Split Any Shape Into Triangles and Its Angles Give Themselves Away
Learn to name polygons by their sides, count their diagonals, work out any interior angle sum, and name every part of a circle from chord to sector.
What is a polygon, and how are polygons named?
A polygon is a simple closed figure made only of line segments. It is named by how many sides it has, and because sides and vertices always come in equal numbers, counting either one tells you the name.
This page covers everything in the ICSE Class 6 Mathematics chapter on polygons and the circle: naming and classifying polygons, counting diagonals, finding interior angle sums, and identifying every named part of a circle.
This page covers everything in the ICSE Class 6 Mathematics chapter on polygons and the circle: naming and classifying polygons, counting diagonals, finding interior angle sums, and identifying every named part of a circle.
How do you classify polygons by sides and by shape?
By number of sides: triangle 3, quadrilateral 4, pentagon 5, hexagon 6, heptagon 7, octagon 8, nonagon 9 and decagon 10.
A polygon is regular when all its sides are equal and all its angles are equal — an equilateral triangle and a square are regular. If either condition fails it is irregular, so a rectangle is irregular despite looking neat, because its sides are not all equal.
A polygon is convex when every interior angle is less than , so no vertex is pushed inwards. It is concave when at least one interior angle is reflex, giving a dented outline.
For example, the cross-section of a pencil is usually a regular hexagon, and a rangoli outline with inward points is concave.
The misconception to clear up: equal sides alone do not make a polygon regular. A rhombus has four equal sides but unequal angles, so it is irregular.
A polygon is regular when all its sides are equal and all its angles are equal — an equilateral triangle and a square are regular. If either condition fails it is irregular, so a rectangle is irregular despite looking neat, because its sides are not all equal.
A polygon is convex when every interior angle is less than , so no vertex is pushed inwards. It is concave when at least one interior angle is reflex, giving a dented outline.
For example, the cross-section of a pencil is usually a regular hexagon, and a rangoli outline with inward points is concave.
The misconception to clear up: equal sides alone do not make a polygon regular. A rhombus has four equal sides but unequal angles, so it is irregular.
Formula
How many diagonals does a polygon have?
A diagonal joins two vertices that are not already joined by a side. From each vertex you can reach every other vertex except itself and its two neighbours, which gives the formula:
Here is the number of sides, and the division by 2 prevents counting each diagonal twice, once from each end.
Worked example for a hexagon, :
And for a quadrilateral, :
Check the boundary case: a triangle has , giving . That is correct — in a triangle every pair of vertices is already joined by a side, so no diagonals exist.
Here is the number of sides, and the division by 2 prevents counting each diagonal twice, once from each end.
Worked example for a hexagon, :
And for a quadrilateral, :
Check the boundary case: a triangle has , giving . That is correct — in a triangle every pair of vertices is already joined by a side, so no diagonals exist.
Formula
How do you find the sum of a polygon's interior angles?
Pick one vertex and draw every diagonal from it. The polygon splits into triangles, and each triangle contributes . A polygon with sides splits into triangles, so
Worked example for a pentagon, :
For an octagon, :
For a regular polygon you can go further and find each angle, since they are all equal. Each interior angle of a regular pentagon is
That last step works only for regular polygons. In an irregular pentagon the five angles still total , but dividing by 5 tells you nothing about any individual angle.
Worked example for a pentagon, :
For an octagon, :
For a regular polygon you can go further and find each angle, since they are all equal. Each interior angle of a regular pentagon is
That last step works only for regular polygons. In an irregular pentagon the five angles still total , but dividing by 5 tells you nothing about any individual angle.
What are the parts of a circle?
A circle is the set of all points at a fixed distance from a fixed point, and each of its parts has a precise name.
The centre is that fixed point. The radius is a segment from the centre to the circle. The diameter passes through the centre with both ends on the circle, making it the longest chord, and
A chord joins any two points on the circle. An arc is a part of the circle itself. A semicircle is half the circle, cut off by a diameter. A segment is the region between a chord and its arc, while a sector is the region between two radii and their arc.
To draw a circle, set the compasses to the radius and turn them about the fixed centre. Given a diameter of 8 cm, set the compasses to 4 cm.
Concentric circles share a centre but have different radii. The interior is everything closer to the centre than the radius; the exterior is everything further away.
The pair students confuse is segment and sector: a segment is cut by a straight chord, a sector by two radii — so a slice of cake is a sector.
The centre is that fixed point. The radius is a segment from the centre to the circle. The diameter passes through the centre with both ends on the circle, making it the longest chord, and
A chord joins any two points on the circle. An arc is a part of the circle itself. A semicircle is half the circle, cut off by a diameter. A segment is the region between a chord and its arc, while a sector is the region between two radii and their arc.
To draw a circle, set the compasses to the radius and turn them about the fixed centre. Given a diameter of 8 cm, set the compasses to 4 cm.
Concentric circles share a centre but have different radii. The interior is everything closer to the centre than the radius; the exterior is everything further away.
The pair students confuse is segment and sector: a segment is cut by a straight chord, a sector by two radii — so a slice of cake is a sector.
Exam tip
The mistake most students make with regular polygons
Asked for "each interior angle", students apply to any polygon. That division is valid only when the angles are known to be equal — that is, for a regular polygon.
Read the question for the word regular, or for markings showing equal sides and angles. Without it, you can state the total but not the individual angles.
The same care applies in reverse. If a question gives you one interior angle of a regular polygon and asks for the number of sides, set equal to that angle and solve for — do not guess from the shape's appearance.
Read the question for the word regular, or for markings showing equal sides and angles. Without it, you can state the total but not the individual angles.
The same care applies in reverse. If a question gives you one interior angle of a regular polygon and asks for the number of sides, set equal to that angle and solve for — do not guess from the shape's appearance.
Did you know
Why is the diameter the longest chord in a circle?
Every chord joins two points on the circle, and the greatest possible separation between two such points is achieved by passing straight through the centre.
Any chord that misses the centre takes a shorter route across, so it must be shorter. That is why the diameter is both a chord and the maximum one — and why it always measures exactly twice the radius.
Any chord that misses the centre takes a shorter route across, so it must be shorter. That is why the diameter is both a chord and the maximum one — and why it always measures exactly twice the radius.
Key takeaways
Polygons and circles in 30 seconds
- Polygons are simple closed figures of line segments, named by their number of sides from triangle to decagon.
- Regular means equal sides and equal angles; convex means every interior angle is under , concave means one is reflex.
- The number of diagonals is , which correctly gives 0 for a triangle.
- The interior angles total , because the polygon splits into triangles; divide by for each angle only if the polygon is regular.
- In a circle, the diameter is twice the radius and is the longest chord; a segment is bounded by a chord, a sector by two radii.
You will remember all of this far better after answering five questions on it than after reading it twice.
- Regular means equal sides and equal angles; convex means every interior angle is under , concave means one is reflex.
- The number of diagonals is , which correctly gives 0 for a triangle.
- The interior angles total , because the polygon splits into triangles; divide by for each angle only if the polygon is regular.
- In a circle, the diameter is twice the radius and is the longest chord; a segment is bounded by a chord, a sector by two radii.
You will remember all of this far better after answering five questions on it than after reading it twice.