What is the triangle inequality theorem?

The sum of any two sides of a triangle must be greater than the third side: a+b > c, a+c > b, b+c > a. If this is violated, no triangle can be formed. For example, sides 2, 3, 6 cannot form a triangle because 2+3 = 5 < 6.

What is Heron's formula and when do I use it?

Heron's formula calculates triangle area from all three side lengths: Area = sqrt(s(s-a)(s-b)(s-c)) where s = (a+b+c)/2. Use it when you know all three sides but not the height -- common in surveying, land measurement, and construction.

What is the difference between ASS and AAS triangle solving?

AAS (two angles and a non-included side) always gives a unique triangle. SSA (two sides and non-included angle) is the "ambiguous case" -- it may produce zero, one, or two triangles depending on the angle and side lengths.

How do I find the height of a triangle from just the sides?

Use Heron's formula to find the area first, then use h = 2 x Area / base. For the height to side a: h_a = 2 x Area / a. Each side has its own corresponding height.

Why do all triangle angles add up to 180 degrees?

This is a consequence of Euclidean geometry and the parallel postulate. In non-Euclidean geometries like on a sphere, angles can sum to more than 180 degrees.

Can any three lengths form a triangle?

No. The triangle inequality must hold: the sum of any two sides must be greater than the third side. Otherwise no triangle is possible.

What is Heron's formula?

Heron's formula calculates triangle area from all three sides: A equals the square root of s times (s minus a) times (s minus b) times (s minus c), where s is the semi-perimeter.

Triangle Calculator — Calculover

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Area

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Triangle Type Reference

Type	Definition	Key Formula
Equilateral	All 3 sides equal, all angles 60°	`A = (√3/4)s²`
Isosceles	Two sides equal, two base angles equal	`A = (b/4)√(4a²−b²)`
Scalene	All sides and angles different	Heron's: `A=√(s(s-a)(s-b)(s-c))`
Right	One 90° angle; a²+b²=c²	`A = ½ab`
Acute	All angles < 90°; a²+b² > c²	Law of cosines applies
Obtuse	One angle > 90°; a²+b² < c²	Heron's or law of cosines

Key Formulas

Formula	Expression
Heron's Area	`s=(a+b+c)/2; A=√(s(s-a)(s-b)(s-c))`
Inradius	`r = A / s` (s = semi-perimeter)
Circumradius	`R = abc / (4A)`
Law of Cosines	`c² = a² + b² − 2ab·cos(C)`
Law of Sines	`a/sin(A) = b/sin(B) = c/sin(C)`

30-60-90 Triangle

Angles	30°, 60°, 90°
Side ratios	1 : √3 : 2
If short leg = x	Long leg = x√3, Hyp = 2x
Area (short leg x)	A = (x²√3)/2
Example (x=1)	sides: 1, 1.732, 2

45-45-90 Triangle

Angles	45°, 45°, 90°
Side ratios	1 : 1 : √2
If leg = x	Hypotenuse = x√2
Area (leg x)	A = x²/2
Example (x=1)	sides: 1, 1, 1.414

Equilateral Triangle

Angles	60°, 60°, 60°
Side ratios	1 : 1 : 1
Area (side s)	A = (√3/4)s²
Height (side s)	h = (√3/2)s
Example (s=2)	A ≈ 1.732, h ≈ 1.732

Golden Gnomon

Angles	36°, 72°, 72°
Side ratios	1 : φ : φ (φ≈1.618)
Related to	Regular pentagon / Penrose tiling
Inradius / Circumradius	r/R = cos(72°) ≈ 0.309
Note	Golden ratio triangle

How to Use This Calculator

Enter Three Side Lengths

Provide the lengths of all three sides (a, b, and c) of the triangle.

Verify the Triangle

The calculator checks whether the sides satisfy the triangle inequality before proceeding.

View Area and Perimeter

The area (via Heron's formula) and perimeter are displayed along with step-by-step workings.

Formula & Methodology

Heron's Formula

A = √(s(s−a)(s−b)(s−c))

Where s is the semi-perimeter. This formula requires only the three side lengths, no angles.

Semi-Perimeter

s = (a + b + c) / 2

Half the perimeter; used as an intermediate value in Heron's formula.

Perimeter

P = a + b + c

The total boundary length of the triangle.

About this calculation · Accuracy & Methodology

Triangle Types by Side Length

Type	Sides	Angles	Area Formula
Equilateral	a = b = c	All 60°	(√3/4)a²
Isosceles	Two sides equal	Two angles equal	Heron's or (b/4)√(4a²−b²)
Scalene	All different	All different	Heron's formula
Right	a²+b²=c²	One 90°	½ab

Key Terms

Semi-Perimeter (s): Half the sum of all three sides; a helper value used in Heron's formula.
Heron's Formula: A method to calculate a triangle's area using only the three side lengths, without needing the height.
Triangle Inequality: A valid triangle requires that the sum of any two sides must exceed the third side.
Scalene Triangle: A triangle with all three sides of different lengths.
Equilateral Triangle: A triangle with all three sides equal; every angle is 60 degrees.

Real-World Examples

Example 1

Land Survey Plot

a = 30 m, b = 40 m, c = 50 m

Area = 600 m² — a right triangle (30-40-50 is a scaled 3-4-5 triple)

Example 2

Triangular Sign

a = 3 ft, b = 3 ft, c = 3 ft

Area = 3.897 ft² — equilateral triangle, about 3.9 square feet

Heron's Formula: Computing Area Without Height

The Elegance of Heron's Formula

Named after Hero of Alexandria (circa 60 CE), this formula computes a triangle's area using only its three side lengths. It avoids the need to measure or calculate the height, which can be difficult for irregular triangles. The formula first computes the semi-perimeter, then uses it in a symmetric expression involving all three sides.

When to Use Heron's Formula

Heron's formula is ideal when you know all three sides but not the height or angles—common in surveying, CAD drawings, and competition math. For right triangles, the simpler formula A = ½ab is faster. For triangles defined by two sides and an included angle, A = ½ab sin(C) is more direct. Choose the formula that matches your available data.

Triangle Calculator