Acute Triangle

In the world of geometry, triangles serve as fundamental building blocks, forming the basis of numerous mathematical concepts and practical applications. Among the various types of triangles, the acute triangle holds a distinct place due to its unique characteristics and its relevance in diverse fields, including mathematics, engineering, and architecture. Characterized by its angles that are all less than 90°, the acute triangle unveils a plethora of fascinating properties and applications, making it a cornerstone of geometric exploration. Let's explore the properties, types, formulas and examples of the acute triangle.

Definition of an Acute Triangle: An acute triangle is a triangle where all three interior angles are acute angles, measuring less than 90°. This fundamental property distinguishes it from other types of triangles, such as right triangles or obtuse triangles.

In the figure above, all the three angles of the △ABC measure less than 90°, hence it is an acute triangle.

Properties of an Acute Triangle

• Interior Angles

  : All three interior angles of an acute triangle are less than 90°. The sum of the measures of these angles always equals 180°.

• Side-Length Relationships

  : Acute triangles adhere to the triangle inequality theorem, which states that the sum of the lengths of any two sides of the triangle must be greater than the length of the third side. Mathematically, this can be represented as a + b > c, b + c > a, and c + a > b, where a, b, and c are the lengths of the sides.

• Altitudes and Orthocenter

  : In an acute triangle, the altitudes (perpendicular lines from each vertex to the opposite side) all lie within the interior of the triangle. The point where all three altitudes intersect is known as the orthocenter.

• Medians and Centroid

  : The medians of an acute triangle (lines from each vertex to the midpoint of the opposite side) intersect at a point called the centroid. The centroid divides each median in a 2:1 ratio, with the longer segment being closer to the vertex.

• Incircle or Inscribed Circle

  : An acute triangle can have an inscribed circle, also known as an incircle, which is a circle inside a triangle just touching all three sides of the triangle and the centre of this circle is called the incentre of the triangle

• Circumcircle

  : The circumcircle of an acute triangle is a circle that passes through all three vertices of the triangle. The center of the circumcircle is the intersection point of the perpendicular bisectors of the sides of the triangle.

Types of an Acute Triangle

Equilateral Triangle

Acute Isosceles Triangle

Acute Scalene Triangle

Acute Triangle Formulas

Perimeter of an Acute Triangle

Area of an Acute Triangle

Area of an Acute Triangle by Heron’s Formula

Solved Examples on Acute Triangle