## What is Acute Triangle?

**A triangle is a basic polygon with three sides and three vertices. Since it has three sides, it has three interior angles.** An angle that measures between 0° and 90° is called an acute angle. An acute triangle is a type of triangle in which all the three internal angles of the triangle are acute. Acute triangles are also called acute-angled triangles. Even though the length of the sides of acute triangles differs, the interior angles are never more than 90°.

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## Types of Acute Triangles

We can categorize acute triangles into three different types based on the side length and angles of the triangles as follows.

**Equilateral Triangle**

All the sides of an equilateral triangle are of equal lengths—each interior angle of this triangle measures 60°. So, an equilateral triangle is always an acute triangle.

In the given figure of an equilateral Triangle,

∠A = ∠B = ∠C = 60°

Side AB = Side BC = Side AC

**Acute Isosceles Triangle**

Two sides of an acute isosceles triangle are of equal length, and angles opposite to those sides measure the same.

In the given figure of an Acute Isosceles Triangle,

∠B = ∠C ≠ ∠A

Side AB = Side AC ≠ Side BC

All three sides and internal angles of a scalene acute triangle are unequal. All angles measure less than 90 degrees.

In the given figure of an Acute Scalene Triangle,

∠A ≠ ∠B ≠ ∠C

Side AB ≠ Side BC ≠ Side AC

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## Acute Triangle Formulas

The basic formulas relating to an acute triangle is to calculate its perimeter and area.

## The Perimeter of An Acute Triangle

Perimeter measures the outer boundary of the given shape. The perimeter of a triangle is the sum of the length of all the sides.

## Area of An Acute Triangle

The area of a triangle means the inner space covered by the three sides of the triangle.

The area of an acute triangle can be calculated using the formula of the area of a triangle.

Area of triangle = (1/2) × b × h

Here, ‘b’ is the base length of an acute triangle, and ‘h’ is the height of an acute triangle.

## Solved Examples on Acute Triangle

**Find the missing angle in the given figure and identify the type of triangle.**

Solution:

Using the angle sum property of triangles, we get ∠A = 180° – 70° – 30° = 80°.

Since all three angles of the given triangle are different acute angles, it is an acute scalene triangle.

**In an acute isosceles triangle ABC, side AB = 6 cm and ∠B = ∠C. If the perimeter of this triangle is 16 cm, then find the length of the side BC.**

Solution:

In the given acute isosceles triangle ABC, ∠B = ∠C. The sides opposite these angles are equal. Therefore, the lengths of the sides AB and AC must be equal.

So, AB = AC = 6 cm

The perimeter of the triangle is 16 cm. Therefore, AB + AC + BC = 16 cm.

So, the length of the side BC must be 16 – 6 – 6 or 4 cm.

**Find the area of the acute triangle whose base is 5 cm long and height is 6 cm?**

Solution:

Area of an acute triangle = Area of a triangle = (1/2) × b × h.

Substituting the value of the base (b) as 5 cm long and height (h) as 6 cm, we get

Area = 1/2 × 5 × 6 = 15 cm^{2}

## Practice Problems On Acute Triangle

## Acute Triangle

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

In an acute triangle, all the three angles are less than 90°. Only option a) satisfies this condition.

### Identify the type of triangle.

All the three interior angles ∆ABC are acute and AB = AC. Therefore, ∆ABC is an acute isosceles triangle.

### If the three interior angles of a triangle measure x, x + 20, and x + 40, then identify the type of the triangle?

Using the angle sum property of a triangle, we get $x + x + 20 + x + 40 = 180$.

So, x must be 40 and the three interior angles of the triangle are 40°, 60° and 80°.

Since all the angles of the triangle are acute but different, the triangle must be an Acute Scalene Triangle.

## Frequently Asked Questions on Acute Triangle

**What is an acute angle?**

An angle that measures between 0° and 90° is called an acute angle. It can be measured using a protractor.

**What are the properties of an acute angled triangle?**

The significant properties of an acute triangle are:

- All the three interior angles of an acute triangle measures less than 90°.
- The angles of an acute triangle add up to 180°.

**Is a scalene triangle always an acute triangle?**

No, a scalene may not always be an acute triangle. It can be a right-angled triangle with the angles of 90°, 40°, and 50°. A scalene triangle can also be an obtuse triangle with angles as 30°, 50°, and 100°. Three interior angles of an acute triangle must be less than 90°.

**Is It Possible for A Right Triangle to Be Acute**?

A right triangle must have an angle that measures exactly 90°. Since the three angles of an acute triangle must be less than 90°, a right triangle cannot be acute.