A triangle is a shape with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are points with no size, while the sides connecting them, called edges, are lines with length. A triangle has three internal angles, each formed between two connected sides. The total of these angles is always 180 degrees or π radians. A triangle is a flat shape, and the area inside it is a flat region. Sometimes, one side is chosen as the base, and the opposite corner is called the apex. The shortest distance from the base to the apex is called the height. The area of a triangle is calculated by multiplying the height by the base length and dividing the result by two.

In Euclidean geometry, any two points define a single line segment that lies on a single straight line. Any three points that are not on the same straight line form a single triangle that lies on a flat plane. More broadly, four points in three-dimensional space form a solid shape called a tetrahedron.

In non-Euclidean geometries, three "straight" lines can form a triangle, such as a spherical triangle or a hyperbolic triangle. A geodesic triangle is a shape on a two-dimensional surface, enclosed by three sides that are straight for that surface. A curvilinear triangle is a shape with three curved sides, such as a triangle with sides shaped like arcs of a circle. (This information is about triangles with straight sides in Euclidean geometry, unless stated otherwise.)

Triangles are grouped into types based on their angles and side lengths. The relationship between angles and side lengths is a key topic in trigonometry. Specifically, the sine, cosine, and tangent functions help connect side lengths and angles in right triangles.

Definition, terminology, and types

A triangle is a shape made up of three line segments, each of which connects to the other two. These line segments are called the sides, and the points where they connect are called the edges. This creates a polygon with three sides and three angles. The way triangles are named has been used for more than two thousand years, as described in Book One of Euclid's Elements. The names used today are either direct translations from Euclid's Greek or their Latin versions.

Triangles are classified based on the lengths of their sides and the sizes of their angles. A triangle with all sides of equal length is called an equilateral triangle. A triangle with two sides of equal length is called an isosceles triangle. A triangle with all sides of different lengths is called a scalene triangle. A triangle with one right angle (a 90-degree angle) is called a right triangle. A triangle with all angles smaller than a right angle is called an acute triangle. A triangle with one angle larger than a right angle is called an obtuse triangle. These definitions have been used since at least the time of Euclid.

• Equilateral triangle
• Isosceles triangle
• Scalene triangle

• Right triangle
• Acute triangle
• Obtuse triangle

Appearances

Triangles are often seen in everyday life and in structures made by people. In buildings, isosceles triangles can be found in the shape of gables and pediments, while equilateral triangles appear on traffic signs like the yield sign. The faces of the Great Pyramid of Giza are sometimes thought to be equilateral triangles, but more precise measurements show they are actually isosceles triangles. Triangles also appear in symbols, such as the flags of Saint Lucia and the Philippines, and in structures studied in molecular geometry.

Triangles are also found in three-dimensional shapes. A polyhedron is a solid object with flat faces, sharp corners called vertices, and straight edges. Some polyhedrons are grouped based on the shapes of their faces. For example, if all faces of a polyhedron are equilateral triangles, it is called a deltahedron. Antiprisms have triangles that alternate on their sides. Pyramids and bipyramids have a base that is a polygon and triangular sides. When these are right pyramids or bipyramids, the triangular sides are isosceles triangles. A Kleetope is a new polyhedron formed by placing a pyramid on each face of another shape, resulting in triangular faces. In higher dimensions, triangles are part of shapes called simplices and polytopes with triangular parts called simplicial polytopes.

Properties

Each triangle has special points inside it, on its edges, or connected to it. These points are found by drawing three lines that are related to the triangle’s sides or corners. These lines meet at one point, and tools like Ceva’s theorem help prove this. Similarly, Menelaus’ theorem helps show when three points on a triangle are on the same straight line. This section explains some common examples.

A perpendicular bisector of a triangle’s side is a line that passes through the middle of the side and forms a right angle with it. The three perpendicular bisectors meet at a single point called the circumcenter. This point is the center of the circumcircle, a circle that goes through all three corners of the triangle. If the circumcenter is on the triangle’s side, the opposite angle is a right angle. If the circumcenter is inside the triangle, the triangle is acute; if it is outside, the triangle is obtuse.

An altitude of a triangle is a line that goes from a corner and is perpendicular to the opposite side. The side where the altitude meets is called the base, and the point where the altitude touches the base is called the foot. The length of the altitude is the distance between the base and the corner. The three altitudes meet at a point called the orthocenter. The orthocenter is inside the triangle only if the triangle is acute.

