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Questions and Answers
What is the condition for a polygon to be convex?
What is the sum of the measure of interior angles of an 8-sided polygon?
What is the measure of each exterior angle of a regular 10-sided polygon?
What is the condition for a figure to be symmetrical?
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What is the number of lines of symmetry of an 8-sided regular polygon?
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What is the term for a polygon in which all vertices lie on a circle?
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What is the condition for a circle to be inscribed or circumscribed about a polygon?
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What is the sum of the measure of interior angles of a 12-sided polygon?
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Study Notes
Polygon Basics
- A polygon is a simple closed plane figure formed by three or more line segments joined end to end, no two of which in succession are collinear.
- The line segments forming the polygon are called sides.
- The common end point of any two sides is called a vertex of the polygon.
- The angles formed inside the polygon are called interior angles.
Types of Polygons
- A convex polygon is a polygon where the measure of each interior angle is less than 180°.
- A concave polygon is a polygon where there is at least one interior angle whose measure is more than 180 degrees.
Angles in Polygons
- The sum of the measure of interior angles of an 𝑛-sided polygon is determined by (𝑛 − 2) × 180°.
- The sum of the measure of exterior angles of a polygon is 360°.
- At each vertex of a polygon, the external and internal angle add up to 180°.
Regular Polygons
- A regular polygon is a convex polygon with all sides equal and all interior angles equal.
- The measure of each interior angle of a regular 𝑛-sided polygon is (𝑛−2)×180°/𝑛.
- The measure of each exterior angle of a regular 𝑛-sided polygon is 360°/𝑛.
- The measure of each central angle of a regular 𝑛-sided polygon is 360°/𝑛.
Symmetry
- If two parts of a figure are identical after folding through the line of symmetry, then it is said to be symmetric.
- A symmetrical figure has at least one line of symmetry.
- An 𝑛-sided regular polygon has 𝑛 lines of symmetry.
Inscribed and Circumscribed Polygons
- An inscribed polygon is a polygon in which all vertices lie on a circle.
- A circumscribed polygon is a polygon in which each side touches a circle.
Apothem and Formulas
- An apothem of a regular polygon is a perpendicular segment from a midpoint of a side of a regular polygon to the centre of the circle circumscribed about the polygon.
- Formulas for the length of side 𝑠, apothem 𝑎, perimeter 𝑃, and area 𝐴 of the regular polygon with 𝑛-sides and radius 𝑟 are given by:
- 𝑠 = 2𝑟 sin (180°/𝑛)
- 𝑎 = 𝑟 cos (180°/𝑛)
- 𝑃 = 2𝑛𝑟 sin (180°/𝑛)
- 𝐴 = (1/2)𝑎𝑃
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Description
Learn about the definition and types of polygons, including convex and concave polygons, and their properties such as sides, vertices, and interior angles.
