Pentagon Shape

In geometry, we study different shapes, and in this article we’ll cover the five-sided polygon, which is called a pentagon.

A polygon is a closed, plane figure made up of at least 3 segments that do not cross.

Examples of polygons include triangle (a three-sided polygon), quadrilateral (a four-sided polygon), pentagon (a five-sided polygon), hexagon (a six-sided polygon), heptagon (a seven-sided polygon), and so on.

Definition of a Pentagon

A pentagon shape is a closed and flat two-dimensional surfaced shape with five angles and five sides.    The word pentagon itself tells you what it is. ‘Penta’ is the Greek word for five and ‘gonia’ is the Greek word for angle.

Pentagon Examples

Madison drew the shapes below. Can you answer the following questions correctly?

  • Which two shapes have the same number of sides?
  • How many sides does each shape have?
  • How many angles does each shape have?
  • Can you name each of these shapes?

Different Types of Pentagon Shapes

Pentagons are classified by certain properties their sides and their angles or their vertices possess.

There are different types of pentagons, namely:

  • Regular or Irregular pentagons
  • Concave or Convex pentagons

Regular or Irregular Pentagons

Types of Pentagons Classified by Sides and angles
  Definition     Name
All five sides of equal lengths, and all the five angles of equal measures.   Regular
The sides aren’t of equal lengths and angles aren’t of equal measures.   Irregular    

In turn, a regular pentagon is a regular polygon with five sides and an irregular pentagon is an irregular polygon with five sides.

Concave and Convex Pentagons

A concave pentagon is the one that has at least one vertex pointing inwards.

A convex pentagon is the one that has all its vertices pointing outwards.

The interior angles of a convex pentagon are always less than 180°.

In turn, a concave pentagon has at least one of its interior angles is greater than 180°.

In a convex pentagon, all diagonals are totally in the interior of the pentagon.

In a concave pentagon, not all diagonals will occur in the interior of the pentagon, as illustrated the example below.

Properties of a Pentagon

To find the sum of the interior angles of a pentagon, divide it up into triangles.

There are three triangles. The sum of the angles of each triangle is 180˚. We get 180 x 3 = 540 . Therefore, the sum of the interior angles of a pentagon is 540 degrees.

Also we can find the measure of the interior angle of a regular pentagon.

We know that, a regular pentagon have all sides are the same length and all interior angles are the same measures.

So, the sum of all the angles is 540 degrees (from above). And there are five angles. We can find the measure of the interior angles.

Therefore, the measure of the interior angle of a regular pentagon is degrees.

To find the measure of the central angle of a regular pentagon, make a circle in the middle. A circle is 360 degrees around. Divide that by five angles. So, the measure of the central angle of a regular pentagon is 360/5 = 72 degrees.

A pentagon has quite a few interesting properties, which are as follows:

All pentagons have five sides, but the sides do not have to be of equal length.

A pentagon can be divided into three triangles.

The sum of the interior angles of a pentagon = 540 degrees.

A regular pentagon has five equal sides and five equal angles.

Each interior angle of a regular pentagon = 108 degrees.

Each exterior angle of a regular pentagon = 72 degrees.

A regular pentagon has five axes of symmetry. Each one of them passes through a vertex of the pentagon and the middle of the opposite edge.

All diagonals of a regular pentagon are equal to each other. Together they form a pentagonal star, also called a pentagram.

The Perimeter of a Pentagon

The perimeter of a closed planar figure is the length of its boundary or outer edge. The perimeter of the adjacent pentagon is:

38 + 29 + 34 + 42 + 22 = 165 mm

As all the sides of a regular pentagon are equal in length, the formula for the perimeter or circumference of a pentagon is as follows:

P = 5a \;units

where ‘a’ is the length of the side of the pentagon.

Area of a Pentagon

The area of a closed planar figure is the space it covers.

Area is measured in units called square units such as square meters (m2), square centimeters (cm2), and square millimeters (mm2).

For instant, if the dimensions of the figure are given in meters then its area is measured in square meters (m2).

In general, whatever the units used for the dimensions of the figure, its area is measured in square units.

The area of a regular pentagon can be calculated by dividing it into five equal triangles. 

We already know that the area of a triangle is given by the formula:

A = \dfrac{1}{2} \times Base \times Height = \dfrac{1}{2}ah_{a}

So, we can find the area of a regular pentagon as: 5×Area of triangle.

A = 5 \times \dfrac{1}{2} \times Base \times Height = \dfrac{5}{2}ah_{a}

The formula to find the area of a regular pentagon with a given side s and apothem length (the line from the centre of the pentagon to a side, which intersects the side at a right angle, i.e., at 90 degrees) is as follows:

A = \dfrac{5}{2}sa

Some geometrical plane figures could be dissected into more familiar shapes such as triangles, rectangles, trapeziums.

The dissection will help us to calculate the area of the original shape.

In order to find the area of the composite figure, it is enough to find the area of the elementary shape then add the total area of all elementary shapes.

Example

Given ABCDE is a regular pentagon with a side length of 10 cm:

  1. Show that EBCD is an isosceles trapezoid.
  2. Find the area of Δ AEB.
  3. Find the height of the trapezoid EBCD.
  4. Find the area of the trapezoid EBCD.
  5. Deduce the area of the pentagon.

Let’s work through the problem and find the solutions to these questions.

  1. ΔEAB is an isosceles triangle. The measure of the interior angle of a regular pentagon is 108 degrees.

    This means that m∠A must equal 108º. We can now use this to show whether EBCD is an isosceles trapezoid:

    m∠1 = \dfrac{180-108}{2} = 36º
    m∠2 = 108 - 36 - 72º

    m∠3 is the same angle as m∠2, 72º.

    So, because EB and DC are parallel, ED and BC are not parallel, and the angles for ED and BC are the same, we know that EBCD is an isosceles trapezoid.
  2. To find the area of Δ AEB, we use the lengths of the two sides and the sine of the included angle. The length of the side (s) is 10cm and the angle is 108º. We can use the formula to calculate the area:

    Δ AEB = \dfrac{1}{2} s \times s \times sin(108)
    Δ AEB = \dfrac{1}{2} 10 \times 10 \times 0.9511 \approx 47.6cm^{2}
  3. To find the height of the trapezoid EBCD, we can look at ΔEHD. It’s a right triangle, with an angle of 72º as previously calculated.

    The sine of an acute angle is defined in the context of a right triangle: for the specified angle, it is the ratio of the length of the side that is opposite that angle, to the length of the longest side of the triangle (the hypotenuse). The formula for this:

    sin(72) = \dfrac{b}{s} = \dfrac{b}{10}

    We can now use algebra to solve the equation:

    b = 10 \times sin(72) = 10 \times 0.9511 \approx 9.5cm
  4. To find the area of the trapezoid EBCD, we multiply the sum of the bases (the parallel sides) by the height (the perpendicular distance between the bases), and then divide by 2.

    We can use the cosine rule formula to calculate the base:

    b_{1}^{2} = 10^2 + 10^2 - 2 \times 10 \times 10 \times cos(108)
    b_{1}^{2} = 100 + 100 - 200 \times 10 \times -0.309 \approx 261.8
    b_1 \approx \sqrt{261.8} \approx 16.2
    EBCD \approx \dfrac{1}{2} \times 9.5 (16.2 + 10) \approx 124.5cm^2
  5. Finally, we can calculate the area of the entire pentagon:

    A \approx 47.6 +124.5 = 172.1cm^2

Hopefully this article helps you to understand the pentagon shape in more detail, and you can use the examples to calculate the area and perimeters of your own pentagons.