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101 LearnersLast updated on September 3, 2025
Properties of Acute Triangle
An acute triangle is a type of triangle where all the interior angles are less than 90 degrees. These properties help students simplify geometric problems related to acute triangles. The properties of an acute triangle include having all angles less than 90 degrees and specific relationships between the sides and angles. These properties help students analyze and solve problems related to angles, side lengths, and area. Now let us learn more about the properties of an acute triangle.
What are the Properties of an Acute Triangle?
The properties of an acute triangle are straightforward, and they help students understand and work with this type of triangle. These properties are derived from the principles of geometry. There are several properties of an acute triangle, and some of them are mentioned below:
Property 1: All angles are acute In an acute triangle, each of the three interior angles is less than 90 degrees.
Property 2: Relationship between sides The sides of an acute triangle must satisfy the triangle inequality theorem, where the sum of the lengths of any two sides is greater than the length of the remaining side.
Property 3: Altitudes All the altitudes of an acute triangle will lie inside the triangle.
Property 4: Circumcenter The circumcenter of an acute triangle, which is the point where the perpendicular bisectors of the sides intersect, lies inside the triangle.
Property 5: Area Formula The formula used to calculate the area of a triangle is given below: Area = ½ x base x height
Tips and Tricks for Properties of an Acute Triangle
Students tend to confuse and make mistakes while learning the properties of an acute triangle. To avoid such confusion, we can follow the following tips and tricks:
All Angles are Acute: Students should remember that in an acute triangle, all the angles are less than 90 degrees. To verify this, students can draw an acute triangle and measure the angles to see that they are all less than 90 degrees.
Altitudes Lie Inside: Students should remember that in an acute triangle, all the altitudes will intersect inside the triangle.
Circumcenter Inside: Students should remember that in an acute triangle, the circumcenter is always located inside the triangle.
Confusing an Acute Triangle with a Right Triangle
Students should remember that an acute triangle has all angles less than 90 degrees, whereas a right triangle has one angle exactly equal to 90 degrees.
Mistake 1
Misinterpreting the Circumcenter Location
Students should know and remember that the circumcenter of an acute triangle lies inside the triangle, unlike in an obtuse triangle, where it lies outside.
Mistake 2
Incorrectly Applying the Area Formula
Students should practice the formula which is used to find the area of a triangle, which is given below. The students must use the correct base and height values. Area = ½ x base x height
Mistake 3
Misunderstanding the Triangle Inequality
Students should remember that in an acute triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
Mistake 4
Forgetting All Angles are Less than 90 Degrees
Students must remember that in an acute triangle, all three interior angles are less than 90 degrees.
Mistake 5
Solved Examples on the Properties of Acute Triangles
In an acute triangle, the sides are labeled as AB, BC, and CA. If AB = 5 cm, BC = 6 cm, and CA = 7 cm, what can you say about the angles of the triangle?
All angles are less than 90 degrees.
Problem 1
In an acute triangle, all the interior angles are less than 90 degrees. Since the given side lengths satisfy the triangle inequality theorem, the triangle is acute.
In an acute triangle XYZ, if angle X = 40 degrees and angle Y = 50 degrees, what is the measure of angle Z?
Explanation
Angle Z = 90 degrees
Problem 2
The sum of angles in a triangle is always 180 degrees. Therefore, angle Z = 180 - 40 - 50 = 90 degrees.
The altitudes of an acute triangle intersect at a point O. What can you conclude about the position of point O?
Explanation
Point O is inside the triangle.
Problem 3
In an acute triangle, all the altitudes intersect at a point called the orthocenter, and it always lies inside the triangle.
In an acute triangle PQR, if the perpendicular bisector of side PQ intersects side PR at point M, where is the circumcenter located?
Explanation
The circumcenter is inside the triangle.
Problem 4
In an acute triangle, the perpendicular bisectors of the sides intersect at a point called the circumcenter, which is located inside the triangle.
An acute triangle has a base of 8 cm and a height of 5 cm. What is the area of the triangle?
Explanation
Area = 20 sq cm.
An acute triangle is a triangle where all the interior angles are less than 90 degrees.
1.How many angles in an acute triangle are less than 90 degrees?
2.Can an acute triangle have a right angle?
3.How do you find the area of an acute triangle?
4.Can an acute triangle have an obtuse angle?
Common Mistakes and How to Avoid Them in Properties of Acute Triangles
Students tend to get confused when understanding the properties of an acute triangle, and they tend to make mistakes while solving problems related to said properties. Here are some common mistakes students tend to make and the solutions to said common mistakes.
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Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
