Acute Triangle

Did you know, there are three types of triangles - acute, right, and obtuse? But if a triangle is acute, it can't be obtuse and right at the same time. The same holds for similar other situations.

Introduction to Acute Triangle | ABC acute triangle

In this mini-lesson, we will be learning about acute triangles. The above image is an example of it. We will be exploring its properties, how to create it, and other interesting facts related to acute triangles.

You can check out the interactive questions to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

Lesson Plan

What Is Acute Triangle?

Each triangle has 3 sides and 3 angles. These can have different values. But for an acute triangle, we say that all three angles of the triangle are from \( 0^o\) to \( 90^o\). So, let us assume a triangle ABC as shown here.

Introduction to Acute Triangle

Now, for the given triangle to be an acute triangle, we need to follow a certain number of restrictions.

So, let A be an interior angle inside the triangle. Thus, we can say that: 

\(A \in (0^o,90^o)\)

Hence, any triangle following the above equality will be known as an acute triangle. 


Types of Acute Triangle

There are 3 types of Acute Triangles: 

1. Acute Equilateral Triangle: All three sides of the triangle are of equal length.

Introduction to types of Acute Triangles | Acute Equilateral Triangle

2. Acute Isosceles Triangle: Any two of the three sides of a triangle are of equal length.

Introduction to types of Acute Triangles | Acute Isosceles Triangle

3. Acute Scalene Triangle: None of the three acute triangle sides are of equal length.

Introduction to types of Acute Triangles | Acute Scalene Triangle


General Properties of Acute Triangle

Given below are a few general properties of acute triangles: 

Property 1

All 3 interior angles of the triangle are acute.

Properties of Acute Triangle | All 3 interior angles of triangle are acute.

As seen above, you can note that all the angles of the triangle are less than \( 90^o\). 

Property 2

Area of a triangle is \(A = {1 \over 2} \times b  \times h \) 

Properties of Acute Triangle | Area of an acute triangle.

The height will be the perpendicular drawn from the vertex to the base of the triangle. And the area will be calculated from height and base as:

\(A = {1 \over 2} \times b  \times h \) 

Property 3

The perimeter of the triangle is \(P=a+b+c \),
And the semi perimeter is \(s = {a+b+c \over 2} \) 

These are the few acute triangle formulas.

Properties of Acute Triangle | perimeter of an acute triangle.

Look at the values of Perimeter and semi - perimeter shown in the diagram given above. 


What Are the Angles of Acute Triangle?

The acute triangle angles can be anything as long as each one of them lies between \( 0^o\) to \( 90^o\). But we also know that the sum of the angles of any triangle will be \( 180^o\). So, overall to be considered as an acute triangle, the angles of the triangle should be following the given two rules: 

  1. Each angle lies between \( 0^o\) to \( 90^o\). 
  2. The sum of all 3 angles of the triangle will be \( 180^o\).

Try inserting values of any two angles in the boxes given above and see what will be the third angle. 

 
important notes to remember
Important Notes
  1. For an acute triangle, all three angles of the triangle are from \( 0^o\) to \( 90^o\).
  2. Area of the acute triangle is \(A = {1 \over 2} \times b  \times h \) 
  3. The Perimeter of the Acute triangle is:
    \(P=a+b+c \)
  4. The types of acute triangles are:
    a) Acute Equilateral Triangle
    b) Acute Isosceles Triangle
    c) Acute Scalene Triangle

Solved Examples

Let us have a look at the acute triangle examples here!

Example 1

 

 

Which of the following angle measures can form an acute triangle?

a) 60°, 70°, 50°

b) 95°, 30°, 55°

c) 90°,45°,45°

d) 90°, 60°, 30°

Solution

An acute triangle has all of the angles  < 90°. 

Among the given options, option (a) satisfies the condition. 

\(\therefore\) Option "a" forms an acute triangle.
Example 2

 

 

Given below is an isosceles triangle \(\text{ABC}\), is this an acute triangle?

two base angles of an isosceles acute triangle are 3x and 2x plus 42. Find the vertex angle

Solution

\[\begin{align}
3x &= x +42 (\because\angle \text{ABC} \! =\! \angle \text{BCA} )\\
\therefore  2x &= 42\\
x &=21\\
\angle \text{ABC} &= x+42\\
&= 63^\circ\\
\Rightarrow \angle \text{BCA} &=63^\circ(\!\because\!3x \!=\!3 \!\times\! 21\! =\!63\! )\\
\therefore \angle\text{BAC} &= (180-(63+63))\\
&=180-126\\
&=54^\circ
\end{align}\]

Hence we finally get : \(\begin{align}\angle \text{ABC}\!=\!\angle \text{BCA}\!=\!63^\circ \text{and} \:\angle\text{BAC}\!=\!54^\circ\end{align}\)

\(\therefore\) ABC is an acute triangle
Example 3

 

 

A triangle with one exterior angle measuring 80° is shown in the image.

Can this triangle be an acute triangle?

One of the exterior angle is 80 degrees. Is the triangle an acute triangle?

Solution

The exterior angle and the adjacent interior angle forms a linear pair (i.e, they add up to 180°).

Hence, the interior angle at vertex B is: 

180° - 80° = 100°

\(\therefore\) The given triangle is obtuse-angled.
Example 4

 

 

Find the area of an acute triangle whose base is 8 in and height is 4 in.

Solution

Area of an acute-angled triangle

= \(\begin{align}\frac{1}{2} \times \text{base} \times \text{height}\end{align}\)

Substituting the values of base and height, we get:

\(\begin {align}
\text{Area} &= \frac{1}{2} \times 8 \times 4 \\
&= 16\: \text{in}^2
 \end{align}\)

\(\therefore\) Area of the triangle = 16 in2
Example 5

 

 

Find the height of an acute triangle whose area = 60 in2 and base = 8 in.

Solution

Area of an acute-angled triangle

= \(\begin{align}\frac{1}{2} \times \text{base} \times \text{height}\end{align}\)

Therefore, height of the acute triangle can be calculated by:

\(\begin{align}\text{Height} = \frac{2 \times \text{Area}}{\text{base}} \end{align}\)

Substituting the values, we get:

\(\begin {align}
\text{Height} &= \frac{2 \times 60}{8}   \\
&= 15\: \text{in}
 \end{align}\)

\(\therefore\) Height  = 15 in
 
Challenge your math skills
Challenging Questions

​If the area of a triangle with vertices (2,3), (5,m), and (5,5) is 12 square units. What are the possible values of "m"?​

Hint : We know that the area of any triangle will be \(A = {1 \over 2} \times b \times h \).

Interactive Questions

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Let's Summarize

The mini-lesson targeted the fascinating concept of acute triangles. The math journey around acute triangles started with the basics of a triangle and went on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. 

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At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

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Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.


Frequently Asked Questions (FAQs)

1. Are isosceles triangles always acute?

No, an isosceles triangle can be acute, right, or obtuse-angled depending upon the measure of the angles it has.

2. What is the equation for an acute triangle?

If A is the measure of any angle in an acute triangle, then we can say that \(A<90^o\).

3. How do you know if a triangle is acute?

A triangle can be acute if all the angles inside it are measuring \(<90^o\).

4. What are the sides of an acute triangle?

There is no limit on the measure of the sides of any triangle. As long as the angles satisfy the acute values, sides can be as long as one wants. 

5. Do acute triangles add up to 180?

Yes, in fact, the angles of any triangle will add up to 180 degrees.

6. Can a triangle be right and acute?

No, a triangle can either be acute or be right-angled. It cannot be both at the same time.

  
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