Mathematics
Grade 10
15 min
Perimeter of polygons with an inscribed circle
Perimeter of polygons with an inscribed circle
What you'll learn
- Identify whether a number is divisible by 2, 3, 5, or 10 by applying the divisibility rules with 80% accuracy on a worksheet.
- Explain in their own words why a number divisible by both 2 and 3 is also divisible by 6, providing at least one example.
- Solve division problems to check if a number is divisible by 4, 6, or 9, and justify their answer using the appropriate divisibility rule with 75% accuracy.
- Apply divisibility rules to determine all the factors of a given number between 1 and 100, and correctly identify at least 3 factors.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Define a tangential polygon, an inscribed circle (incircle), and a tangent segment.
State and apply the Two Tangent Theorem to find unknown segment lengths.
Calculate the length of each side of a tangential polygon by summing its constituent tangent segments.
Calculate the total perimeter of a tangential polygon using the lengths of the tangent segments from its vertices.
Apply Pitot's Theorem to solve for unknown side lengths in tangential quadrilaterals.
Solve multi-step problems involving algebraic expressions for segment lengths in tangential polygons.
Ever wondered how a circular gear fits perfectly inside a square casing or why a hexagonal nut fits a wrench so well? ⚙️ This perfect fit is all about the geometry of polygons with inscribed circle...
2
Key Concepts & Vocabulary
TermDefinitionExample
Inscribed Circle (Incircle)A circle is inscribed in a polygon if every side of the polygon is tangent to the circle. The circle is inside the polygon and touches each side at exactly one point.A circle drawn inside a triangle so that it just touches all three sides.
Tangential PolygonA convex polygon that has an inscribed circle. It is also called a circumscribed polygon because it is 'circumscribed about' a circle.A square with a circle inside it that touches the midpoint of all four sides.
Point of TangencyThe single point where a side of the polygon touches the inscribed circle.If side AB of a square touches its incircle at point P, then P is the point of tangency.
Tangent SegmentThe line segment connecting a vertex of a polygon to one of the two points...
3
Core Formulas
The Two Tangent Theorem
If two tangent segments are drawn to a circle from the same external point (a vertex), then they are congruent (equal in length).
This is the most important rule for this topic. For any vertex of a tangential polygon, the distances from that vertex to the points of tangency on the two adjacent sides are equal. This property is the key to finding side lengths and perimeters.
Perimeter of a Tangential Polygon
P = 2(s_1 + s_2 + s_3 + ... + s_n)
Where s_1, s_2, ..., s_n are the lengths of the unique tangent segments from each of the n vertices. Since each tangent segment length appears twice around the polygon (once on each adjacent side from a vertex), we can sum the unique segment lengths and multiply by two to find the total perimeter.
Pitot's...
5 more steps in this tutorial
Sign up free to access the complete tutorial with worked examples and practice.
Sign Up Free to ContinueSample Practice Questions
Easy
Which of the following best defines a 'tangential polygon'?
A.polygon where all vertices lie on a single circle.
B.polygon where every side is tangent to an inscribed circle.
C.polygon with at least one side tangent to a circle.
D.polygon that is inside a circle.
Easy
According to the Two Tangent Theorem, if two tangent segments are drawn to a circle from the same external point, what must be true about the segments?
A.They are perpendicular to each other.
B.They are parallel to each other.
C.They are congruent (equal in length).
D.Their lengths sum to the radius of the circle.
Easy
A triangle ABC is circumscribed about a circle. The point of tangency on side AB is P. If the tangent segment from vertex A, AP, has a length of 8 cm, what is the length of the other tangent segment from vertex A?
A.4 cm
B.8 cm
C.16 cm
D.Cannot be determined
Want to practice and check your answers?
Sign up to access all questions with instant feedback, explanations, and progress tracking.
Start Practicing FreeMore from Circles
Mathematics for other grades
Frequently asked questions
What grade level is "Perimeter of polygons with an inscribed circle"?
Perimeter of polygons with an inscribed circle is a Grade 10 Mathematics lesson on ExcelOS.
What will I learn in Perimeter of polygons with an inscribed circle?
You'll be able to: Identify whether a number is divisible by 2, 3, 5, or 10 by applying the divisibility rules with 80% accuracy on a worksheet; Explain in their own words why a number divisible by both 2 and 3 is also divisible by 6, providing at….
Is "Perimeter of polygons with an inscribed circle" free to practice?
Yes. You can read the tutorial preview for free, and signing up for a free ExcelOS account unlocks the full tutorial and all practice questions with instant feedback.
How many practice questions are included with Perimeter of polygons with an inscribed circle?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.