Triangle

A triangle is a polygon with three corners and three sides. It is one of the basic shapes in geometry. The corners are called vertices, and the sides connecting them are called edges. A triangle has three internal angles. The sum of angles of a triangle is always 180 degrees.

Triangles are important in many areas of math and science. They are used in Euclidean geometry to study shapes and spaces. In this type of geometry, any three points that do not all lie on the same straight line all lie on the same straight line make a triangle. Triangles can also be found in other types of geometries, like spherical triangle or hyperbolic triangle.

The area of a triangle can be found using its height and base length. This makes triangles useful in real life, such as in building design and mapping. Relations between angles and side lengths are studied in trigonometry.

Definition, terminology, and types

A triangle is a shape with three sides and three corners. The sides are the lines that connect the corners, and the corners are called vertices. Triangles can look different depending on the lengths of their sides and the sizes of their angles.

There are several types of triangles. An equilateral triangle has all three sides the same length. An isosceles triangle has two sides that are the same length. A scalene triangle has three sides of different lengths. A right triangle has one angle that is exactly a right angle. An acute triangle has all angles smaller than a right angle. An obtuse triangle has one angle larger than a right angle.

Appearances

Triangles are everywhere! You can see them in nature and in things people make. For example, the Egyptian pyramids have shapes that look like triangles, and road signs that tell you to slow down are also triangles. Buildings often use triangle shapes for their roofs or decorations above doors.

Triangles are also used in 3D objects. Some solid shapes, called polyhedra, have triangle faces. For instance, pyramids have triangle sides, and special shapes like deltahedra are made up entirely of triangles.

Properties

Triangles are simple shapes in geometry. They have three corners called vertices and three sides called edges. Each corner makes an angle, and all three angles in a triangle always add up to 180 degrees.

Triangles have special points and lines. For example, drawing lines called perpendicular bisectors from the middle of each side meets at a point called the circumcenter. This point is the center of a circle that passes through all three vertices of the triangle. Another important point is the centroid, found where lines called medians meet. This point balances the triangle perfectly.

Triangles can be studied using angles and their side lengths. The sum of the angles in a triangle is always 180 degrees, which helps in finding missing angles. Triangles can be grouped based on their angles and side lengths, such as similar triangles (same shape but different sizes) and congruent triangles (exactly the same size and shape). The area of a triangle can be found in several ways, such as using half the product of a side and its matching height.

Triangles are strong shapes because their three sides keep their angles fixed. This strength is used in buildings like bridges and roofs. By splitting other shapes into triangles, we can study their properties more easily through a process called triangulation.

Location of a point

To find where a point is in or near a triangle, we can use special ways to describe its place.

One way is to put the triangle on a grid and use numbers called coordinates to show where the point is. But this changes if we move or turn the triangle.

There are two better ways that work no matter how we move the triangle.

The first way, called trilinear coordinates, uses how far the point is from each side of the triangle. These distances tell us the point's position.

The second way, called barycentric coordinates, uses how much weight would need to be placed on each corner of the triangle to keep it balanced at that point.

Related figures

Figures inscribed in a triangle

Every triangle has a special circle inside it called an incircle that touches all three sides. There is also a special shape called a Steiner inellipse that fits inside the triangle and touches the middle points of each side.

From any point inside a triangle, you can make a new triangle called a pedal triangle by connecting the closest points on each side. If the point is in the center, this new triangle will have its points at the middle of each side.

Figures circumscribed about a triangle

Every triangle has a circle that goes through all three corners, called a circumcircle. There is also a special stretched circle shape called a Steiner circumellipse that goes through the three corners and has the smallest area of all such shapes.

Miscellaneous triangles

Circular triangles

Main article: Circular triangle

A circular triangle is a special triangle with curved sides, like parts of circles. These curves can bend out or in. A common example is the Reuleaux triangle, made by joining three circles of the same size. You can make this shape with just a compass, no ruler needed.

Another interesting shape is the pseudotriangle, which has three smooth, curved sides that meet at points called cusps. These shapes can be split into smaller parts in special ways.

Triangle in non-planar space

Main articles: Hyperbolic triangle and Spherical triangle

Triangles aren’t just flat — they can be on curved surfaces too! For example, on a sphere, like Earth, the angles of a triangle add up to more than 180°. This is called a spherical triangle. On a saddle-shaped surface, the angles add up to less than 180°, known as a hyperbolic triangle.

Fractal geometry

Fractal patterns based on triangles include the Sierpiński gasket and the Koch snowflake. These are shapes with detailed patterns that look the same no matter how much you zoom in.