Mathematics
Grade 5
15 min
Construct the inscribed or circumscribed circle of a triangle (Tutorial Only)
Construct the inscribed or circumscribed circle of a triangle (Tutorial Only)
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1
Introduction & Learning Objectives
Learning Objectives
Identify an inscribed circle within a triangle.
Identify a circumscribed circle around a triangle.
Explain the difference between an inscribed and a circumscribed circle.
Visually estimate the approximate center of an inscribed circle.
Visually estimate the approximate center of a circumscribed circle.
Draw a circle that is approximately inscribed or circumscribed in a given triangle.
Have you ever wondered how a perfectly round pizza could fit inside a triangular box? 🍕 Or how a triangular park could be perfectly surrounded by a circular fence? 🏞️
Today, we'll explore special circles that relate to triangles: circles that fit perfectly inside (inscribed) and circles that perfectly surround (circumscribed). For Grade 5, 'constructing' mea...
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Key Concepts & Vocabulary
TermDefinitionExample
TriangleA flat shape with three straight sides and three corners (called vertices).A slice of pizza or a road sign that looks like a yield sign.
CircleA perfectly round shape where all points on its edge are the same distance from its center.A wheel, a coin, or a hula hoop.
Inscribed CircleA circle drawn *inside* a triangle that touches all three of the triangle's sides exactly once.Imagine a ball perfectly snuggled into the corner of a triangular box, touching all three sides.
Circumscribed CircleA circle drawn *around* a triangle that passes through all three of the triangle's corners (vertices).Imagine a triangular island in the middle of a perfectly circular lake, with the island's points touching the water's edge.
Center of a CircleThe exact...
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Core Formulas
Inscribed Circle Property
An inscribed circle is always found *inside* a triangle and touches each of its three sides.
This means the circle fits snugly within the triangle, like a ball in a corner, making contact with every side without crossing them.
Circumscribed Circle Property
A circumscribed circle is always found *around* a triangle and passes through all three corners (vertices) of the triangle.
This means the triangle's corners are all on the edge of the circle, and the circle encloses the entire triangle.
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Challenging
If point 'P' is the center of a triangle's inscribed circle, what must be true about the distance from 'P' to each of the three sides?
A.The distances are all different.
B.The distance to the longest side is the greatest.
C.The distances to all three sides are exactly equal.
D.The distance to the corners is zero.
Challenging
If point 'Q' is the center of a triangle's circumscribed circle, what must be true about the distance from 'Q' to each of the three vertices?
A.The distances are all different.
B.The distances to all three vertices are exactly equal.
C.The distance to the sides are all equal.
D.The distance to the highest vertex is the smallest.
Challenging
Imagine a triangle where all three sides have the exact same length (an equilateral triangle). If you correctly draw both the inscribed and circumscribed circles, what would you notice about their centers?
A.They are in the exact same spot.
B.The inscribed center is higher than the circumscribed center.
C.The circumscribed center is outside the triangle.
D.They are on opposite sides of the triangle.
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