Obtuse Triangle

A triangle whose one of the angle is Obtuse angle is known as Obtuse Triangle

But what is Obtuse Angle?
Angle whose measurement is above 90 degree but below 180 degree

Hence Obtuse triangle has one angle between 90 to 180 degree

Examples of Obtuse Triangle
Given below are two examples of Obtuse triangle
⟹ Note one of the angles is between 90 and 180 degree
⟹ Also, sum of all angles in triangle = 180

What is Obtuse Triangle

Obtuse Triangle Property

(1) In Obtuse Triangle, one angle is obtuse and other two angles are acute angles

(02) A triangle cannot have more than one obtuse angle

If x, y and z are the three angles of triangle.
Angle property of triangle says: Sum of all three angle = 180 degree
i.e. x + y + z = 180

If two angles are obtuse angle ( x & y = 91 degree)
91 + 91 + z will be greater than 180 degree.
The angle property will not be satisfied.

Hence, there can be only one Obtuse angle in the triangle

(03) The side opposite to obtuse angle is the longest side

Obtuse Triangle Example

In the above example, note angle B is an obtuse angle (also the largest one)
So, the side opposite to the largest angle is the longest
Hence AC is the longest side of triangle ABC

(04) Circumcenter of Obtuse Triangle
The circumcenter of obtuse triangle lies outside the triangle

What is Circumcenter?
It is the center of circle which crosses all the vertex of triangle
Is is located at the intersection of perpendicular bisector of triangle

Locating Circumcenter of Obtuse triangle

Step 01
draw perpendicular bisector of side BC

Circumcenter of Obtuse Triangle

Step 02
Now draw perpendicular bisector of side AC

Examples of Obtuse Triangle

Point O is the point of intersection of two perpendicular bisector, hence circumcenter of triangle ABC

From the above diagram we concluded that:
(a) In obtuse triangle, the circumcenter is located outside the triangle
(b) The circumcenter is located towards the longest side of the triangle

(05) Centroid of Obtuse Triangle
The centroid of obtuse triangle lies inside the triangle
Infact, the centroid of any triangle lies inside the body

But what is centroid?
Centroid is the central point of the geometrical figure.
It is also the center of mass of any object, where the entire mass is concentrated.
It is located at the intersection of median.

Locating centroid of Obtuse triangle

Step 01
Find the midpoint of side AB of triangle and join it with the opposite vertex

Centroid of Obtuse Triangle

Step 02
Similarly find the midpoint of side BC and join it with opposite vertex

Where is the centroid in Obtuse Triangle

The point O is the intersection of two medians and is the centroid of the triangle

The centroid of obtuse triangle lies inside the body

Area of Obtuse Triangle

There are two formulas for calculating area
(a) Formula when base and height lengths are given
(b) Formula when length of all sides are given

Formula when Base and Height lengths are given

Given is the obtuse triangle ABC with following information:
⟹ length of base is b cm
⟹ perpendicular distance of base and opposite vertex is h cm

Note that we have to extend the triangle (as shown in dotted line) to find the perpendicular height (h cm )

Calculating area of Obtuse Triangle

If above information is provided then use the following formula

Area=\ \frac{1}{2} \times \ base\ \times height

Formula when side lengths are given

Given below is the obtuse triangle with sides a, b and c cm
Follow the below step to calculate the triangle area

Formula for Obtuse Triangle Area

Step 01
Calculate semi perimeter of triangle
S\ =\ \frac{a+b+c}{2}

Step 02
Now use the area formula
Area\ =\sqrt{S\ ( S-a)( S-b)( S-c)}

Frequently Asked Question : Obtuse Triangle

(01) Check if the below triangle is Obtuse Triangle

Obtuse Triangle definition
Read Solution

This is not an obtuse angle.
In Obtuse angle, the angle measurement should be above 90 degree

(02) What is the location of circumcenter, centroid and orthocenter of Obtuse Triangle?

Read Solution

Circumcenter : Outside the triangle
Centroid : Inside the triangle
Orthocenter : Outside the triangle

(03) Can we apply Pythagoras theorem to Obtuse triangle

Read Solution

Pythagoras Theorem only applied to right triangle

(04) How to know if the triangle is Obtuse Triangle?

Read Solution

There are two methods to find out
(a) Check Angles
If any of the angle measurement is above 90 degree then the triangle is Obtuse

(b) Check Side Lengths
If the sides follows following rule, then it is obtuse angle
c^{2} >\ \ a^{2} +b^{2}

Where, c is the longest side of the triangle

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