An Introduction to Acute Angled Triangles
Triangle is a three sided-polygon with three edges, three vertices and three interior angles. In other words, a triangle is a closed two-dimensional figure with three sides and three angles.
Triangles are classified into different types on the basis of their sides and angles. Each of them has its own individual properties.
Classification of Triangles on the Basis of Their Sides is as Follows
Equilateral Triangle: A triangle whose all the three sides are of equal length.
Isosceles Triangle: A triangle whose two sides are of equal length.
Scalene Triangle: A triangle whose all three sides are unequal.
Classification of Triangles on the Basis of Their Angles is as Follows
Acute Angled Triangle: A triangle whose all interior angles are less than 90°.
Right Angled Triangle: A triangle whose one of the interior angles is 90°.
Obtuse Angled Triangle: A triangle whose one of the interior angles is more than 90°.
In this article, you will learn more about Acute angled triangles like its definition, properties, the formula of perimeter and area along with some solved examples.
Definition of Acute Angled Triangle
An acute triangle is a triangle with three acute angles Acute and obtuse triangles are the two different types of oblique triangles — triangles that are not right triangles because they have no 90° angle.
Acute Triangle Formulas
There are two basic formulas of an acute triangle, which are given below;
Area of an acute triangle
Perimeter of an acute triangle
Identification of the Acute Angled Triangle
According to Pythagoras theorem the triangle is acutely angled if the square of the longest side is less than the sum of the squares of two smaller sides. Let 'a', 'b' and 'c' are the length of sides of a given triangle, in which side 'a' is longest, then the given triangle is acutely angled if a2 < b2 + c2.
Area of an Acute Angled Triangle
The area of an acute triangle is the amount of space that it occupies in a two-dimensional surface. So, the area of an acute triangle can be calculated if the length of its base and corresponding altitude (height) is known or the length of its three sides is known or length of two sides and angle between them is given.
Hence, the area of an acute triangle = \[ \frac{1}{2} \] × (base) × (height) sq. units
= \[ \frac{1}{2} \] × (b) × (h) sq. units
or,
The area of acute triangle using Heron’s formula = \[ \sqrt{s(s-a)(s-b)(s-c)} \] sq. units
Where, ‘a’, ‘b’ and ‘c’ are the length of sides of the acute triangle
And, s = semi-perimeter of triangle =\[ \frac{a+b+c}{2} \]
or,
The area of the acute triangle if the length of its two sides and angle between them is given.
Area of acute triangle = \[\frac{1}{2}\] × a × b × sinC
where ‘a’ and ‘b’ are the length of two sides and C is the angle between them.
Perimeter of an Acute Angled Triangle
The perimeter of an acute triangle is the sum of the length of its sides. So, if the length of the sides of the acute triangle are a, b and c units, then its perimeter is given by:
Perimeter of an acute triangle = (a + b + c) units.
Method to Check if a Given Triangle is Acute Angled or Not
Method 1: If the measure of angles of the triangle are given, then check the measure of its angles. If all the three angles of the triangle measure less than 90° degrees, then the given triangle is an acute angled triangle.
Method 2: If the length of three sides of a triangle are given, then using Pythagora's identity we can easily determine if a given triangle is acutely angled or not.
According to Pythagoras identity, the triangle is acutely angled if the square of the longest side is less than the sum of the squares of two smaller sides.
Let ‘a’, ‘b’ and ‘c’ are the length of sides of a given triangle, in which side ‘a’ is longest, then the given triangle is acutely angled if a2 < b2 + c2.
Properties of Acute Triangle
There are a few important properties that make the acute triangle different from other types of triangles. The properties of an acute-angled triangle are listed below:
An acute triangle has all interior angles measured less than 90° degrees.
The side opposite to the smallest angle is the smallest side of the triangle.
The square of the longest side is less than the sum of the squares of two smaller sides.
The points of concurrency, Centroid, Incenter, Circumcenter and Orthocenter lie inside the triangle.
Solved Examples
Q.1. Two angles of an acute angled triangle are of measure 750 and 350. Find the measure of the third angle.
Solution: Let the third angle be ∠A and the ∠B = 750 and ∠C = 350. Then,
By interior angle sum property of triangles,
∠A + ∠B + ∠C = 1800
⇒ ∠A + 750 + 350 = 1800
⇒ ∠A + 1100 = 1800
⇒ ∠A = 180 -1100
⇒ ∠A = 700
So, the measure of the third angle of the given triangle is 700.
Q.2. Find the area of an acute angled triangle whose base is 10cm and the corresponding altitude is 8cm.
Solution: Given, the base of the acute angled triangle = 10cm, and height = 8cm.
Since, the area of an acute triangle = \[\frac{1}{2}\] × (base) × (height) sq. units
= \[\frac{1}{2}\] × 10 × 8 sq. cms
= 40 sq. cms
Conclusion
This is all about the meaning and formulas of acute angled triangles. Learn how these formulas are derived to understand the basic concept. Practice solving questions related to these formulas and develop your conceptual foundation in this topic.
FAQs on Acute Angled Triangle
1. What is the difference between an acute and an obtuse angled triangle?
In an acute angle triangle, all the angles will be below 90 degrees. In an obtuse angled triangle, one of the three angles will be more than 90 degrees and the other two will be less than 90 degrees.
2. What is the difference between a right angled triangle and an acute angled triangle?
The prime difference between a right angled and an acute angled triangle is that one angle in the former will be equal to 90 degrees whereas all the angles in the latter will be less than 90 degrees. The sum of all the angles will be equal to 180 degrees. The right angled triangle will also have a base, a perpendicular, and a hypotenuse.