Acute Angle Triangle- Definition, Properties, Formulas, Questions

Edited By Team Careers360 | Updated on Jul 02, 2025 05:28 PM IST

when the angles of the triangle are measured to be less than 90° then they are termed to be the acute angle triangle. Equilateral triangles are considered to be acute triangles as they have a sum total of 60° + 60° + 60 degrees. The longest side of the acute triangle is considered to be that side which is opposite to the highest angle present in the Triangle. Isosceles triangles and Scalene triangles can also be considered acute triangles. An acute angle triangle always contains an interior angle to be less than 90°. The line constructed from the base of the triangle and up to the opposite vertex is found to be perpendicular to the base in the acute angle triangle.

Types of Acute angle triangle

Scalene acute angle triangle

The angle whose angle is found to be less than 90° degrees but has no equal sides is termed to be a scalene acute angle triangle.

Isosceles acute angle triangle

The angle whose angle is found to be less than 90° and possesses two equal sides are termed to be an isosceles acute angle triangle.

Equilateral acute angle triangle

The angle whose angle is found to be less than 90° and possesses three equal sides are termed to be an equilateral acute angle triangle. Equilateral triangles are considered to be acute triangle as they have a sum total of 60° + 60° + 60 degrees.

Degrees of acute angle triangle

An acute angle triangle degree measures less than that of the 90°.

There are many examples of acute angle triangle degrees such as 65° and 80°.

Commonly we can say that the acute angle triangle lies between 0° and 90°.

Properties of the acute angle triangle

When we talk about the acute angle triangle then we find out that the measurement of the interior angles of the acute angle triangle is found to be 90 degrees.
In an acute angle triangle, the angle that is opposite to the largest side is considered to be the largest angle.
In an acute angle triangle, the angle that is opposite to the smallest side is considered to be the smallest angle.
In an acute angle triangle, the square of the largest side of the triangle is found to be equal to the sum of squares of the other smaller sides of the triangle.
The orthocentre and Circumcentre in the case of the acute angle triangle are observed to be present in the interior of the triangle.

The perimeter of acute angle triangle

The perimeter of the acute angle triangle is defined as to be the sum of the total sides of an acute angle triangle. consider there are three different heights of an acute triangle namely a, b and c, then the perimeter is found out to be the sum of a, b and c sides

Perimeter = a+b+c

Frequently Asked Questions (FAQs)

1. Define acute angle parallelogram.

A parallelogram is defined as a quadrilateral that has opposite sides to be parallel. In parallelograms, opposite angles are always equal. In a parallelogram, there are observed to be too acute angles and two obtuse angles.

2. DefineAcute angle trapezoid.

The trapezoid is termed to be a parallelogram that has the parallel sides of one pair. In an acute angle trapezoid the interior angles in the trapezoid measure to be about less than 90°

3. What are the different types of angles present?

There are a total of five types of angles present namely

acute angles are defined as those angles whose sum is less than 90°
A right angle whose sum is equal to the 90 degree
Obtuse angles are defined as those whose measure is greater than the 90 degree
The straight angle which is equal to 180 degrees and
Reflex angles are defined as triangles whose measure is greater than 180 degrees.

4. What is the difference between an acute angle and an obtuse angle?

Acute angle triangle measures less than 90° whereas an obtuse angle triangle measures greater than 90°. A triangle can possess more than one acute angle but a triangle can never have more than one obtuse angle.

5. What is an acute angle triangle?

An acute angle triangle is a triangle where all three internal angles are less than 90 degrees. This means that each angle in the triangle is "acute," or sharp. The sum of all angles in any triangle is always 180 degrees, so in an acute triangle, the three angles must add up to 180° while each being less than 90°.

6. How does an acute angle triangle differ from other types of triangles?

An acute angle triangle differs from right and obtuse triangles. In a right triangle, one angle is exactly 90 degrees. In an obtuse triangle, one angle is greater than 90 degrees. An acute angle triangle has all angles less than 90 degrees, making it unique among triangle types.

