Acute and obtuse triangles — Safekipedia Adventurer

An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). All the corners of these triangles are pointed, and none of the angles are flat or square. Because every angle is smaller than 90°, the triangle looks very "pointy."

An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. This means one corner of the triangle is stretched out more than a square corner, while the other two corners stay pointed.

Since a triangle's angles must add up to 180° in Euclidean geometry, a triangle cannot have more than one obtuse angle. This makes obtuse triangles different from right triangles, which have exactly one 90° angle.

Acute and obtuse triangles are two types of oblique triangles—triangles that are not right triangles because they do not have any right angles (90°). Knowing about these triangles helps us see shapes in many places, such as buildings, art, and nature.


Right	Obtuse	Acute
	⏟ {\displaystyle \quad \underbrace {\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad } _{}}
	Oblique

Properties

In any triangle, special points like the centroid and incenter are always inside the triangle.

In an acute triangle—all angles less than 90°—the orthocenter and circumcenter are also inside the triangle.

In an obtuse triangle—one angle greater than 90°—these points lie outside the triangle.

Acute triangles can fit three small squares inside them, while obtuse triangles can only fit one such square. The longest side of an obtuse triangle is always opposite the largest angle.

Inequalities

Acute triangles and obtuse triangles have different properties.

In an acute triangle, all three angles are smaller than 90°. In an obtuse triangle, one angle is bigger than 90° and the other two are smaller than 90°. The angles in any triangle always add up to 180°, so there can only be one obtuse angle in a triangle.

These differences in angles mean that the sides and other parts of the triangle follow different rules. For example, in an obtuse triangle with a long side, the lengths of the sides follow one rule. In an acute triangle, they follow a different rule. Similar ideas apply to the area of the triangle and to lines called "medians" that connect a corner to the middle of the opposite side.

Examples

Some triangles have special names based on their angles and sides. For example, an equilateral triangle with all angles measuring 60° is acute. The golden triangle, an isosceles triangle where the ratio of two sides matches the golden ratio, is also acute.

Other triangles, like the heptagonal triangle, are obtuse, meaning one of their angles is larger than 90°.