Acute Triangle

A triangle whose all sides are acute angle is known as Acute Triangle

What is Acute Angle?
Angle below 90 degree measurement is known as acute angle

Hence in acute angle triangle, all the angle measurement is below 90 degrees

Examples of Acute Triangle
Given above are the example of acute angle triangle.
Note all the angles of triangle are below 90 degrees

Types of Acute Angle

There are three types of acute angle possible:
(a) Scalene Acute Triangle
(b) Isosceles Acute Triangle
(c) Equilateral Acute Triangle

The definition of each angle is self explanatory:

Scalene Acute Triangle
Acute Triangle in which length of all sides are different are called Scalene Acute Triangle
Here all the angles are also measured differently

Isosceles Acute Triangle
It’s an Acute Triangle in which length of two sides are equal
Here two angles are of same measurement

Equilateral Acute Triangle
Its an Acute triangle in which length of all sides are equal
All angles measures 60 degrees

Examples of all the triangles are shown in below image

Acute Angle Properties

(01) In the acute triangle, there will be no angle measuring exact or more than 90 degree

(02) The circumcenter of acute triangle lies inside the triangle

What is Circumcenter?
Its a center of the circle through which circle can be drawn which will touch all the vertex of triangle

Example

Given above is acute triangle ABC and circumcenter O through which circle can be drawn touching all the vertex of the triangle.

How to locate the circumcenter in acute triangle?
Circumcenter is a point formed by the intersection of two perpendicular bisectors of triangle

Let us find circumcenter of the below triangle using following steps:

Step 01
Take compass rounder and draw perpendicular bisector of side BC

Step 02
Similarly draw perpendicular bisector of line AB

Step 03
The point where two perpendicular bisector meet is the circumcenter of the triangle.
Label the point as O

Conclusion
The circumcenter of acute triangle lies inside the triangle

(03) Centroid of acute triangle lies inside the triangle

What is centroid?
Centroid is the midpoint of the geometrical figure.
It can be located at the point of intersection of the medians

Let us find the centroid of the given acute triangle HGI using following steps

Step 01
Find the midpoint of side GI and draw straight line touching the midpoint and vertex of opposite side
The line MN is the median of side GI

Step 02
Similarly draw the median on other two sides

The point O is the intersection of median.
This point is also the centroid of the triangle.

Conclusion
In acute triangle, the centroid lies inside the triangle

(04) Similar to the circumcenter and centroid, other points like orthocenter and incenter also lines inside the triangle

Area of Acute Triangle

There are two formulas for calculation of Area of Acute Triangle.
(a) Formula when height and base length is given
(b) Formula when length of all sides are given

Area formula when base length & height is given

Area of triangle = \frac{1}{2} \times \ b\ \times h\

Area Formula when all sides are given

Given below is an acute triangle with side length a, b and c
Follow the below steps for area calculation

Step 01
Calculation of semi perimeter (S)
S\ =\ \frac{a+b+c}{2}

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

This formula is also known as Heron’s formula
Remember both the formulas as it will help to solve different math problems.

Frequently Asked Questions – Acute Triangle

(01) Is the below image is of acute triangle?

(02) Can acute triangle has two equal sides?

Yes
In acute isosceles triangle, two sides are equal to each other and all angles are acute angle

(03) In acute triangle, can all the angle be equal to each other

Yes, in equilateral acute triangle, all the angles are equal measuring 60 degree

(04) In acute triangle, is circumcenter and centroid lie on same location?