Exploring Plane Figures: Types, Properties, and Examples

A plane figure is a simple but important concept in geometry. Plane figures refer to shapes that are situated on a level, two-dimensional surface, like squares, circles, and triangles. These geometric entities carry substantial mathematical significance and find real-world applications in fields such as architecture, engineering, and design.

This article aims to cover the basic concept of plane figures, exploring their properties and common types. We'll even solve some examples to help you grasp the concept better.

What is a Plane Figure?

A plane figure is a flat 2-dimensional shape that lies entirely within a single plane, such as a sheet of paper. These figures are characterized by having length and width but no depth.

Properties of Plane Figures

Plane figures have many different properties that can be used to classify them and solve problems involving them. Some of the most important properties of plane figures include:

• Sides: Plane figures may possess any number of sides ranging from three to infinite. Polygons have three or more sides, while circles have no sides.
• Angles: Plane figures can have any number of angles from zero to infinity. Polygons have at least three angles, while circles have no angles.
• Perimeter: The perimeter of a plane figure is the total length of all of its sides.
• Area: The area of a plane figure is the amount of space that it encloses.
• Symmetry: Some plane figures have symmetry, which means that they can be divided into two or more parts that are mirror images of each other.

Classification of Plane Figures

Closed shapes and open shapes are two categories used to classify plane figures based on whether their boundaries are connected or not. Here is a simple explanation of each:

Closed Shapes

1. Circle: A perfectly round closed shape with a continuous boundary.
2. Square: A geometric shape featuring four sides of equal measure and all internal angles as right angles.
3. Triangle: A closed figure comprised of three straight sides and three interior angles.
4. Rectangle: A four-sided quadrilateral with all angles measuring 90 degrees, where the lengths of opposite sides are equal.
5. Pentagon: A closed figure composed of five straight sides and five internal angles.
6. Hexagon: A six-sided polygon with six angles.
7. Octagon: An eight-sided polygon with eight angles.
8. Ellipse: A shape that appears similar to a stretched-out circular curve.
9. Rhombus: A quadrilateral with all sides equal in length but not necessarily right angles.
10. Regular Polygon: A geometric shape characterized by equalizing its side lengths and angles.

Open Shapes

1. Line Segment: A portion of a line that connects two specific points.
2. Ray: A line segment that begins at a single point and continues indefinitely in one direction along a straight path.
3. Arc: A curved segment of a circle's circumference, not forming a closed loop.
4. Curves: Various curves like the letter C or S have disconnected endpoints.
5. Semicircle: Half of a circle's circumference, forming a curved open shape.
6. Polyline: A series of connected line segments, where the endpoints of adjacent segments meet but do not form a closed shape.
7. L-shape: An open shape that resembles the letter L, formed by two perpendicular line segments meeting at a right angle.
8. Zigzag Line: A line that changes direction multiple times, creating a series of connected line segments with distinct endpoints.

Area and perimeter

Below are formulas to calculate the area and perimeter of different shapes.

Circle:

• Perimeter (Circumference): C = 2πr (where r is the radius).
• Area: A = πr².

Square:

• Perimeter: P = 4S (where S is the length of one side).
• Area: A = S².

Triangle:

• Perimeter: Sum of the lengths of all three sides.
• Area: A = ½ × base × height.

Rectangle:

• Perimeter: P = 2L + 2W (where L is the length and W is the width).
• Area: A = L × W.

Rhombus:

• Perimeter: P = 4S (where S is the length of one side).
• Area: A = (½) × d₁ × dâ‚‚, where d₁ and dâ‚‚ are the lengths of the diagonals.

Regular Polygon:

• Perimeter: P = n S (where n is the number of sides and S is the length of one side).
• Area: A = (1/2) × n × S × a (where a is the apothem, the distance from the center to the midpoint of a side).

Examples of plane Shapes

Example 1:

Calculate the perimeter and area of a square with a side length of 6 cm.

Solution:

• Perimeter (P) = 4 × Side Length = 4 × 6 cm = 24 cm
• Area (A) = Side Length × Side Length = 6 cm × 6 cm = 36 cm²

Example 2:

Find the circumference and area of a circle with a radius of 5 meters (use π ≈ 3.14).

Solution:

• Circumference (C) = 2πr = 2 × 3.14 × 5 m ≈ 31.4 m
• Area (A) = πr² = 3.14 × 5 m × 5 m ≈ 78.5 m²

Example 3:

Determine the perimeter and area of a rectangle with a length of 8 inches and a width of 4 inches.

Solution:

• Perimeter (P) = 2 × (Length + Width) = 2 × (8 in + 4 in) = 24 in
• Area (A) = Length × Width = 8 in × 4 in = 32 in²