Obtuse angle triangles are triangles in which one angle of the triangle measures greater than 90 degrees. As the name suggests one angle of an obtuse angle triangle is an obtuse angle. A triangle is a closed, two-dimensional geometric figure with three angles, and the sum of all the angles of a triangle is 180 degrees. On the basis of a measure of angles, we divide the triangle into three categories i.e.
Now, let’s learn more about obtuse angled triangles, their properties, formulas, examples, and others in detail in this article.
Obtuse Angled Triangle Definition
An obtuse-angled triangle is defined as a triangle whose one interior angle measures more than 90°. According to the angle sum property of a triangle, the sum of all the interior angles of an obtuse-angled triangle is 180°. As one interior angle in an obtuse-angled triangle i.e. it measures more than 90°, the other two interior angles are acute, and their sum is than 90° so that the triangle sum property holds true.
In an obtuse angle triangle, the side opposite to the obtuse angle is the longest side. The figure given below shows an obtuse-angled triangle whose interior angles are 110°, 35°, and 35°. Since the given triangle has one angle greater than 90°, it is an obtuse-angled triangle.
What is Obtuse Angle?
Before learning further about obtuse angled triangles we must first learn about what is an obtuse angle. As we know that we measure angles from 0 degrees to 360 degrees and the angles that are greater than 90 degrees but less than 180 degrees are called obtuse angles. Some examples of obtuse angles are,
If a triangle has one obtuse angle it is called an obtuse angle triangle.
Note: A triangle can not have more than one angle as obtuse angle as its than fails the angle sum property of triangle.
Types of Obtuse-Angled Triangle
Obtuse-angled triangles are classified into two types depending on the length of their sides and measures of angles. That includes
- Scalene Obtuse Triangle
- Isosceles Obtuse Triangle
Now let’s learn more about them.
Isosceles Obtuse Triangle
An obtuse-angled triangle whose two sides are equal is called Isosceles Obtuse Triangle. It also has two angles equal. A triangle with angles 100°, 40°, and 40° is an Isosceles Obtuse Triangle.
Scalene Obtuse Triangle
An obtuse-angled triangle whose all sides are unequal is called Scalene Obtuse Triangle. A triangle with angles 100°, 60°, and 20° is a Scalene Obtuse Triangle.
Properties of an Obtuse-Angled Triangle
The following are some important properties of an obtuse-angled triangle:
- The side opposite to the obtuse angle in the obtuse-angled triangle is the largest side of the triangle.
- The sum of the interior angles of an obtuse-angled triangle is always equal to 180°.
- A triangle can have a maximum of one Obtuse angle as the sum of all the interior angles of triangles should not exceed 180°.
- In an obtuse-angled triangle, the circumcentre and the orthocentre lie outside the triangle, whereas the centroid and the incenter lie inside the triangle.
How to know if a Triangle is Obtuse?
A triangle can be determined if it is Obtuse or not by just following any of the methods discussed below.
Method 1: If at least any two angles of the triangle are given then by triangle sum property we can find the third angle of the triangle and finally observing the three angles of the triangle and looking for the obtuse angle we can tell whether the triangle is obtuse or not.
Method 2: If the three side lengths of a triangle are given, then the triangle is said to be obtuse if the sum of the squares of the smaller sides is less than the square of the longest side. If the sides of a triangle are a, b, and c, and c is the longest side, the triangle is said to be obtuse if a2 + b2 < c2.
Can a Triangle have Two Obtuse Angles?
The simple answer to this question is NO. Now let’s learn why a triangle can not have two obtuse angles. Suppose we have a triangle with two obtuse angles say 95°, 100° and 50°. Now according to the angle sum property of the triangle, the sum of all the interior angles must be 180°. In this case, the sum exceeds the value of 180°
95° + 100° + 50° = 245° > 180°
So this triangle is not possible. Similarly, any two obtuse angles taken together exceed the sum of 180° and hence we can not have a triangle with two obtuse angles.
