Triangle

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are points while the sides connecting them, also called edges, are line segments. A triangle has three internal angles, and the sum of angles of a triangle always equals a straight angle (180 degrees or π radians).

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 determine a unique triangle. Triangles can also be found in other types of geometries, like spherical triangle or hyperbolic triangle.

The area of a triangle can be calculated using its height and base length. This makes triangles useful in real life, such as in building design, mapping, and many other fields. Relations between angles and side lengths are studied in a branch of math called 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 be 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 all around us! You can see them in many places, both in nature and in things people make. For example, the Egyptian pyramids have shapes that look like triangles, and the 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 three-dimensional 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. Triangles can even help describe shapes in more complex geometry.

Properties

Triangles are basic shapes in geometry, having three corners called vertices and three sides called edges. Each corner forms an angle, and the total of all three angles inside any triangle always adds up to 180 degrees.

Triangles have special points and lines connected to them. 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 also 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 sorted into groups 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 calculated in several ways, such as using half the product of a side and its matching height.

Triangles are very strong shapes because their three sides fix their angles, making them rigid. This strength is used in building structures like bridges and roofs. By dividing 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 just 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 right 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 also 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 kind of triangle where the sides are curved like parts of circles. These curved sides can bend outward or inward. One common example is the Reuleaux triangle, made by joining three circles of the same size. You can create this shape using just a compass, without needing a ruler.

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

Triangle in non-planar space

Main articles: Hyperbolic triangle and Spherical triangle

Triangles aren’t just flat shapes — they can also exist on curved surfaces! For example, on a sphere, like Earth, the angles of a triangle add up to more than 180°. This type of triangle 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 that show detailed patterns no matter how much you zoom in.