An angle bisector is a line that splits an angle into two equal parts. The three angle bisectors meet at a point called the incenter, which is the center of the incircle. The incircle is the largest circle that fits inside the triangle and touches all three sides. The radius of the incircle is called the inradius. Other circles, called excircles, are outside the triangle and touch one side and the extensions of the other two. The centers of these circles form an orthocentric system. The midpoints of the sides and the feet of the altitudes all lie on a circle called the nine-point circle. This circle also includes the midpoints of the segments between the corners and the orthocenter. The nine-point circle’s radius is half the circumcircle’s radius. It touches the incircle and the three excircles. The orthocenter, nine-point circle center, centroid, and circumcenter all lie on a line called Euler’s line. The nine-point circle’s center is halfway between the orthocenter and circumcenter, and the distance between the centroid and circumcenter is half the distance between the centroid and orthocenter. Usually, the incenter is not on Euler’s line.

A median is a line that connects a corner to the midpoint of the opposite side, splitting the triangle into two equal areas. The three medians meet at a point called the centroid, which is the triangle’s center of mass. The centroid divides each median into a 2:1 ratio, with the longer part between the corner and the centroid. Reflecting a median over the angle bisector from the same corner creates a symmedian. The three symmedians meet at a point called the symmedian point.

In Euclidean geometry, the sum of the interior angles of a triangle is always 180 degrees. This is related to Euclid’s parallel postulate. This rule helps find the third angle of a triangle if two angles are known. An exterior angle is formed by extending one side of the triangle. The measure of an exterior angle equals the sum of the two non-adjacent interior angles. The total of the three exterior angles of any triangle is 360 degrees, a rule that also applies to all convex polygons.

Trigonometric functions like sine and cosine are defined using the ratios of sides in a right triangle. These functions help find unknown sides or angles in scalene triangles using the law of sines and the law of cosines.

Any three angles that add up to 180 degrees can form a triangle. Many triangles can have the same angles but different sizes. A degenerate triangle, where the corners are on a straight line, has angles of 0° and 180°, though it may not be considered a triangle in some cases. A condition for three angles to form a triangle can be checked using trigonometric equations, such as $cos^2alpha + cos^2beta + cos^2gamma + 2cosalphacosbetacosgamma = 1$.

Two triangles are similar if their corresponding angles are equal. Their sides are in the same proportion, and this is enough to prove similarity.

Similar triangles follow these rules:
– If two pairs of angles are equal, the triangles are similar.
– If two pairs of sides are in the same proportion and the included angles are equal, the triangles are similar.
– If all three pairs of sides are in the same proportion, the triangles are similar.

Congruent triangles are identical in size and shape. All congruent triangles are similar, but not all similar triangles are congruent. For two triangles to be congruent, all corresponding angles and sides must match. Three conditions are enough to prove congruence:
– SAS Postulate: Two sides and the included angle of one triangle match those of another.
– ASA: Two angles and the included side of one triangle match those of another.

Location of a point

One way to find the positions of points inside or outside a triangle is to place the triangle anywhere on the Cartesian plane and use x and y coordinates. While this method is useful, it has a problem: the numbers used to describe points depend on where the triangle is placed on the plane.

Two other systems solve this issue. These systems make sure that the numbers describing a point’s location stay the same even if the triangle is moved, rotated, reflected (like in a mirror), or resized (making a similar triangle):

• Trilinear coordinates describe a point’s location based on how far it is from each side of the triangle. For example, numbers like x : y : z show the ratio of the point’s distance from the first side compared to the second side, and so on.
• Barycentric coordinates, written as α : β : γ, describe a point’s location by showing how much weight would need to be placed on each corner of the triangle to balance it perfectly on that point.

Related figures

Every triangle has a special circle called an incircle that fits perfectly inside the triangle and touches all three sides. It is the only circle that can do this. Another special shape inside a triangle is the Steiner inellipse, which is an oval that touches the middle of each side. Marden's theorem helps find the exact points called foci that define this ellipse. This ellipse has the largest possible area of any oval that can touch all three sides of a triangle. The Mandart inellipse is another oval inside the triangle, but it touches the sides at the same points where the triangle’s excircles (special circles outside the triangle) touch the sides. For any oval inside a triangle, if the foci are labeled P and Q, a specific formula involving distances from these points to the triangle’s corners and sides always equals 1.

A point inside a triangle can form a new triangle called a pedal triangle by connecting the closest points on the triangle’s sides to that point. If the point is the triangle’s circumcenter (the center of the circle passing through all three corners), the pedal triangle’s corners are the midpoints of the triangle’s sides, and this triangle is called the medial triangle. The medial triangle divides the original triangle into four smaller triangles that are all the same size and shape as the original.

