The area of a triangle equals one-half the product of height and base length.

More generally, four points in three-dimensional Euclidean space determine a solid figure called tetrahedron.

In particular, the sine, cosine, and tangent functions relate side lengths and angles in right triangles.

The terminology for categorizing triangles is more than two thousand years old, having been defined in Book One of Euclid's Elements.

[2] The names used for modern classification are either a direct transliteration of Euclid's Greek or their Latin translations.

[10] The faces of the Great Pyramid of Giza are sometimes considered to be equilateral, but more accurate measurements show they are isosceles instead.

A polyhedron is a solid whose boundary is covered by flat polygonals known as the faces, sharp corners known as the vertices, and line segments known as the edges.

An important tool for proving the existence of these points is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent.

[22] An altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side.

The three angle bisectors intersect in a single point, the incenter, which is the center of the triangle's incircle.

There are three other important circles, the excircles; they lie outside the triangle and touch one side, as well as the extensions of the other two.

[27] The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the orthocenter.

The three medians intersect in a single point, the triangle's centroid or geometric barycenter.

The centroid of a rigid triangular object (cut out of a thin sheet of uniform density) is also its center of mass: the object can be balanced on its centroid in a uniform gravitational field.

If one reflects a median in the angle bisector that passes through the same vertex, one obtains a symmedian.

[31] The sum of the measures of the interior angles of a triangle in Euclidean space is always 180 degrees.

[34] The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees, and indeed, this is true for any convex polygon, no matter how many sides it has.

[35] Another relation between the internal angles and triangles creates a new concept of trigonometric functions.

[42] Some individually necessary and sufficient conditions for a pair of triangles to be congruent are:[43] In the Euclidean plane, area is defined by comparison with a square of side length ⁠

Because the ratios between areas of shapes in the same plane are preserved by affine transformations, the relative areas of triangles in any affine plane can be defined without reference to a notion of distance or squares.

⁠ for the vertices of a triangle, its relative oriented area can be calculated using the shoelace formula,

Unlike a rectangle, which may collapse into a parallelogram from pressure to one of its points,[50] triangles are sturdy because specifying the lengths of all three sides determines the angles.

Tessellated triangles still maintain superior strength for cantilevering, however, which is why engineering makes use of tetrahedral trusses.

[citation needed] Triangulation means the partition of any planar object into a collection of triangles.

While convenient for many purposes, this approach has the disadvantage of all points' coordinate values being dependent on the arbitrary placement in the plane.

[65] The Kiepert hyperbola is unique conic that passes through the triangle's three vertices, its centroid, and its circumcenter.

[69] A pseudotriangle is a simply-connected subset of the plane lying between three mutually tangent convex regions.

These sides are three smoothed curved lines connecting their endpoints called the cusp points.

[72] In particular, it is possible to draw a triangle on a sphere such that the measure of each of its internal angles equals 90°, adding up to a total of 270°.

[76] Fractal shapes based on triangles include the Sierpiński gasket and the Koch snowflake.