Obtuse Triangle

An obtuse triangle is a triangle with one interior angle measuring greater than 90 degrees. In geometry, triangles are considered as 2D closed figures with three sides of the same or different lengths and three angles with the same or different measurements. Based on the length, angles, and properties, there are six kinds of triangles that we learn in geometry i.e. scalene triangle, right triangle, acute triangle, obtuse triangle, isosceles triangle, and equilateral triangle.

If one of the interior angles of the triangle is more than 90°, then the triangle is called the obtuse-angled triangle. Let's learn more about obtuse triangles, their properties, the formulas required, and solve a few examples to understand the concept better.

An obtuse-angled triangle or obtuse triangle is a type of triangle whose one of the vertex angles is bigger than 90°. An obtuse-angled triangle has one of its vertex angles as obtuse and other angles as acute angles i.e. if one of the angles measure more than 90°, then the sum of the other two angles is less than 90°. The side opposite to the obtuse angle is considered the longest. For example, in a triangle ABC, three sides of a triangle measure a, b, and c, c being the longest side of the triangle as it is the opposite side to the obtuse angle. Hence, the triangle is an obtuse-angled triangle where a² + b² < c²

An obtuse-angled triangle can be a scalene triangle or isosceles triangle but will never be equilateral since an equilateral triangle has equal sides and angles where each angle measures 60°. Similarly, a triangle cannot be both an obtuse and a right-angled triangle since the right triangle has one angle of 90° and the other two angles are acute. Therefore, a right-angle triangle cannot be an obtuse triangle and vice versa. Centroid and incenter lie within the obtuse-angled triangle while circumcenter and orthocenter lie outside the triangle.

The triangle below has one angle greater than 90°. Therefore, it is called an obtuse-angled triangle or simply an obtuse triangle.

There are separate formulas to calculate the perimeter and the area of an obtuse triangle. Let's learn each of the formulas in detail.

Obtuse Triangle Perimeter

The perimeter of an obtuse triangle is the sum of the measures of all its sides. Hence, the formula for the perimeter of an obtuse-angled triangle is:

Perimeter of obtuse-angled triangle = (a + b + c) units.

Area of Obtuse Triangle

To find the area of an obtuse triangle, a perpendicular line is constructed outside of the triangle where the height is obtained. Since an obtuse triangle has a value of one angle more than 90°. Once the height is obtained, we can find the area of an obtuse triangle by applying the formula mentioned below.

In the given obtuse triangle ΔABC, we know that a triangle has three altitudes from the three vertices to the opposite sides. The altitude or the height from the acute angles of an obtuse triangle lies outside the triangle. We extend the base as shown and determine the height of the obtuse triangle.

Area of ΔABC = 1/2 × h × b where BC is the base, and h is the height of the triangle.

Area of an Obtuse-Angled Triangle = 1/2 × Base × Height

Obtuse Triangle Area by Heron's Formula

The area of an obtuse triangle can also be found by using Heron's formula. Consider the triangle ΔABC with the length of the sides a, b, and c.

Heron's formula to find the area of an obtuse triangle is: \(\sqrt {s(s - a)(s - b)(s - c)}\), where, (a + b + c) is the perimeter of the triangle and S is the semi-perimeter which is given by (s): = (a + b + c)/2

Each triangle has its own properties that define them. An obtuse triangle has four different properties. Let's see what they are:

Property 1: The longest side of a triangle is the side opposite to the obtuse angle. Consider the ΔABC, side BC is the longest side which is opposite to the obtuse angle ∠A. See the image below for reference.

Property 2: A triangle can only have one obtuse angle. We know that the angles of a triangle sum up to 180°. Consider the obtuse triangle shown below. We can observe that one of the angles measures greater than 90°, making it an obtuse angle. For instance, if one of the angles is 91°, the sum of the other two angles will be 89°. Hence, a triangle cannot have two obtuse angles because the sum of all the angles cannot exceed 180 degrees. Observe the image given below to understand the same with an illustration.

Property 3: The sum of the other two angles in an obtuse triangle is always smaller than 90°. We just learned that when one of the angles is an obtuse angle, the other two angles add up to less than 90°.

In the above triangle, ∠1 > 90°. We know that by angle sum property, the sum of the angles of a triangle is 180°. Therefore, ∠1 + ∠2 + ∠3 = 180° and ∠1 > 90°

Subtracting the above two, we have, ∠2 + ∠3 < 90°.

Property 4: The circumcenter and the orthocenter of an obtuse-angled triangle lie outside the triangle. The orthocenter (O), the point at which all the altitudes of a triangle intersect, lies outside in an obtuse triangle. As seen in the image below:

Circumcenter (H), the median point from all the triangle vertices, lies outside in an obtuse triangle. As seen in the image below:

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What Is an Obtuse-Angled Triangle?

An obtuse triangle is a triangle in which one of the interior angles is greater than 90°. It has one of its vertex angles as obtuse and other angles as acute angles i.e. when one angle measures more than 90°, the sum of the other two angles is less than 90°. An obtuse triangle can also be called an obtuse-angled triangle. In general, an obtuse triangle can be a scalene triangle or isosceles triangle but not an equilateral triangle. The circumcenter and orthocenter lie outside the triangle while the centroid and incenter come inside the obtuse triangle.

Is it Possible to Have a Triangle With One Obtuse-Angle and One Right-Angle?

No, a triangle cannot have one obtuse angle and one right-angle together since the right triangle has one angle of 90° and the other two angles are acute. Therefore, a right-angle triangle cannot be an obtuse triangle and vice versa.

Can a Triangle Have Two Obtuse Angles?

No, a triangle cannot have more than one obtuse angle as angle > 90° + angle > 90° = angle >180°. Because the sum of the angles of a triangle in Euclidean geometry must be 180°, no Euclidean triangle can have more than one obtuse angle.

How Can You Tell If a Triangle is Obtuse by the Side Lengths?

The longest side of a triangle is considered to be the opposite side of the obtuse angle. If a, b, c are three sides of a triangle such that a² + b² < c², then the triangle will have an obtuse angle and it will be an obtuse triangle.

How Do You Know If a Triangle Is Obtuse?

To find if a triangle is obtuse, we can look at the angles mentioned. If one angle is greater than 90° and the other two angles are lesser along with their sum being lesser than 90°, we can say that the triangle is an obtuse triangle. For example, ΔABC has these angle measures ∠A = 120°, ∠A = 40°, ∠A = 20°. This triangle is an obtuse triangle because ∠A = 120°.

What Are the Properties of an Obtuse Triangle?

Here are the properties of obtuse triangles:

• The longest side of the triangle is the side opposite to the obtuse angle.
• A triangle cannot have more than one obtuse angle.
• The sum of the other two angles in an obtuse triangle is always smaller than 90°.
• The circumcenter and the orthocenter of an obtuse triangle lie outside the triangle.
• The sum of the squares of the two sides is less than the square of the third side.