7. Can an equilateral triangle be an acute angle triangle?

Yes, all equilateral triangles are acute angle triangles. In an equilateral triangle, all three angles are equal and measure 60 degrees each. Since 60° is less than 90°, equilateral triangles always satisfy the definition of an acute angle triangle.

8. What is the largest possible angle in an acute angle triangle?

The largest possible angle in an acute angle triangle is just under 90 degrees. If any angle were to reach or exceed 90 degrees, the triangle would no longer be classified as an acute angle triangle. The exact maximum depends on the other two angles, but it must be less than 90°.

9. Can an isosceles triangle be an acute angle triangle?

Yes, an isosceles triangle can be an acute angle triangle. An isosceles triangle has two equal sides and two equal angles. If these equal angles are less than 90 degrees, and consequently the third angle is also less than 90 degrees, then the isosceles triangle is also an acute angle triangle.

10. What is the relationship between the sides of an acute angle triangle?

In an acute angle triangle, the longest side is always opposite the largest angle. However, unlike in a right triangle where the longest side (hypotenuse) is always opposite the right angle, in an acute triangle, the longest side can be opposite any of the three angles, as long as that angle is the largest in the triangle.

11. How does the Pythagorean theorem apply to acute angle triangles?

The Pythagorean theorem (a² + b² = c²) does not directly apply to acute angle triangles as it does to right triangles. However, a modified version called the law of cosines can be used: c² = a² + b² - 2ab cos(C), where C is the angle opposite the side c. This law works for all triangles, including acute ones.

12. What is the centroid of an acute angle triangle?

The centroid of an acute angle triangle, like any triangle, is the point where the three medians intersect. A median is a line segment that connects a vertex to the midpoint of the opposite side. The centroid divides each median in a 2:1 ratio, with the longer segment closer to the vertex.

13. How do you calculate the area of an acute angle triangle?

There are several ways to calculate the area of an acute angle triangle:

14. What is the inscribed circle of an acute angle triangle?

The inscribed circle, or incircle, of an acute angle triangle is the largest circle that can fit inside the triangle, touching all three sides. The center of this circle is called the incenter, which is the point where the angle bisectors of the triangle intersect.

15. How do you find the circumcenter of an acute angle triangle?

The circumcenter of an acute angle triangle is the point where the perpendicular bisectors of all three sides intersect. It is equidistant from all three vertices and forms the center of the circumscribed circle (the circle that passes through all three vertices of the triangle).

16. What is special about the circumcenter in an acute angle triangle?

In an acute angle triangle, the circumcenter is always located inside the triangle. This is different from right triangles (where it's on the hypotenuse) and obtuse triangles (where it's outside the triangle). This property can help identify acute triangles visually.

17. What is the orthocenter of an acute angle triangle?

The orthocenter of an acute angle triangle is the point where the three altitudes intersect. An altitude is a line segment from a vertex perpendicular to the opposite side (or its extension). In an acute triangle, the orthocenter always lies inside the triangle.

18. How does the location of the orthocenter differ in acute, right, and obtuse triangles?

In an acute triangle, the orthocenter is inside the triangle. In a right triangle, it coincides with the right angle vertex. In an obtuse triangle, it lies outside the triangle. This difference in location can help distinguish between these triangle types.

19. What is the Euler line in an acute angle triangle?

The Euler line is a straight line that passes through several important points in a triangle, including the centroid, orthocenter, and circumcenter. In an acute angle triangle, all these points lie inside the triangle, and the Euler line passes through them in a specific order: the orthocenter, then the centroid, then the circumcenter.

20. How do the angles in an acute angle triangle relate to its side lengths?

In an acute angle triangle, as in all triangles, the largest angle is opposite the longest side, and the smallest angle is opposite the shortest side. This relationship is described by the law of sines: a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are side lengths and A, B, C are the opposite angles.

