Grade 6 Math: Area of Polygons on the Coordinate Plane
TL;DR: When a polygon sits on a coordinate plane, the gridlines do half the measuring for you. Plot the vertices, look at what shape you have, and reach for the formula you already know. For oddly-shaped polygons, slice them into rectangles and triangles, find each area, and add them up. Count side lengths straight off the gridlines, no ruler needed. Once you can read coordinates as measurements, polygons on the plane stop being mysterious and start being routine.
Free tool: try the problems from this lesson in our Area & Perimeter Calculator.
Quick answer: To find the area of a polygon in 6th grade, split it into shapes you know, such as rectangles and triangles, find each area, and add them. On a coordinate plane, get side lengths by subtracting coordinates. Example: a rectangle from (2, 1) to (7, 5) is 7 − 2 = 5 wide and 5 − 1 = 4 tall, so its area is 20 square units.
Key takeaways:
- Plot each vertex carefully – the ordered pair tells you the column (x) and row (y).
- Identify the shape: rectangle, triangle, trapezoid, or composite (a mix).
- Use the right formula: rectangle \(=l \times w\); triangle \(=\dfrac{1}{2} \times b \times h\); trapezoid \(=\dfrac{1}{2}(b_1+b_2)h\).
- For composite shapes, split into pieces, find each area, then add them up.
- Count whole units along the gridlines; don’t eyeball.
Grade 6 focus: On the coordinate plane, you can find the area of polygons whose vertices are given as ordered pairs \((x,y)\). Common approaches: surround with a rectangle and subtract triangles, decompose into rectangles and right triangles, or use the shoelace formula when appropriate (often introduced later).
Video lesson: Watch this Anywhere Math lesson on polygons in the coordinate plane.
Rectangle method
- Plot the vertices in order.
- Draw a bounding rectangle whose sides align with the grid.
- Subtract areas of right triangles or rectangles outside the polygon but inside the bounding box.
Decomposition method
Split the polygon into shapes with known area formulas (rectangles, right triangles). Add the areas.
Why coordinates help
Side lengths can be found from differences in \(x\) and \(y\) values when sides are horizontal or vertical.
Worked idea
For a rectangle with vertical/horizontal sides on the grid, width = difference in \(x\), height = difference in \(y\); area = width \(\times\) height.
Common mistakes
- Plotting points in the wrong order, crossing segments.
- Confusing \(x\) and \(y\) when finding side lengths.
- Using non-perpendicular “slanted” sides as if they were base/height without a proper height segment.
Practice tip
Use graph paper or a grid: sketch lightly, label every length you use in your calculations.
Recommended EffortlessMath Books
For a workbook that pairs with this page, Mastering Grade 6 Math walks your sixth grader through every grade-6 topic with worked examples and plenty of practice. For more story-problem reps, Mastering Grade 6 Math Word Problems is the matching word-problem book.
Area Formulas You Need in 6th Grade
| Shape | Formula | Example |
|---|---|---|
| Rectangle | A = length × width | 8 × 3 = 24 square units |
| Triangle | A = ½ × base × height | ½ × 6 × 5 = 15 square units |
| Parallelogram | A = base × height | 7 × 4 = 28 square units |
| Trapezoid | A = ½ × (b₁ + b₂) × height | ½ × (4 + 8) × 3 = 18 square units |
| Any other polygon | Split into the shapes above and add | See the example below |
Height always means the straight up-and-down distance, at a right angle to the base, not the length of a slanted side.
Worked Example: A Five-Sided Polygon From Its Vertices
Find the area of the pentagon with vertices A(1, 1), B(7, 1), C(7, 4), D(4, 6) and E(1, 4).
- Plot and look. The bottom part, ABCE, is a rectangle. The top part, ECD, is a triangle.
- Rectangle. Width: 7 − 1 = 6. Height: 4 − 1 = 3. Area: 6 × 3 = 18 square units.
- Triangle. Its base EC runs from x = 1 to x = 7, so it is 6 units long. Its height is the distance from y = 4 up to D at y = 6, which is 2 units. Area: ½ × 6 × 2 = 6 square units.
- Add. 18 + 6 = 24 square units.
Try the box method too. For a triangle with no horizontal side, such as (1, 1), (5, 2) and (2, 6), draw a box around it from x = 1 to 5 and y = 1 to 6. The box is 4 × 5 = 20. Cut off the three right triangles in the corners: ½ × 4 × 1 = 2, ½ × 3 × 4 = 6, and ½ × 1 × 5 = 2.5. The triangle’s area is 20 − (2 + 6 + 2.5) = 9.5 square units.
Frequently Asked Questions
How do you find the area of polygons on a coordinate plane?
Plot the vertices, connect them in order, and decide which shape you have. Find side lengths by subtracting coordinates along horizontal and vertical sides. Then use the right formula, or split an odd shape into rectangles and triangles and add the pieces. Another option is to draw a rectangle around the shape and subtract the corner triangles.
How do you find the area of a triangle on a coordinate plane?
Choose a horizontal or vertical side as the base, then count straight across to the opposite vertex for the height. For the triangle with vertices (1, 1), (7, 1) and (4, 5), the base runs from x = 1 to x = 7, so it is 6 units. The height is 5 − 1 = 4 units. The area is ½ × 6 × 4 = 12 square units.
How do you find the area of a polygon in 6th grade?
Sixth graders use formulas they know, such as rectangle l × w, parallelogram b × h, triangle ½ × b × h and trapezoid ½(b₁ + b₂)h, and split any other polygon into those shapes. For example, a parallelogram with a base of 8 units and a height of 3 units has area 8 × 3 = 24 square units. Always give the answer in square units.
How do you find the area of a rectangle from its vertices?
Subtract the x-coordinates to get the width and the y-coordinates to get the height, then multiply. A rectangle with vertices (−2, 1), (4, 1), (4, 6) and (−2, 6) is 4 − (−2) = 6 units wide and 6 − 1 = 5 units tall, so its area is 6 × 5 = 30 square units. Subtracting a negative coordinate means adding its distance from zero.
Is area of polygons on the coordinate plane taught in 6th grade?
Yes. In the Common Core standards, 6.G.1 asks students to find the area of triangles, quadrilaterals and other polygons by composing and decomposing shapes, and 6.G.3 asks them to draw polygons on the coordinate plane from their vertices and use coordinates to find side lengths. Together, these skills let students find areas without a ruler.
Related EffortlessMath Lessons
If a topic on this page feels rusty, these short lessons go deeper:
- How to graph points on the coordinate plane
- How to identify polygons
- How to find the area of rectangles
- How to find the area of shapes (resource currently unavailable)
- How to find perimeter and area
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