Finding Area with Subtraction
Finding area with subtraction means calculating the area of an irregular composite shape by finding the area of a larger enclosing shape and then subtracting the areas of the parts that are cut out or excluded. For example, to find the area of a frame, calculate the area of the outer rectangle and subtract the area of the inner rectangle. This Grade 7 math skill from Saxon Math, Course 2 is a powerful problem-solving strategy for any irregular region and is essential for floor plan calculations, material estimation in construction, and geometry problem solving.
Key Concepts
Property See a complex shape as a big rectangle with a piece missing. Find the area of the whole rectangle, then subtract the area of the missing chunk to get your final answer.
Examples A $10 \times 12$ rectangle with a $3 \times 4$ piece missing: $(10 \cdot 12) (3 \cdot 4) = 120 12 = 108 \text{ m}^2$. A $20 \times 8$ shape missing a $5 \times 5$ square: $(20 \cdot 8) (5 \cdot 5) = 160 25 = 135 \text{ m}^2$.
Explanation It's the donut method! You find the area of the whole thing as if it were solid, then just subtract the empty hole in the middle.
Common Questions
How do I find area using subtraction?
Find the area of the largest simple shape that contains the figure, then subtract the areas of any parts that are removed or absent. The result is the area of the remaining region.
When should I use the subtraction method for finding area?
Use subtraction when a shape has a piece removed from a larger shape — like a square with a circular hole, a rectangular frame, or an L-shaped room missing a corner.
How do I find the area of a frame or border?
Calculate the area of the outer rectangle and subtract the area of the inner rectangle. The difference is the area of the frame or border.
What other shapes can the subtraction method be used for?
The subtraction method works for any composite shape: a semicircle removed from a rectangle, a triangle cutout from a square, a circular pond in the middle of a rectangular lawn.
When do students learn to find area with subtraction?
Area by subtraction is a Grade 7 strategy. Saxon Math, Course 2 covers it in Chapter 9 alongside other composite area methods.
What are common mistakes when finding area by subtraction?
Students sometimes subtract the wrong dimensions or forget to use the correct formulas for each sub-region. Always sketch the problem and label each region before calculating.
How does finding area with subtraction connect to real life?
A painter calculating how much paint is needed for a wall with windows uses subtraction: total wall area minus window areas = area to be painted.