Acute and obtuse triangles — Safekipedia Discoverer

An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). These triangles have all their corners pointed, and none of the angles are flat or square. Because all angles are less than 90°, the triangle looks very "pointy" overall.

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 remain pointed.

Since a triangle's angles must sum to 180° in Euclidean geometry, no Euclidean triangle can 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 the two different types of oblique triangles—triangles that are not right triangles because they do not have any right angles (90°). Understanding these triangles helps us recognize shapes in many areas, from architecture to 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. However, in an acute triangle—all angles less than 90°—the orthocenter and circumcenter are also inside. In an obtuse triangle—one angle greater than 90°—these points lie outside the triangle.

Acute triangles have three small squares that fit perfectly 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 when it comes to their sides, angles, and other measurements.

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

These differences in angles lead to different rules for things like the lengths of the sides and the sizes of certain lines drawn inside the triangle. For example, in an obtuse triangle with a long side, certain sums of squares of side lengths follow one rule, while in an acute triangle they follow a different rule. Similar patterns show up in formulas involving the triangle’s area, the lines called "medians" that connect vertices to the middle of the opposite side, and other triangle measurements.

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, with angles of 36°, 72°, and 72°.

Other triangles, like the heptagonal triangle, are obtuse, meaning one of their angles is larger than 90°. There are also triangles with whole-number side lengths that are either acute or obtuse, such as the triangle with sides (2, 3, 4), which is obtuse.