21. Can an acute angle triangle have two equal sides?

Yes, an acute angle triangle can have two equal sides. This type of triangle is called an isosceles acute triangle. In this case, the angles opposite the equal sides are also equal, and both of these angles, as well as the third angle, must be less than 90 degrees.

22. What is the difference between the incenter and the centroid in an acute angle triangle?

The incenter is the point where the angle bisectors intersect and is the center of the inscribed circle. The centroid is the point where the medians intersect and represents the triangle's center of mass. While both points are always inside the triangle, they generally do not coincide except in equilateral triangles.

23. How does the concept of similarity apply to acute angle triangles?

Two acute angle triangles are similar if their corresponding angles are equal. This means that the triangles have the same shape but may be different sizes. In similar triangles, the ratios of corresponding sides are constant, and this principle applies to all triangles, including acute ones.

24. What is the relationship between an acute angle triangle's perimeter and its area?

There's no fixed relationship between a triangle's perimeter and area, but for acute triangles, we can use the isoperimetric inequality: Area ≤ (s(s-a)(s-b)(s-c))^(1/2), where s is the semi-perimeter and a, b, c are the side lengths. The equality holds only for equilateral triangles.

25. How does the concept of congruence apply to acute angle triangles?

Two acute angle triangles are congruent if they have exactly the same shape and size. This means all corresponding sides and angles are equal. Congruence can be proven using various criteria like SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), and AAS (Angle-Angle-Side).

26. What is the nine-point circle in an acute angle triangle?

The nine-point circle, also called the Euler circle, is a circle that passes through nine significant points of a triangle: the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the line segments from the orthocenter to each vertex. In acute triangles, all these points are well-defined and distinct.

27. How do you determine if a triangle is acute without measuring angles?

You can determine if a triangle is acute using the Pythagorean theorem. If a² + b² > c² (where c is the longest side), then the triangle is acute. This is because the square of the longest side is less than the sum of squares of the other two sides in an acute triangle.

28. What is the relationship between an acute angle triangle's angles and its area?

The area of an acute angle triangle can be calculated using any two sides and the included angle: Area = 1/2 × a × b × sin(C). This formula shows that for a given pair of sides, the area increases as the included angle increases (up to 90°), demonstrating the relationship between angles and area.

29. How does the concept of concurrent lines apply to acute angle triangles?

In acute angle triangles, several sets of lines are concurrent (intersect at a single point):

30. What is the Fermat point in an acute angle triangle?

The Fermat point (or Torricelli point) is a point in an acute angle triangle where the total distance to the three vertices is minimized. It has the property that the lines joining it to the vertices form three 120° angles. In obtuse triangles, the Fermat point coincides with the obtuse angle vertex.

31. How does the law of cosines apply to acute angle triangles?

The law of cosines (c² = a² + b² - 2ab cos(C)) applies to all triangles, including acute ones. In acute triangles, all angles have cosines between 0 and 1, so the term -2ab cos(C) is always negative, ensuring that c² < a² + b², which is characteristic of acute triangles.

32. What is the relationship between an acute angle triangle's inradius and circumradius?

In any triangle, including acute ones, the relationship between the inradius (r) and circumradius (R) is given by the formula: r = (a + b - c)R / (a + b + c), where a, b, and c are the side lengths. In acute triangles, the inradius is always smaller than the circumradius.

33. How does the concept of triangle inequality apply to acute angle triangles?

The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the third side. This applies to all triangles, including acute ones. In acute triangles, this inequality is always strict: each side is less than the sum of the other two sides.

34. What is the significance of the medial triangle in an acute angle triangle?

The medial triangle is formed by connecting the midpoints of the sides of the original triangle. In acute triangles, the medial triangle is always acute and similar to the original triangle. It has half the linear dimensions and one-fourth the area of the original triangle.

