How to Calculate Polygon Curve Angles Easily in Geometry
What is Curve in Maths?
Geometry is a study of shapes. It is broadly classified into two types: plane geometry and solid geometry. Plane geometry deals with one dimensional and two-dimensional figures like curves, lines, polygons, square, circle, rectangle, triangle and many more. Whereas Solid geometry deals with the study of three- dimensional shapes like cube, cuboid, cylinder, cone, sphere, and many more.
Here in this article we will be studying plain geometry concepts: what is curve in maths, polygonal curve, angles and polygons.
A curve is a line which is not straight; it bends and changes its direction at least once.
A curve is a continuous and smooth flowing line that you can draw on a paper without using a ruler, without any sharp turns. Curves bends and changes their direction at least once.
Examples of Curves Around us
A race track is an example of a curve.
Roads on hills and mountains are curvy.
Different Types of the Curve are as Follows:
1. Upward Curve: A curve that faces in the upward direction is called an upward curve. It is also known as a concave upward. Concave Upward also called or “Convex Downward”
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2. Downward Curve: A curve that faces in the downward direction is called a downward curve. It is also known as a concave downward. Concave Downward also called or “Convex Upward”.
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3. Open Curve Shape: An open curve shape is a curve with two different end points which does not enclose any area within itself. Examples of some of the open curve shape are given in the figure below.
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4. Closed Curve: A closed curve is a curve whose endpoints meet each other i.e. it has no end points and encloses an area (or a region).In other words a closed curve is formed by joining the end points of an open curve together. A closed curve can form a well defined geometric shape. For example circles , ellipses are formed from closed curves. Below figure displays some of the shapes that have curves, closed curves.
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5. Simple Curve: A simple curve is a curve that changes direction but does not intersects itself at any point while changing direction. A simple curve can be an open curve and closed curve.
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6. Non-simple Curves: A curve that crosses its own path while changing the direction is called a non-simple curve.
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Angle
The intersection of two lines having a common vertex form an angle. Parts of an angle are described as vertex, arms, interior, and exterior of angles. When two lines intersect at a point they form four angles at the point of intersection. An angle is denoted by the symbol ∠.
From the figure, ∠ ABC is an angle. B is the point of intersection of the two arms AB and BC called the vertex and AB and BC are the sides of the angle. Angles are commonly measured in terms of degree.
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Angles are Classified According to their Sizes as Follows
Acute Angle
The measure between the two arms is less than 900 then an acute angle is formed.
Obtuse Angle
When the measure between the two arms is above 900 it forms an obtuse angle.
Right Angle
When the two arms make an angle of 900 it forms a right angle.
Reflex Angle
When the two arms of an angle, makes an angle more than 1800 and less than 3600 it is called a reflex angle.
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Polygon
A polygon is a simple closed curve. A polygon is a polygonal curve that is the union of three or more line segments whose endpoints meet. A two-dimensional closed figure bounded with three or more than three straight lines is called a polygon. Triangles, square, rectangle, pentagon, hexagon, are some examples of polygons.
The segments are referred to as the sides of the polygon. The points at which the segments meet are called vertices. Segments that share a vertex are called adjacent sides. A segment whose endpoints are nonadjacent vertices is called a diagonal.See the picture below.
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The below figures show some of the examples of polygons or polygonal curves( a closed curve that is not a polygon).
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In figure 1 you can see that all the shapes are polygons, as all the shapes are drawn joining the straight lines only. There are no curved lines. Even the shape in orange color is a polygon because it is formed by joining straight lines and it is a closed figure.
But in figure 2 the shape is not a polygon because it is not fully connected and also has curved lines and such shapes that have curves are called a closed curve that is not a polygon.Circle is the best example of a closed curve that is not a polygon.
The name of the polygon itself implies the number of segments in it. For instance, the triangle is a polygon having three sides, a quadrilateral is a polygon having four sides, etc.
Based on the polygon sides and angles polygons are classified into different types of polygons.
Types of Polygons
Different types of polygons are
Regular Polygon
Irregular Polygon
Convex Polygon
Concave polygon
Solved Examples
1. Identify the type of curves
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Answer:A closed curve that is not a polygon.
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Answer: A open curve shape
FAQs on Polygon Curve Angle: Meaning, Types & Solved Examples
1. What is a curve in mathematics, and how is it different from a straight line?
In geometry, a curve is a continuous line that is not straight. You can think of it as any shape you can draw without lifting your pen from the paper. The key difference is that a straight line maintains a single direction, whereas a curve continuously changes direction. Therefore, a polygon, which is made of straight line segments, is considered a type of simple closed curve.
2. What is the difference between a simple curve and a complex curve?
The primary difference lies in whether the curve intersects itself.
- A simple curve is a curve that does not cross its own path. A circle or the outline of a triangle are good examples.
- A complex curve, or a curve that is not simple, is one that crosses over itself at one or more points. The figure '8' is a classic example of a complex (non-simple) closed curve.
3. How is a polygon different from a general curve?
A polygon is a very specific kind of curve. While a general curve can be smooth and flowing, a polygon is a simple closed curve that is made up exclusively of straight line segments that meet at endpoints called vertices. For instance, a square is a polygon, but a circle is a curve that is not a polygon because its boundary is not made of straight lines.
4. What are the main classifications of polygons?
Polygons are chiefly classified based on their sides and angles:
- Convex vs. Concave: A convex polygon has all interior angles measuring less than 180°, with all vertices pointing outwards. A concave polygon has at least one interior angle greater than 180°, which creates an inward-pointing vertex or a 'dent' in the shape.
- Regular vs. Irregular: A regular polygon has all sides of equal length and all angles of equal measure (equilateral and equiangular). A square is a regular polygon. An irregular polygon does not have all equal sides and angles.
5. How do you calculate the sum of the interior angles of any convex polygon?
You can find the sum of the interior angles for any convex polygon using the formula: Sum of angles = (n - 2) × 180°. In this formula, 'n' represents the number of sides the polygon has. For example, for a hexagon (which has 6 sides), the sum of its interior angles is (6 - 2) × 180° = 4 × 180° = 720°.
6. Why is the sum of the exterior angles of any convex polygon always 360°?
This is because walking along the perimeter of any convex polygon completes one full rotation. Imagine starting at a vertex and walking along each side. The turn you make at each corner corresponds to the exterior angle. By the time you return to your starting point, you have turned a full circle, which is 360°. This principle is true for a triangle, a pentagon, or any other convex polygon.
7. Is it true that a polygon's total angle sum is always 180° or 360°?
This is a common misconception. Only a triangle (a 3-sided polygon) has an interior angle sum of 180°. A quadrilateral's interior angles sum to 360°, but for polygons with more sides, the sum increases. The correct way to find the sum is with the formula (n - 2) × 180°. For example, a pentagon's (5 sides) angles sum to 540°. The only sum that is always 360° is the sum of the exterior angles of a convex polygon.
8. Can a polygon have curved sides?
No, a polygon cannot have curved sides. By its strict mathematical definition, a polygon is a two-dimensional closed figure formed by a finite number of straight line segments. If any part of a shape's boundary is curved, it is not classified as a polygon. For instance, a shape like a semi-circle is not a polygon because it has a curved edge.
