Find the Perimeter of a Polygon on the Coordinate Plane
Finding the perimeter of a polygon on the coordinate plane means calculating the total length of all sides by using the coordinates of the vertices. For sides that are horizontal or vertical, subtract the corresponding coordinate values. For diagonal sides, use the distance formula. The perimeter is the sum of all side lengths. This 6th grade geometry skill from enVision Mathematics Grade 6 integrates coordinate geometry, measurement, and number sense, and prepares students for more complex area and distance calculations in middle and high school.
Key Concepts
Property The perimeter, $P$, of a polygon on the coordinate plane is the sum of the lengths of all its sides. For a polygon with $n$ sides of lengths $s 1, s 2, ..., s n$, the perimeter is calculated as: $$P = s 1 + s 2 + ... + s n$$ The lengths of horizontal and vertical sides are found by calculating the distances between their vertices.
Examples Find the perimeter of a polygon with vertices $A( 2, 1)$, $B(3, 1)$, $C(3, 3)$, and $D( 2, 3)$. The side lengths are $AB = |3 ( 2)| = 5$, $BC = |1 ( 3)| = 4$, $CD = |3 ( 2)| = 5$, and $DA = | 3 1| = 4$. The perimeter is $P = 5 + 4 + 5 + 4 = 18$ units.
Find the perimeter of a polygon with vertices $P(1, 4)$, $Q(5, 4)$, $R(5, 2)$, $S(3, 2)$, $T(3, 1)$, and $U(1, 1)$. The side lengths are $PQ=4$, $QR=2$, $RS=2$, $ST=1$, $TU=2$, $UP=3$. The perimeter is $P = 4 + 2 + 2 + 1 + 2 + 3 = 14$ units.
Common Questions
How do you find the perimeter of a polygon on the coordinate plane?
Calculate the length of each side using the coordinates of its endpoints, then add all side lengths. For horizontal sides: |x2 - x1|. For vertical sides: |y2 - y1|.
How do you find the length of a diagonal side on the coordinate plane?
Use the distance formula: d = sqrt((x2-x1)^2 + (y2-y1)^2). For points (0,0) and (3,4): d = sqrt(9+16) = sqrt(25) = 5.
Do you need the distance formula for all polygon sides?
Only for diagonal sides. Horizontal sides are found by |x2-x1| and vertical sides by |y2-y1|. The distance formula simplifies to these for horizontal and vertical cases.
What grade finds polygon perimeter on the coordinate plane?
Finding polygon perimeter on the coordinate plane is a 6th grade geometry skill in enVision Mathematics Grade 6, combining coordinate plane understanding with perimeter concepts.
How does this skill connect to the Pythagorean theorem?
The distance formula is derived from the Pythagorean theorem. The diagonal distance is the hypotenuse of a right triangle formed by the horizontal and vertical distances.
What is the perimeter of a triangle with vertices at (0,0), (4,0), and (0,3)?
The horizontal side has length 4, the vertical side has length 3, and the hypotenuse = sqrt(16+9) = sqrt(25) = 5. Perimeter = 4 + 3 + 5 = 12 units.