Montana Math MT.CORE.GEOM.MEAS.2

MathGrades 9–12Measurement, Problem-Solving, and Geometric Modeling

The Standard

Derive the equation of a circle of a given center and radius using the Pythagorean Theorem.

2026 Montana Mathematics Content Standards · Geometric Reasoning

Teacher's field guide

What Students Need to Do: Students draw a right triangle from a circle’s center to a point on its edge. They use the Pythagorean theorem to write an equation for every point on that circle.
What Mastery Looks Like: Given a center of (-2, 3) and radius 4, a student writes (x + 2)² + (y - 3)² = 16 and explains each term. They can check whether a point lies on the circle.
Common Misconceptions: Students may write x - 2 for a center whose x-coordinate is -2. They may put 4 instead of 16 on the right, or use x and y as leg lengths without measuring from the center.
How to Assess It: Exit ticket: A circle has center (3, -1) and radius 5. Sketch a right triangle to a point (x, y), label its legs, and derive the circle’s equation.

Lesson moves

On graph paper, plot center (1, 1) and point (4, 5), draw the right triangle, then replace its leg lengths with expressions using x and y.
Ask students to sketch center (-2, 3) and explain in writing why its horizontal leg is x + 2, not x - 2.
Give pairs center-and-radius cards and equation cards to match, then have each pair justify one match with a right triangle.
Map a sprinkler at (2, 1) with a five-meter reach, derive its boundary equation, and check whether a bench at (5, 5) is on it.

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