Polygons on the Coordinate Plane

Geometry · Axiom Academy

LESSON Polygons on the Coordinate Plane Finding perimeter and area straight from the coordinates of the vertices. The backbone of coordinate geometry is the distance formula — it is just the Pythagorean theorem in disguise. The horizontal gap and the vertical gap are the legs of a right triangle, and the segment connecting the two points is its hypotenuse. Length of a segment between two points The four formulas we'll lean on The length of any segment between two points. The center point of a segment — average the coordinates. Tells you whether sides are parallel (equal slopes) or perpendicular ( m_1 m_2 = -1 ). Find the perimeter of triangle ABC with vertices A(1, 2) , B(5, 2) , and C(3, 6) . Apply the distance formula to each side, then add the three lengths. Watch each side trace onto the grid one at a time — the right triangle behind side BC shows where its length comes from, and the running total builds the perimeter. The worked lengths appear once it finishes. AB = |5 - 1| = 4 (or by the formula: ) 3. Finding Area: the Shoelace Formula To find the area of any polygon from its coordinates, use the Shoelace Formula. List the vertices in order around the polygon, multiply diagonally one way, multiply diagonally the other way, subtract, and halve. The crisscross pattern of those products is why it's called the "shoelace." Go clockwise or counterclockwise, repeating the first point at the end. Multiply diagonally down-right ( x_i y_ i+1 ) and add them all.

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