Mathematics
Grade 10
15 min
Construct an equilateral triangle inscribed in a circle (Tutorial Only)
Construct an equilateral triangle inscribed in a circle (Tutorial Only)
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1
Introduction & Learning Objectives
Learning Objectives
Accurately construct an equilateral triangle inscribed in a circle using only a compass and a straightedge.
Define and identify key geometric terms such as 'inscribed', 'equilateral', 'radius', and 'central angle'.
Explain the geometric principles that prove the constructed triangle is equilateral.
Calculate the side length of an inscribed equilateral triangle given the radius of the circle.
Determine the measure of the central angles and arcs formed by the vertices of the inscribed triangle.
Recognize and avoid common errors in the construction process.
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Key Concepts & Vocabulary
TermDefinitionExample
Inscribed PolygonA polygon drawn inside a circle such that all of its vertices lie on the circle's circumference.A triangle with vertices A, B, and C is inscribed in circle O if points A, B, and C are all on the circle.
Equilateral TriangleA triangle with three sides of equal length. Consequently, all three of its interior angles are equal (60° each).A triangle with side lengths of 5 cm, 5 cm, and 5 cm.
Radius (plural: radii)A straight line segment extending from the center of a circle to any point on its circumference.In a circle with center O and a point P on the circumference, the line segment OP is a radius.
CompassA technical drawing instrument used for drawing circles or arcs and for transferring measurements.Setting the compass to a width of 4 cm allows y...
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Core Formulas
Side Length of Inscribed Equilateral Triangle
s = r \sqrt{3}
This formula relates the side length (s) of an inscribed equilateral triangle to the radius (r) of the circle. It is derived from the properties of 30-60-90 triangles formed by the radius, the apothem, and half of the triangle's side.
Central Angles of a Regular Inscribed Polygon
\theta = \frac{360^{\circ}}{n}
The central angles formed by connecting the center to consecutive vertices of a regular n-sided inscribed polygon are all equal. For an equilateral triangle (n=3), each central angle is 360°/3 = 120°.
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Challenging
A student claims to have a simpler method: 'Draw a circle. Use a protractor to measure and mark three 120° central angles. Connect the points on the circumference.' Why is this method NOT a valid geometric construction based on the tutorial's principles?
A.It is impossible to accurately measure 120° with a protractor.
B.This method creates an isosceles triangle, not an equilateral one.
C.It relies on a measuring tool (protractor) rather than only a compass and an un-marked straightedge.
D.This method is valid but is not taught in the tutorial.
Challenging
What is the ratio of the area of an inscribed equilateral triangle to the area of the circle that circumscribes it?
A.3√3 : 4π
B.√3 : π
C.3 : 2Ï€
D.4√3 : 3π
Easy
In the context of geometric constructions, what does it mean for a polygon to be 'inscribed' in a circle?
A.The circle is drawn inside the polygon.
B.All vertices of the polygon lie on the circle's circumference.
C.The polygon's sides are all tangent to the circle.
D.The center of the polygon is the same as the center of the circle.
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