How To Calculate The Area Of An Irregular Shape

When you first start calculating area, you get easy shapes that have clearly defined formulas for finding their area: circles, triangles, squares and rectangles, for example. But what happens when you're faced with a shape that doesn't fit easily into those categories? Until you enter the brave new world of calculus integrals, the best way to find the area of irregular shapes is by subdividing them into shapes you're already familiar with.

1. Assemble Your Tools

Collect the area formulas for shapes you're already familiar with. The most common shapes and their formulas include:

\(\text{Area of a square or rectangle } = l × w\)

where ​l​ is length and ​w​ is width.

\(\text{Area of a triangle } = \frac{1}{2}(b × h)\)

where ​b​ is the triangle's base and ​h​ is its vertical height.

\(\text{Area of a parallelogram } = b × h\)

where ​b​ is the parallelogram's base and ​h​ is its vertical height.

\(\text{Area of a circle } = πr^2\)

where ​r​ is the radius of the circle.

2. Subdivide the Irregular Shape

Use your imagination to subdivide the irregular shape you have into more familiar shapes. Sometimes drawing the shape out, then adding lines for the subdivisions, helps you visualize it, and track the appropriate measurements for each dimension. For example, imagine that you have to find the area of a five-sided shape that isn't a hexagon but has three perpendicular sides opposite the "point." With a little thinking, you can subdivide this into a rectangle that butts up against a triangle, with the triangle forming the "point" of the shape.

3. Find the Dimensions of the Subdivided Shapes

Refer back to your area formulas for the dimensions you'll need to calculate the area of each subdivided shape. In this case, you'll need the base and vertical height of the triangle and the length and width (or two adjacent sides) of the rectangle. If you're working a math problem in school, you'll probably get at least some of these measurements and may need to use some basic algebra or geometry to find any missing measurements. If you're working in the real world, you might be able to fill in some of the dimensions by physically measuring.

4. Calculate the Area of Each Subdivided Shape

Fill the dimensions into the area formula for each subdivided shape. For example, if the triangle has a base of 6 inches and a vertical height of 3 inches, its area formula is:

\(\frac{1}{2}(b × h) = \frac{1}{2} (6 \text{ in} × 3 \text{ in}) = \frac{1}{2} (18 \text{ in}^2) = 9 \text{ in}^2\)

If the rectangle has a length of 6 inches (which is also the side that makes up the base of the triangle) and a height of 4 inches, its area formula is:

\(l × w = 6 \text{ in} × 4 \text{ in} = 24 \text{ in}^2\)

5. Total the Areas of the Subdivided Shapes

Add the areas of the subdivided shapes; the total is the area of the irregular shape you started with. To conclude this example, the area of the triangle is 9 in², and the area of the rectangle is 24 in². So your total area is:

\(9 \text{ in}^2 + 24 \text{ in}^2 = 33 \text{ in}^2\)