel Lines and Transversals

Angle Relationships

When a transversal line intersects two parallel lines, several pairs of angles are formed with specific relationships:

L1 L2 || L1 T a b c d e f g h
  • Vertically Opposite Angles: Equal. (e.g., $a=d$, $b=c$, $e=h$, $f=g$)
  • Corresponding Angles: Equal. (e.g., $a=e$, $b=f$, $c=g$, $d=h$)
  • Alternate Interior Angles: Equal. (e.g., $c=f$, $d=e$)
  • Alternate Exterior Angles: Equal. (e.g., $a=h$, $b=g$)
  • Consecutive Interior Angles (Same-Side Interior): Supplementary (sum to $180^\circ$). (e.g., $c+e=180^\circ$, $d+f=180^\circ$)

Types of Triangles

Equilateral

All three sides are equal, and all three angles are $60^\circ$.

$a$ $a$ $a$

Isosceles

Two sides are equal, and the angles opposite those sides are equal.

$a$ $b$ $b$

Scalene

All three sides and all three angles are different.

$a$ $b$ $c$

Acute Triangle

All three angles are acute (less than $90^\circ$).

$<90^\circ$ $<90^\circ$ $<90^\circ$

Right Triangle

One angle is a right angle ($90^\circ$). The side opposite the right angle is the hypotenuse.

$b$ $a$ $c$ $90^\circ$

Obtuse Triangle

One angle is obtuse (greater than $90^\circ$).

$>90^\circ$ $a$ $b$ $c$

Key Triangle Properties

  • Sum of Angles: The sum of the interior angles of any triangle is always $180^\circ$. $A + B + C = 180^\circ$.
  • Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side. (e.g., $a+b > c$, $a+c > b$, $b+c > a$).
  • Law of Sines: Relates the sides of a triangle to the sines of its opposite angles. $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$ (where R is the circumradius).
  • Law of Cosines: Relates the lengths of the sides of a triangle to the cosine of one of its angles.
    • $a^2 = b^2 + c^2 - 2bc \cos A$
    • $b^2 = a^2 + c^2 - 2ac \cos B$
    • $c^2 = a^2 + b^2 - 2ab \cos C$
  • Area of a Triangle:
    • Given base $b_s$ and height $h_s$: Area $= \frac{1}{2} b_s h_s$.
    • Given two sides $a, b$ and the included angle $C$: Area $= \frac{1}{2} ab \sin C$.
    • Heron's Formula (given sides $a,b,c$ and semi-perimeter $s = \frac{a+b+c}{2}$): Area $= \sqrt{s(s-a)(s-b)(s-c)}$.
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