Hello, and welcome to today's mathematics lesson. Today, we are going to explore shapes and polygons. By the end of this lesson, you will understand what curves and polygons are, how to classify them, and how to calculate the angles inside and outside these fascinating shapes.
Let us begin with something fundamental — curves. In everyday language, when we say "curve," we usually mean something that is not straight. But in mathematics, the word has a broader meaning. A curve is simply any figure drawn on a plane, and this includes straight lines too.
Curves can be divided into different types. First, we have open curves. An open curve is one that does not cut itself — imagine drawing a gentle arc or a wavy line that starts at one point and ends at another, without ever crossing its own path.
Next, we have closed curves. A closed curve is one that closes on itself, forming a complete loop where the starting and ending points coincide. Think of a circle or an oval — these are closed curves because you can trace them completely and return to your starting point.
Now, within closed curves, there is a special category called simple closed curves. A simple closed curve is a closed curve that does not pass through any point more than once. In other words, it does not cross itself at any point. A circle is a perfect example of a simple closed curve.
Here is something important to visualize. When you draw a simple closed curve on a flat surface, it divides that surface into three distinct parts. First, the interior — all the points lying inside the curve. Second, the boundary — the points that actually lie on the curve itself. And third, the exterior — all the points lying outside the curve. When we combine the interior with the boundary, we call this the region of the curve.
Now let us move to the heart of this chapter — polygons.
A polygon is defined as a closed plane figure bounded by straight line segments. But there are two crucial conditions. First, the line segments must intersect only at their endpoints. Second, each endpoint must be shared by exactly two line segments — no more, no less.
Let me clarify with some examples of what is NOT a polygon. If two line segments cross each other at a point that is not an endpoint, the figure is not a polygon. If three or more line segments meet at a single point, that point is shared by more than two segments, so again, it is not a polygon. And obviously, if the figure is not closed, it cannot be a polygon.
The straight line segments that form a polygon are called its sides. The points where two sides meet are called vertices.
Polygons are named according to how many sides they have. A three-sided polygon is a triangle. A four-sided polygon is a quadrilateral. Five sides make a pentagon. Six sides make a hexagon. Eight sides make an octagon. And ten sides make a decagon.
Now, let us classify polygons based on their angles.
A convex polygon is one where every interior angle is less than 180°. Imagine a regular hexagon — all its angles point outward, and none of them bend inward.
A concave polygon, on the other hand, has at least one interior angle that is greater than 180°. This creates a "caved-in" appearance at that vertex. Unless specified otherwise, when we say "polygon," we usually mean a convex polygon.
Here is a useful term — diagonal. A diagonal of a polygon is a line segment joining any two non-consecutive vertices. In a pentagon ABCDE, if you join vertex A to vertex C, skipping vertex B, then AC is a diagonal. Similarly, AD is another diagonal from vertex A.
Now we come to one of the most powerful results in this chapter — the sum of interior angles of a polygon.
Here is a beautiful technique. Pick any single vertex of a polygon, and draw all possible diagonals from that vertex. You will notice that these diagonals divide the polygon into triangles. Count carefully, and you will find that the number of triangles formed is always two less than the number of sides.
So if a polygon has n sides, the number of triangles formed is n − 2.
Since each triangle has angles summing to 180°, we can write: The sum of interior angles of a polygon with n sides equals (n − 2) × 180°.
This can also be written as (2n − 4) × 90°, or (2n − 4) right angles.
Let us verify this with a quadrilateral. Here, n = 4. The sum should be (4 − 2) × 180° = 360°. And indeed, we know that the four angles of any quadrilateral always add up to 360 degrees.
Now for something equally remarkable — the sum of exterior angles.
Imagine walking around the perimeter of a polygon. At each vertex, you turn by some angle to follow the next side. If you walk all the way around and return to your starting point, facing your original direction, you will have made one complete rotation.
This complete rotation equals 360°, or 4 right angles. Therefore, no matter how many sides a polygon has, if you produce all its sides in order and measure the exterior angles formed, their sum will always be 360°.
This is a constant — it does not depend on the number of sides. A triangle, a hexagon, a decagon — all have exterior angles summing to exactly 360°.
Let us apply these formulas with a worked example.
Suppose the sides of a pentagon are produced in order, and the exterior angles are x°, 2x°, 3x°, 4x°, and 5x°. We need to find each exterior angle.
Since the sum of exterior angles is always 360 degrees, we write: x° + 2x° + 3x° + 4x° + 5x° = 360°. This simplifies to 15x° = 360°, so x = 24.
Therefore, the five exterior angles are 24°, 48°, 72°, 96°, and 120°.
Here is another example. A seven-sided polygon has one angle of 114°, and the other six angles are all equal to x°. Find x.
First, the sum of interior angles of a heptagon equals (2 × 7 − 4) × 90° = 900°.
The sum of all seven angles is 114° + 6x°. Setting this equal to 900 degrees, we get 6x° = 786°, so x = 131. Each of the six equal angles measures 131°.
Finally, let us discuss regular polygons — the most symmetric and elegant of all polygons.
A regular polygon satisfies three conditions: all its interior angles are equal, all its sides are equal, and consequently, all its exterior angles are equal.
For a regular polygon with n sides, we have these important formulas.
The sum of interior angles is (2n − 4) × 90°. Therefore, each interior angle equals (2n − 4) × 90° / n.
The sum of exterior angles is always 360 degrees. Therefore, each exterior angle equals 360° / n.
From this, we can derive a very useful result: the number of sides equals 360° divided by the exterior angle.
Also, remember that at every vertex of any polygon, regular or not, the exterior angle plus the interior angle equals 180 degrees, because they form a straight line.
Let us work through an example. If each interior angle of a regular polygon is 144°, how many sides does it have?
Using the relationship between interior and exterior angles: exterior angle equals 180° − 144° = 36°. Then, number of sides equals 360° / 36° = 10. This is a regular decagon.
Here is a final example to challenge your thinking. The sum of interior angles of a regular polygon equals six times the sum of its exterior angles. Find the number of sides.
We know the sum of exterior angles is always 360 degrees. Six times this is 2160 degrees. Setting the sum of interior angles equal to 2160 degrees: (2n − 4) × 90° = 2160°. Dividing both sides by 90: 2n − 4 = 24, so 2n = 28, and n = 14. This regular polygon has 14 sides.
Let us now recap the key takeaways from today's lesson.
First, in mathematics, a curve includes both straight and non-straight lines. Simple closed curves divide the plane into interior, boundary, and exterior regions.
Second, a polygon is a closed figure made of straight line segments that meet only at endpoints, with exactly two segments meeting at each endpoint.
Third, the sum of interior angles of a polygon with n sides is (n − 2) × 180°, or (2n − 4) right angles.
Fourth, the sum of exterior angles of any polygon, produced in order, is always 360° or 4 right angles, regardless of the number of sides.
Fifth, a regular polygon has equal sides, equal interior angles, and equal exterior angles. The number of sides equals 360° divided by the exterior angle.
And sixth, at each vertex of any polygon, the interior angle plus the exterior angle equals 180°.
That brings us to the end of today's lesson on Understanding Shapes and Polygons. I hope you can now see the elegant patterns that govern these geometric figures. Remember, every polygon tells a story through its angles — the interior angles reveal its complexity, while the exterior angles always complete the full circle of 360°. Keep exploring, keep questioning, and enjoy the beauty of mathematics. Until next time, goodbye and happy learning!