Obtuse-Angled Triangle Formulas
The area and perimeter are the two basic formulas of an obtuse-angled triangle which are discussed below:
Perimeter of Obtuse-Angled Triangle
The perimeter of an obtuse-angled triangle is equal to the sum of its three side lengths. If a, b, and c are the side lengths of an obtuse-angled triangle, then its perimeter is given as (a + b + c) units.
Perimeter of an obtuse-angled triangle = (a + b + c) units
Where a, b, and c are the side lengths of the triangle.
Learn more about, Perimeter of a Triangle
Area of Obtuse-Angled Triangle
The area of a triangle is defined as the total space enclosed by the three sides of any triangle in a two-dimensional plane. As one interior angle of an obtuse triangle is greater than 90°, a perpendicular is drawn outside the triangle as shown in the figure given below to determine its height.
Area of Obtuse-Angled Triangle = ½ × b × h
Where
- “b” is the base length, and
- “h” is the height of the triangle.
Learn more about, Area of a Triangle
Obtuse Triangle Area by Heron’s Formula
If the three side lengths of an obtuse-angled triangle are given, then its area can be calculated using Heron’s Formula.
Area of Obtuse-Angled Triangle = 
Where
- “s” is the semi-perimeter,
- s = (a + b + c)/2, and
- a, b, and c are the side lengths of the triangle.
Solved Examples on Obtuse-Angled Triangle
Example 1: Calculate the area of an obtuse triangle whose height is 8 cm and the base is 6 cm.
Solution:
Given,
Height of Triangle (h) = 8 cm
Base of Triangle (b) = 6 cm
We know that,
Area of the triangle (A) = ½ × b × h
⇒ A = ½ × 6 × 8
⇒ A = ½ × 48
⇒ A = 24 sq. cm.
Hence, the area of the given obtuse triangle is 24 sq. cm.
Example 2: What is the area of an obtuse triangle whose sides are AB = 14 cm, BC = 9 cm, and AC = 7 cm?
Solution:
Given,
Sides of Obtuse Triangle are,
AB = c = 14 cm
BC = a = 9 cm
AC = b = 7 cm
We know that,
Area of Triangle =
and s = (a+b+c)/2
⇒ s = (14+9+7)/2
⇒ s = 30/2 = 15
Thus, A = 
⇒ 
⇒ A = √720 sq. cm
⇒ A = 26.833 sq. cm
Hence, the area of the given obtuse triangle is 26.833 sq. cm.
Example 3: Check if the angles 95°, 40°, and 45° constitute an Obtuse Angled Triangle.
Solution:
We know that in an Obtuse-Angled Triangle an angle greater than 90° must exist.
Here,
One Angle of triangle = 95° > 90°
Also,
95° + 40 + 45° = 180°
The Sum of all angles of triangle is 180°, i.e. this triangle follow angle sum property of traingle.
Thus, the given traingle is an Obtuse Angled Triangle.
Example 4: What is the perimeter of an obtuse triangle PQR whose sides are PQ = 12 units, QR = 10 units, and PR = 6 units?
Solution:
Given.
Sides of Obtuse Triangle
PQ = r = 12 units
QR = p = 10 units
PR = q = 6 units
Perimeter of Obtuse Triangle (P) = p + q + r
⇒ P = (10 + 6 + 12) units
⇒ P = 28 units
Hence, the perimeter of the obtuse triangle is 28 units.
FAQs on Obtuse-Angled Triangle
Q1: What is an Obtuse-Angled Triangle?
Answer:
An obtuse-angled triangle is defined as a triangle whose one interior angle is an obtuse angle, i.e. its measures is more than 90°.
Q2: How many Obtuse Angles are possible in a triangle?
Answer:
A triangle can have a maximum of one obtuse angle. If it has more than one obtuse angle then it fails the angle sum property of the triangle and thus, the triangle does not exist.
Q3: Can a triangle have two Obtuse Angles?
Answer:
No, a triangle can not have two obtuse angles as in this case, it does not verify the angle sum property of traingle and the triangle does not exist.
Q4: What are the different Types of Obtuse-Angled Triangles?
Answer:
Obtuse-angled triangles are classified into two types depending on the side lengths and measures of angles, i.e.,
- Scalene Obtuse Triangle
- Isosceles Obtuse Triangle
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