To calculate result you have to disable your ad blocker first.

×

Triangles are one of the simplest and most versatile shapes in geometry, forming the basis for many other geometric concepts. A triangle is a fundamental geometric shape composed of three-line segments that intersect at three distinct points.

This article aims to cover the definition, types, properties, formulas for calculating area & perimeter, and examples of triangles.

A **triangle **is a closed two-dimensional shape with three straight sides and three interior angles. The sum of the interior angles of a triangle always equals **180 degrees**. The points of a triangle are called **vertices**, while the line segments are the **sides **of the triangle.

Tringles are classified based on sides and angles:

- Equilateral Triangle.
- Isosceles Triangle.
- Scalene Triangle.

The lengths of the three sides are equal in an equilateral triangle. Each of its angles is **60 degrees**. It is a special case of an acute triangle (all angles less than **90 degrees**).

An isosceles triangle has two sides of equal length and one side of a different length. The angles opposite the equal sides are also equal in measure. The angle opposite the longer side is typically larger than the other two angles.

A** scalene triangle** has all three sides of different lengths. Since its sides are unequal; its angles are also unequal. It can be an acute; obtuse; or right triangle; depending on the measures of its angles.

Triangles can also be classified based on the measurements of their angles.

- Acute Triangle:
- Obtuse Triangle:
- Right Triangle:

An acute triangle has all three angles less than **90 degrees**. In other words, all its angles are “acute” (smaller than a right angle).

An **obtuse triangle** has one angle greater than 90 degrees; known as an “obtuse” angle. The other two angles are acute (smaller than **90 degrees**).

A **right triangle** has one angle exactly equal to **90 **degrees; which is known as a “right” angle. The sum of the other two angles is also **90 **degrees.

These two classifications can be combined to describe triangles more precisely:

**Equilateral Acute Triangle:**

All sides are equal in length, and all angles are < 90 degrees.

**Isosceles Obtuse Triangle:**

Two sides are equal in length; one angle is > 90 degrees.

**Scalene Right Triangle:**

All sides are of different lengths; one angle is exactly 90 degrees.

Triangles possess a range of properties. Some of the key properties include:

**Angle Sum Property:**The sum of the three interior angles of a triangle always adds up to**180**degrees.**Exterior Angle Property:**The exterior angle of a triangle is equal to the sum of its two non-adjacent interior angles.**Pythagorean Property:**In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.

The perimeter of a triangle is the sum of the lengths of all its sides.

`P = a + b +c`

Here are a couple of common methods to calculate the area of a Triangle:

**Using Base and Height:**

If you know the length of the base (b) of the triangle and its corresponding height (h) – the perpendicular distance from the base to the opposite vertex, then the area (A) can be calculated using the formula:

**Area = 1/2 × Base × Height**

The perimeter and area of a triangle can be solved by using our triangle calculator to get results in a fraction of a second.

If you know the lengths of all three sides of the triangle you can use Heron's formula to calculate the area. Let “s” be the semi-perimeter of the triangle (half of the perimeter):

`S = a + b + c / 2`

So, the area can be calculated using Heron's formula:

`A = [s×(s−a) ×(s−b) ×(s−c)]`

^{1 / 2}

Let’s solve some examples to gain more understanding of the triangle to solidify our understanding.

**Example 1: Perimeter Calculation**

Find the perimeter of a triangle with the following side lengths:

- Side a = 12 units
- Side b = 15 units
- Side c = 18 units

**Solution:**

To find the perimeter of a triangle with the given side lengths, simply add up the lengths of all three sides:

Perimeter (P) = Side a + Side b + Side c

Perimeter (P) = 12 units + 15 units + 18 units

`Perimeter (P) = 45 units`

Therefore, the perimeter of the triangle is **45 **units.

**Example 2: Area Calculation using Base and Height**

Given a triangle with

base (b) = 16 units

height (h) = 9 units,

Calculate its area.

**Solution:**

To calculate the area of the triangle using the given base and height, you can use the formula:

**Given:**

base (b) = 16 units

height (h) = 9 units

**Formula:**

Area (A) = 0.5 × base × height

**= **0.5 × 16 × 9 = 72 square units.

**Example 3: Heron's Formula for Area**

A triangle has side lengths:

- Side a = 7 units
- Side b = 9 units
- Side c = 12 units

Calculate the area of the triangle using Heron's formula.

**Solution:**

All three sides are available so we use Heron's formula.

**Step 1: **Find the perimeter

S = a + b + c / 2

S = (7 + 9 + 12) / 2

`S = 28 / 2 s = 14 units`

**Step 2**: Plug the values into Heron's Formula

Area = √ (s × (s - a) × (s - b) × (s - c))

Area = √ (14 × (14 - 7) × (14 - 9) × (14 - 12))

Area = √ (14 × 7 × 5 × 2)

`Area = √ (980) Area ≈ 31.30495 square units`

Therefore, the area of the triangle is approximately **31.30495 **square units.

ADVERTISEMENT

AdBlocker Detected!

To calculate result you have to disable your ad blocker first.