Area of Irregular Polygons: The Box Method
The box method for finding the area of irregular polygons is a Grade 6 geometry strategy in Reveal Math, Course 1. To use the box method, enclose the irregular polygon in a rectangle (the box), calculate the rectangle's area, then subtract the areas of any right triangles or other simple shapes that are inside the box but outside the polygon. The result is the area of the irregular shape. This approach works whenever the polygon has vertices on a grid and avoids the need to decompose the polygon into smaller pieces directly.
Key Concepts
Property For complex or irregular polygons, the easiest way to find the area is the Enclosure Method (The Box Method) : 1. Draw a perfect rectangle (a box) around the entire irregular polygon. 2. Calculate the area of this large rectangle. 3. Calculate the areas of the "empty" spaces (usually right triangles) inside the box but outside your polygon. 4. Subtract the empty spaces from the large rectangle's area.
Examples You draw a large rectangle around an irregular polygon. The large rectangle has an area of 20 square units. There are three empty right triangles in the corners with areas of 6, 3, and 2.5 square units. Total empty space = 6 + 3 + 2.5 = 11.5 square units. Area of the polygon = 20 11.5 = 8.5 square units.
Explanation Sometimes a shape is too weird to slice up into nice pieces. When that happens, think outside the box by drawing a box! Imagine your weird shape is a cookie cut out of a rectangular sheet of dough. If you find the area of the whole rectangular sheet, and subtract the scraps of dough you don't need, you are left with the exact area of your cookie.
Common Questions
What is the box method for finding area?
Enclose the irregular polygon in the smallest rectangle that fits around it. Calculate the rectangle's area (l x w). Identify the corner triangles or other shapes inside the box but outside the polygon. Subtract those areas. The remainder is the polygon's area.
When is the box method better than decomposition for finding area?
The box method works well when the polygon fits neatly in a rectangle with simple right triangles in the corners, which is common for polygons on a coordinate grid. Decomposing is better when the polygon splits naturally into familiar shapes.
How do you find the area of the corner triangles in the box method?
Each corner is a right triangle. Use A = (1/2) x base x height, where base and height are the legs of the right triangle (the sides along the rectangle edges). Subtract all corner triangle areas from the rectangle.
Does the box method work for all polygons?
The box method works best for polygons with horizontal and vertical sides or for polygons plotted on a coordinate grid. For polygons with angled sides that do not form right triangles in the corners, decomposition or other methods may be needed.
How is the box method related to the coordinate plane?
On a coordinate plane, polygon vertices have integer coordinates, making it easy to identify rectangle dimensions and corner triangle leg lengths. The box method takes direct advantage of this grid structure.
When do students learn the box method for area?
The box method is taught in Grade 6 as part of the area of polygons unit in Reveal Math, Course 1, often alongside decomposition strategies.
Which textbook covers the box method for irregular polygons?
Reveal Math, Course 1, used in Grade 6, covers the box method in the area chapter as an alternative strategy for complex shapes.