Triangle Geometry: Area, Perimeter, and Angle Formulas
Calculate triangle properties including area using base-height, Heron formula, trigonometric methods, perimeter, and angle sums for various triangle types.
Triangle Geometry Fundamentals
The triangle is the simplest polygon — three sides, three angles, and a set of properties that make it the most fundamental shape in geometry. Triangles form the basis of trigonometry, are the fundamental building block of more complex polygons, and appear everywhere from bridge trusses to computer graphics to GPS positioning. Understanding triangle formulas is essential for construction, engineering, design, navigation, and countless practical applications.
Every triangle has three sides (typically labeled a, b, and c) and three interior angles (A, B, and C), where angle A is opposite side a, angle B opposite side b, and angle C opposite side c. The sum of all three interior angles is always 180 degrees — a universal property of triangles in Euclidean geometry.
Triangle Classification
Triangles are classified in two ways: by side lengths and by angles. By sides: equilateral (all three sides equal, all angles 60°), isosceles (two sides equal, base angles equal), and scalene (no sides equal, no angles equal). By angles: acute (all angles less than 90°), right (one angle exactly 90°), and obtuse (one angle greater than 90°). A triangle can belong to both classifications — for example, a right isosceles triangle has one 90° angle and two equal sides.
Swipe sideways to compare columns.
| Type | Side Properties | Angle Properties | Example |
|---|---|---|---|
| Equilateral | All sides equal | All angles 60° | Equal-sided |
| Isosceles | Two sides equal | Base angles equal | Same leg lengths |
| Scalene | No equal sides | No equal angles | All different |
| Right | Any side lengths | One angle = 90° | Forms an L shape |
| Acute | Any side lengths | All angles < 90° | Compact shape |
| Obtuse | Any side lengths | One angle > 90° | Wide shape |
Triangle Perimeter
The perimeter of a triangle is simply the sum of its three side lengths: Perimeter = a + b + c. For an equilateral triangle with side length 6, the perimeter is 18. For a right triangle with sides 3, 4, and 5, the perimeter is 12. Perimeter is always measured in linear units (inches, feet, meters, etc.) and is essential for applications like fencing, framing, and material estimation.
Area of a Triangle: Multiple Methods
Triangle area can be calculated using several different formulas depending on what information you have available. The most common formula uses base and height: Area = ½ × base × height. The base can be any side, and the height is the perpendicular distance from that base to the opposite vertex.
For triangles where you cannot easily measure height, other formulas become useful. Heron formula requires only the three side lengths: s = (a + b + c) ÷ 2 (the semi-perimeter), then Area = √(s(s-a)(s-b)(s-c)). The trigonometric method uses two sides and the included angle: Area = ½ × a × b × sin(C).
Swipe sideways to compare columns.
| Triangle | Given Information | Method | Area Calculation | Result |
|---|---|---|---|---|
| Base-height | base=10, height=6 | Standard | ½ × 10 × 6 | 30 sq units |
| Heron | sides 5,6,7 | Heron (s=9) | √(9×4×3×2) | 14.70 sq units |
| Trigonometric | sides 8,12, angle=30° | ½ab sin C | ½ × 8 × 12 × 0.5 | 24 sq units |
| Right triangle | legs 3,4 | ½ × legs | ½ × 3 × 4 | 6 sq units |
The Pythagorean Theorem
The Pythagorean theorem is the most famous relationship in triangle geometry, applying exclusively to right triangles: a² + b² = c², where c is the hypotenuse (the side opposite the right angle) and a and b are the two legs. If you know any two sides of a right triangle, you can find the third: c = √(a² + b²), a = √(c² — b²), or b = √(c² — a²).
Beyond simple side-finding, the Pythagorean theorem is the foundation for distance calculations between any two points in a plane. The distance formula d = √((x₂-x₁)² + (y₂-y₁)²) is derived directly from the theorem. Construction workers use it to verify square corners, GPS systems use it for position calculations, and it appears in physics, engineering, and computer graphics.
Law of Sines and Law of Cosines
For non-right triangles, the law of sines and law of cosines extend the relationships between sides and angles. The law of sines states that the ratio of a side length to the sine of its opposite angle is constant: a ÷ sin(A) = b ÷ sin(B) = c ÷ sin(C) = 2R (where R is the circumradius). This is useful when you know two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).
The law of cosines generalizes the Pythagorean theorem to any triangle: c² = a² + b² — 2ab × cos(C). When angle C is 90°, cos(90°) = 0 and the formula reduces to the Pythagorean theorem. The law of cosines is used when you know two sides and the included angle (SAS) or all three sides (SSS).
Swipe sideways to compare columns.
| Known Values | Formula to Use | What You Can Find |
|---|---|---|
| Two sides + right angle | Pythagorean theorem | Third side |
| Two sides + included angle | Law of cosines | Third side |
| Two angles + one side | Law of sines | Other two sides |
| Three sides | Law of cosines | All three angles |
| Base + height | Area = ½ × b × h | Area |
| Three sides | Heron formula | Area |
Calculate Triangle Properties
Triangle CalculatorUse our Triangle Calculator to compute area, perimeter, angles, and missing side lengths for any triangle type using multiple methods.Frequently Asked Questions
Can any three side lengths make a triangle?
No. The triangle inequality theorem states that the sum of any two sides must be greater than the third side for a triangle to exist. For sides a, b, and c: a + b > c, a + c > b, and b + c > a. If one side is equal to the sum of the other two, all three points are collinear — a degenerate triangle with zero area.
How do I find the height of a triangle if I only know the side lengths?
Use Heron formula to find the area first, then rearrange the area formula: height = (2 × Area) ÷ base. The height will be different for each choice of base, corresponding to the altitude to that specific side.
What is the difference between an altitude, median, and angle bisector?
An altitude is a perpendicular line from a vertex to the opposite side (used for area calculations). A median connects a vertex to the midpoint of the opposite side. An angle bisector divides the vertex angle into two equal angles. In an equilateral triangle, all three are the same line. In other triangles, they are different.