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**Height|Definition & Meaning**

**Definition**

**Height **of a geometrical shape represents the vertical distance from the base to its opposite vertex. Since the distance must be measured vertically, the line joining the base and the vertex must be **perpendicular** (angle of 90$^\circ$).

Height is usually defined for polygons only. Polygons are geometrical shapes that have 3 or more sides. A triangle is the simplest possible polygon.

**Height of Triangle **

There are different types of triangles, each of which have different procedures of finding height.

**Height of Right Angle Triangle**

A **right-angle** triangle is a special triangle where one of the **interior angles** is **90**$^\circ$. The three sides of the right angle triangles are Base, Perpendicular, and Hypotenuse as shown in the figure below.

It can be seen that the perpendicular represents the distance from the base of the triangle to its vertex. Hence, in the case of a right-angle triangle, the height is given by **perpendicular**. The lengths of the right angle triangle’s three sides are related by the **Pythagoras Theorem** given by:

Hypotenuse^{2} = Base^{2} + Perpendicular^{2}

Hence, by knowing the length of any **two sides** we can compute the length of the third **unknown** **side** through the use of Pythagoras’ theorem.

**Height of Other Triangles**

In triangles with no right angle, the height is computed by **joining** the **base** and the **opposite vertex** through a **perpendicular**. This perpendicular then represents the height of that triangle.

In the following triangle ABC, **BC** is the **base** of the triangle. The height is found by joining the base BC to the opposite vertex A through a **perpendicular** AD. Thus, **AD** is the **height** of the triangle.

The **area** of a triangle **directly** depends on its **height:**

Area = (1/2) * b * h

Where** ‘****b****’** and **‘h’** respectively represent the length of the base and the height of the triangle.

**Height of Parallelogram**

A **parallelogram** is a **four-sided** polygon consisting of a **pair** of **parallel sides** equal in length. There are three types of parallelograms: **Square**, **Rectangle**, and **Rombous**.

**Height of Square**

A** **parallelogram in which **all four sides** and **angles** are **equal **is called a **square**. All the angles are **right angles**. Since all the sides are equal, the **height** of square is the same as **length** of any side.

**Height of Rectangle**

A parallelogram in which both **pairs** of opposite **sides** are **equal,** and **all angles** are **equal, is a rectangle**. All the angles are **right angles**. The dimensions of the rectangle are represented by its width and height. The **vertical dimension** is the **height,** and the **horizontal **dimension is the **length**.

**Height of Rhombus**

**A** type of parallelogram in which all **four sides** and a **pair** of **opposite angles** are **equal **is called a** rhombus**. In the case of Rhombus, height is not directly given and is computed by joining the **base** to the **opposite vertex** through a **perpendicular**.

**Height of Trapezium**

A **trapezium** is a four-sided polygon with a pair of parallel sides. The bases in this context are the **parallel sides**. The** height** of the trapezium is the **vertical distance** between the two **bases** and is obtained by joining them through a **perpendicular line**.

In the trapezium ABCD shown below, the two parallel sides are AD and BC. These sides also form the bases of the trapezium. The perpendicular distance between these bases represents the height of the trapezium and is shown by AE in the figure.

Height of the trapezium is required to compute its area:

**Area **= 1/2 * (Sum of parallel sides) * height

**A Few Examples of Calculating Heights of Polygons**

**Example 1**

A triangle ABC with two sides length is shown. Compute the height of the triangle.

**Solution**

In the given triangle ABC, note that there is a 90-degree angle. This means that the triangle is a **right-angle** triangle. As discussed earlier, the height of right angle triangle is given by the perpendicular of the triangle. In the given triangle, two sides’ lengths are given. The given sides’ length are:

Base = 12 and Hypotenuse = 13

Using the **Pythagoras **theorem:

13^{2} = 12^{2} + Perpendicular^{2}

Computing the squares gives:

169 = 144 + Perpendicular^{2}

Solving the equation for perpendicular produces:

Perpendicular = $\mathsf{\sqrt{25}}$

And thus, the length of the perpendicular is:

Perpendicular = 5 units

Hence, the **height **of the given triangle is **5 units**.

**Example 2**

A **trapezium **has an **area **of **35 square units**. The two **parallel sides **are **6** and **8** units. Calculate the **height **of the trapezium.

**Solution**

Note that the two parallel sides of the trapezium are called its bases. Thus, the lengths of the bases of the trapezium are given as 6 and 8 units. Recall that the **area **of the trapezium is given by:

Area = (1/2) * (Sum of parallel sides) * height

**Substitute **the given value of **area **and **lengths **of parallel sides:

35 = (1/2) * (6 + 8) * height

35 = 7 * height

**Solve **the equation to compute the height:

Height = 5 units

Hence, the **height **of the given trapezium is **5 units**.

**Example 3**

An **equilateral triangle **has a base of length 6 units. Calculate the **height **of the triangle.

**Solution**

Consider an **equilateral **triangle ABC as shown in the figure. The height of the triangle will be the perpendicular distance from base BC to vertex A. Thus, **AD **represents the **height **of the triangle.

Note that the lengths of all sides of an equilateral triangle are equal. This means that the length of **AC **is also **6 units**. Also, note that **AD** is a **perpendicular bisector** to the base BC. Thus, the length of **DC **is **3 units** – half of the length of BC.

Now consider the right angle triangle ADC. The length of two sides AC and DC is known and the length of the third side AD is to be found which represents the perpendicular of the triangle.

Using **Pythagoras’ theorem**, the equation will be:

6^{2} = 3^{2} + Perpendicular^{2}

Solving the equation results in:

Perpendicular = 3$\mathsf{\sqrt{3}}$

Thus, the **height **of the **equilateral **triangle is $\mathsf{3\sqrt{3}}$ units.

*All images/mathematical drawings were created with GeoGebra.*