35. How does the concept of cyclic quadrilaterals relate to acute angle triangles?

Any four points on the circumcircle of an acute angle triangle form a cyclic quadrilateral. The opposite angles of this quadrilateral are supplementary (add up to 180°). This property can be used to prove various theorems about triangles and circles.

36. What is the relationship between an acute angle triangle's area and its semiperimeter?

The area of an acute angle triangle can be calculated using Heron's formula: Area = √(s(s-a)(s-b)(s-c)), where s is the semiperimeter (half the perimeter) and a, b, c are the side lengths. This formula shows a direct relationship between the area and the semiperimeter.

37. How does the concept of triangle centers apply to acute angle triangles?

Acute angle triangles have several important centers:

38. What is the Euler's theorem for acute angle triangles?

Euler's theorem states that in any triangle, including acute ones, the distance between the circumcenter (O) and orthocenter (H) is related to the radius of the circumcircle (R) by the equation: OH² = 9R² - (a² + b² + c²), where a, b, and c are the side lengths of the triangle.

39. How does the concept of triangle cevians apply to acute angle triangles?

A cevian is any line segment in a triangle from a vertex to the opposite side. In acute triangles, important cevians include medians, angle bisectors, and altitudes. The concept of cevians leads to many theorems, such as Ceva's theorem, which applies to all triangles including acute ones.

40. What is the relationship between an acute angle triangle's angles and its circumcenter?

In an acute angle triangle, the circumcenter always lies inside the triangle. The closer an angle is to 90°, the closer the circumcenter is to the midpoint of the side opposite that angle. If all angles are equal (60° in an equilateral triangle), the circumcenter is equidistant from all sides.

41. How does the concept of triangle congruence criteria apply specifically to acute angle triangles?

The standard triangle congruence criteria (SSS, SAS, ASA, AAS) apply to acute triangles. However, the Side-Side-Angle (SSA) criterion, which is ambiguous for general triangles, can sometimes guarantee congruence for acute triangles if the given angle is opposite the longer of the two given sides.

42. What is the significance of the Euler line in acute angle triangles?

In acute angle triangles, the Euler line (which contains the centroid, orthocenter, and circumcenter) always lies entirely inside the triangle. The centroid divides the segment between the orthocenter and circumcenter in a 1:2 ratio, with the centroid closer to the circumcenter.

43. How does the concept of triangle area apply to acute angle triangles in coordinate geometry?

In coordinate geometry, the area of an acute angle triangle (or any triangle) can be calculated using the formula: Area = |1/2(x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2))|, where (x1, y1), (x2, y2), and (x3, y3) are the coordinates of the three vertices.

44. What is the relationship between an acute angle triangle's angles and its orthocenter?

In an acute angle triangle, the orthocenter always lies inside the triangle. The location of the orthocenter is influenced by the angles: as one angle approaches 90°, the orthocenter moves closer to that vertex. In an equilateral triangle (all angles 60°), the orthocenter coincides with the other triangle centers.

45. How does the concept of triangle inequality theorem extend to angles in acute triangles?

In acute angle triangles, not only does the sum of any two sides exceed the third side, but also the sum of any two angles is greater than the third angle. This is because in acute triangles, all angles are less than 90°, and their sum is always 180°.

46. What is the significance of the Nagel point in acute angle triangles?

The Nagel point is the point where the lines from each vertex to the point of tangency of the opposite side with the excircle intersect. In acute triangles, the Nagel point always lies inside the triangle and has interesting properties related to areas and distances within the triangle.

47. How does the concept of power of a point apply to acute angle triangles?

The power of a point theorem states that for any point P and a circle, the product of the lengths of the segments of any line through P intersecting the circle is constant. This concept applies to the circumcircle of acute triangles and can be used to prove various geometric properties.

48. What is the relationship between an acute angle triangle's area and the radius of its inscribed circle?

The area of an acute angle triangle (or any triangle) is related to its inradius r and semiperimeter s by the formula: Area = rs. This shows a direct relationship between the area and the size of the inscribed circle.