Use sides and angles to solve any triangle. Calculate the area, perimeter and any missing values.
F(x) = Ax2 + Bx + C
sin(x) = opposite/hyp
Visualizing Mathematical Functions
Triangle Calculator
The Triangle Calculator provides a free, precise online math tool for engineers, students and teachers. This tool uses high precision trigonometric algorithms to calculate all side lengths and internal angles. It also computes area, perimeters, medians, circumradii, and inradii.
This tool is part of the Math Calculators Collection. It works alongside Area Calculator and Right Triangle Calculator.
Triangles are classified according to their shape
The triangle ($Delta$), a three-sided fundamental polygon, is defined by its three vertices and the three line segments that connect them (edges). The relative lengths of the sides or internal angles are used to classify triangles.
The classification of Sides
- Equilateral : all sides of the are 60 degrees.
- isosceles : The two sides and angles opposite each other are equal.
- Scalene All sides are different in length and the angles of all internal faces are unbalanced.
Angles of classification
- Right Triangle: Includes one angle with an interior of 90deg. Hypotenuse is the side that faces the right angle.
- (Acute) : The angles of all three inner sides are smaller than 90deg.
- (Obtuse): An interior angle that is greater than 90 degrees.
Theorems and Laws of the Fundamental Triangle
In order to solve a triangle, you must apply fundamental geometric laws in order to determine unknown properties. This calculator uses these mathematical core principles.
1. Inside Angle Sum Theorem
Sum of three angles inside a Euclidean triangular always equals 180deg. ($A +B +C = 180circ$). A triangle cannot have more than 90deg of angle.
2. Triangle Inequality Theorem
The sum of two sides of a valid triangle must be greater than that of the third side. ($a +b >c$), $a +c >b$ and $b +c >a$ are all examples.
3. The Pythagorean theorem
The square of the hypotenuse in right-angled triangular shapes is the same as the sum of squares on the two other sides. The Law of Cosines is applied in this case when $C=90circ$.
$a^2 + b^2 = c^2$
4. Law of Sines
This ratio is the same for all sides in a triangle. It is especially useful when solving oblique triangular problems and you only know two angles (AAS/ASA), or two sides (SSA), and an angle that’s not included.
$\fraca\sin(A) = \fracb\sin(B) = \fracc\sin(C)$
5. Cosine Law
The Pythagorean extension is used to solve triangles when two sides are given and the angle included (SAS), or the three side lengths combined (SSS).
$c^2 = a^2 + b^2 – 2ab \cos(C)$
To find angles when all sides are known: $C = \arccos\left(\fraca^2 + b^2 – c^22ab\right)$
Triangle Area Calculation Formula
The calculator will automatically calculate the area formula for you based on what inputs you provide.
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Standard Height and Base
Useful when you know the height perpendicular ($h$), and base ($b$).
Formula : $textArea=frac12btimesh$ -
Trigonometric Method
When two sides are known, and the angle they include.
Formula: $\textArea = \frac12 ab \sin(C)$ -
Heron’s Formula SSS
Useful when you only know the length of three sides ($a$,$b$ and $c$).
Formula : $textArea=sqrts (s-a)(s-b)(s-c)$ where $s$ represents the semiperimeter : $s = $fraca+b+c2$
The Farmer And His Daughter – Triangle Daze
A farmer sold his rectangular land to build a pool for his entire family. The farmer’s daughter wants the swimming pool to be in an equilateral triangular shape, and all sides must measure exactly 77 feet. This is her seventh birthday. The farmer uses Heron’s Formula to calculate the area needed for the footprint of the pool:
s = (77+77+77)/2 = 115.5
Area = [115.5 x (115.5 – 77) x (115.5 – 77) x (115.5 – 77)] 2,567.33 sq ft
The farmer can confirm that he has enough space for a new build by calculating the footprint.
The Advanced Properties of Medians and Inradius
This high precision calculator will resolve several geometric properties that are advanced for your triangle input:
1. The Medians in a Triangle
A line segment that extends from the vertex of a triangular side to its midpoint is called the median. Three medians always meet at the same point, called centroid. These are the lengths of each median ($m_a$), $m_b$ and $m_c$:
$m_a = \frac12\sqrt2b^2 + 2c^2 – a^2$ | $m_b = \frac12\sqrt2a^2 + 2c^2 – b^2$ | $m_c = \frac12\sqrt2a^2 + 2b^2 – c^2$
2. Inradius ($r$)
Inradius: The radius of the biggest circle which can be completely contained within the triangle. The incenter is the center of a circle that is perpendicular to the three triangle sides. The inradius is calculated by using the semiperimeter and area.
$r = \frac\textAreas$
3. Circumradius ($R$)
Circumradius: The radius of a circle which passes through the three points of the circumcircle (triangle). Circumcenter is the point at which the perpendicular bisectors of each side intersect. The side product or Law of Sines method is used to calculate it.
$R = \fraca2\sin(A) = \fracabc4 \times \textArea$
Related Mathematical Calculators
Notices on Precision and Accuracy
- The results are calculated by using floating point arithmetic to eight significant figures.
- You can enter angle values directly using mathematical constants like pi/2 and Pi/4.
- Check your inputs. A single typographical mistake in the side lengths could trigger an “inequality violation” meaning that values can’t physically form a triangle.
The Authoritative Resource
Triangle Calculations: Frequently Asked questions
What is the minimum number of values required to solve a triangular problem?
At least three values are required, at least one of which must be a length. It is not enough to know three angles, because infinitely many triangles have the same angle but different size.
Is it possible for a triangle to have two angles that are right?
No. No. The third interior angle of a triangle with two right angles (90circ +90circ =180circ) would need to be zero degrees, but this is impossible because the shape will collapse into a line.
What’s the difference between circumradius and inradius?
Inradius is defined as the diameter of the biggest circle possible that fits inside of a triangle and touches all of its three sides. Circumradius ($R$), on the other hand, is the diameter of a circle which passes through the three outer vertices of a triangle.