The intouch triangle has corners where the triangle’s incircle touches its sides. The extouch triangle has corners where the triangle’s excircles touch its sides (not extended sides).

Squares that fit inside a triangle with their corners touching the triangle’s sides are a special case of a math problem about squares inside shapes. A famous example is the Calabi triangle, where three squares can be placed so their corners touch all sides of the triangle. Every acute triangle (a triangle with all angles less than 90 degrees) has three such squares. In a right triangle, two of these squares overlap at the right angle, so only two distinct squares exist. An obtuse triangle (a triangle with one angle greater than 90 degrees) has only one such square, placed along its longest side. The size of the square depends on the triangle’s side lengths and area. A formula connects the square’s side length, the triangle’s side length, the triangle’s height, and its area. The largest possible area of an inscribed square compared to the triangle is 1/2, which happens in specific cases. The smallest ratio of square side lengths in non-obtuse triangles is 2√2/3, and this occurs in isosceles right triangles.

The Lemoine hexagon is a six-sided shape inside a triangle. Its corners are where the triangle’s sides intersect lines that are parallel to the sides and pass through the triangle’s symmedian point (a special point inside the triangle). This hexagon can be drawn in two forms, both inside the triangle with two corners on each side.

Any convex polygon (a shape with straight sides and no indentations) that has an area T can fit inside a triangle with an area no larger than 2T. This maximum area is only possible if the polygon is a parallelogram (a four-sided shape with opposite sides equal and parallel).

The tangential triangle of a reference triangle (not a right triangle) is a triangle whose sides are tangent to the reference triangle’s circumcircle (the circle passing through its corners) at those corners.

Every triangle has a unique circumcircle that passes through all three corners, with its center at the intersection of the triangle’s perpendicular bisectors (lines that split sides into equal parts at right angles). Every triangle also has a unique Steiner circumellipse, an oval that passes through the triangle’s corners and is centered at the triangle’s centroid (the point where the triangle’s medians intersect). Of all ovals that pass through the triangle’s corners, this one has the smallest area.

The Kiepert hyperbola is a special curved shape that passes through a triangle’s three corners, its centroid, and its circumcenter (the center of the circumcircle).

Among all triangles that can fit inside a given convex polygon, one with the largest possible area can be found quickly. The corners of this triangle are always three of the polygon’s own corners.

Miscellaneous triangles

A circular triangle is a triangle with edges that are curved lines called circular arcs. These edges can curve outward (convex) or curve inward (concave). When three circular areas called disks overlap, they can form a circular triangle with all convex sides. A common example of a circular triangle with three convex sides is a Reuleaux triangle. This shape can be created by overlapping three circles of the same size. It can be drawn using only a compass, as shown by the Mohr–Mascheroni theorem. Another way to make it is by rounding the sides of an equilateral triangle.

A special type of concave circular triangle is called a pseudotriangle. A pseudotriangle is a flat shape that lies between three rounded areas that touch each other. The curved sides of a pseudotriangle connect points called cusp points. A pseudotriangle can be divided into smaller pseudotriangles using the edges of rounded shapes and lines that touch two points on the same shape, a process called pseudo-triangulation. If there are n rounded shapes in a pseudotriangle, the division creates 2n − 2 pseudotriangles and 3n − 3 bitangent lines. The outer shape formed by connecting the farthest points of a pseudotriangle is a triangle.

A non-planar triangle is a triangle that is not drawn in flat, Euclidean space. Triangles can also exist in other types of spaces, such as hyperbolic space (a space with negative curvature, like a saddle) and spherical geometry (a space with positive curvature, like the surface of a sphere). A triangle in hyperbolic space is called a hyperbolic triangle, and one in spherical geometry is called a spherical triangle.

Triangles in these spaces have different properties than those in flat, Euclidean space. In Euclidean space, the sum of the internal angles of a triangle is always 180°. In hyperbolic space, the sum is less than 180°, and in spherical geometry, the sum is more than 180°. For example, a triangle on a sphere can have three right angles (90° each), totaling 270°. According to Girard’s theorem, the sum of the angles of a triangle on a sphere is 180° × (1 + 4f), where f is the fraction of the sphere’s area covered by the triangle.

In more complex spaces, rules called comparison theorems relate the properties of triangles in those spaces to triangles in model spaces like hyperbolic or spherical geometry. For example, a CAT(k) space is defined using such comparisons.

Fractals based on triangles include the Sierpiński gasket and the Koch snowflake.