Practical examples of finding the area of irregular shapes

Real-world examples of finding the area of irregular shapes

Let’s skip the formal definitions and jump straight into real examples. The best examples of finding the area of irregular shapes usually come from everyday life: yards, rooms, playgrounds, and paths that are almost simple shapes but not quite.

Think of these irregular shapes as mash-ups of familiar ones. If you can find the area of rectangles, triangles, and circles, you already have the tools. The trick is learning how to combine those tools.

We’ll walk through several examples of examples of finding the area of irregular shapes using different strategies:

• Breaking a shape into smaller standard shapes
• Subtracting missing pieces
• Using a grid for sketchy, hand-drawn shapes
• Using coordinates when you have a map or graph
• Using digital tools and apps in 2024–2025

Example of breaking an irregular yard into rectangles

Imagine you want to lay sod or artificial turf in a backyard that looks like a big “L” instead of a neat rectangle. This is one of the best examples of finding the area of irregular shapes by splitting the shape.

Picture this:

• The long part of the yard is 30 feet by 20 feet.
• On one side, there’s a smaller rectangular section sticking out that’s 10 feet by 15 feet.

The yard is an irregular shape overall, but you can treat it as two rectangles.

1. Find the area of the big rectangle:

   • \(A_1 = 30 \times 20 = 600\) square feet.

2. Find the area of the smaller rectangle:

   • \(A_2 = 10 \times 15 = 150\) square feet.

3. Add them:

   • Total area \(= 600 + 150 = 750\) square feet.

So, in this example of an irregular shape, the strategy is: split → solve → add.

You can apply this same idea to floor plans, parking lots, or any layout that looks like an “L”, “T”, or plus sign.

Examples of finding the area of irregular shapes by subtracting cut-outs

Sometimes the shape is almost simple, but there’s a bite taken out of it. These examples include decks around pools, patios with notches, or rooms with closets.

A deck with a missing corner

Say you’re building a wooden deck shaped like a big 20 ft by 16 ft rectangle, but there’s a 6 ft by 4 ft corner cut out for a basement entrance.

1. Start with the full rectangle:

   • \(A_\text{big} = 20 \times 16 = 320\) square feet.

2. Find the area of the missing rectangle:

   • \(A_\text{cutout} = 6 \times 4 = 24\) square feet.

3. Subtract the missing part:

   • Deck area \(= 320 - 24 = 296\) square feet.

This is one of the cleanest examples of examples of finding the area of irregular shapes: start big, subtract the holes.

An office with a built-in closet

Office floor plans are another great example of irregular shapes. Suppose an office is roughly 18 ft by 14 ft, but a 4 ft by 3 ft closet sticks into the room from one wall.

If you want the usable floor area (ignoring the closet space you can’t walk on):

1. Area of whole rectangle:

   • \(A_\text{room} = 18 \times 14 = 252\) square feet.

2. Area of closet footprint:

   • \(A_\text{closet} = 4 \times 3 = 12\) square feet.

3. Subtract:

   • Usable floor area \(= 252 - 12 = 240\) square feet.

This same subtracting approach is used in architecture and construction planning. Many high school geometry standards, like those described by Illustrative Mathematics, use similar real examples in classroom tasks.

Examples include shapes made of rectangles and triangles

Not every irregular shape is made of rectangles only. Roofs, ramps, and some garden beds mix rectangles and triangles.

A slanted garden bed

Imagine a garden bed along a fence. It’s 12 ft long at the front, 8 ft deep at one end, and 4 ft deep at the other. If you connect those points, the bed is a trapezoid, but you can treat it as a rectangle plus a right triangle.

• Front edge: 12 ft
• Left side depth: 8 ft
• Right side depth: 4 ft

Split it so that you have:

• A rectangle 12 ft by 4 ft (the shallow part)
• A right triangle with base 12 ft and height \(8 - 4 = 4\) ft

1. Rectangle area:

   • \(A_\text{rect} = 12 \times 4 = 48\) square feet.

2. Triangle area:

   • \(A_\text{tri} = \tfrac{1}{2} \times 12 \times 4 = 24\) square feet.

3. Total area:

   • \(48 + 24 = 72\) square feet.

This is a classic example of finding the area of an irregular shape by splitting it into one rectangle and one triangle.

A house roof face

Roofing cost estimates give great real examples of finding the area of irregular shapes. Suppose one face of a roof looks like a rectangle with a triangular peak on top:

• The rectangular part is 24 ft wide and 10 ft tall.
• The triangular peak has the same 24 ft base and a height of 4 ft.

1. Rectangle:

   • \(A_\text{rect} = 24 \times 10 = 240\) square feet.

2. Triangle:

   • \(A_\text{tri} = \tfrac{1}{2} \times 24 \times 4 = 48\) square feet.

3. Roof face area:

   • \(240 + 48 = 288\) square feet.

Roofers routinely use this kind of breakdown, and many building and construction courses (for example, those described by community college programs listed on sites like ed.gov) train students with similar problems.

Mixed-shape examples of finding the area of irregular shapes

Sometimes an irregular shape is a mix of rectangles and circles, or circle segments. Think of playgrounds, round patios attached to straight paths, or semi-circular steps.

A patio with a semicircular extension

You have a rectangular patio, 16 ft by 10 ft, with a half-circle on one of the 16 ft sides.

• Rectangle: 16 ft by 10 ft
• Semicircle diameter: 16 ft, so radius \(r = 8\) ft

1. Rectangle area:

   • \(A_\text{rect} = 16 \times 10 = 160\) square feet.

2. Full circle area with radius 8 ft:

   • \(A_\text{circle} = \pi r^2 = \pi \times 8^2 = 64\pi\) square feet.

3. Semicircle area:

   • \(A_\text{semi} = \tfrac{1}{2} \times 64\pi = 32\pi\) square feet.

4. Total patio area:

   • \(160 + 32\pi\) square feet.
   • Using \(\pi \approx 3.14\), this is about \(160 + 100.48 = 260.48\) square feet.

This example of an irregular shape shows how you can mix straight-edge formulas with circle formulas in one problem.

A playground with rounded corners

Imagine a small playground that’s mostly a 40 ft by 30 ft rectangle, but the four corners are rounded off by quarter-circles of radius 5 ft. The overall boundary is an irregular shape, but you can think of it as:

• Start with the big rectangle.
• Subtract four quarter-circles (which together make one full circle).

1. Rectangle area:

   • \(A_\text{rect} = 40 \times 30 = 1200\) square feet.

2. One full circle with radius 5 ft:

   • \(A_\text{circle} = \pi \times 5^2 = 25\pi\) square feet.

3. Playground area:

   • \(1200 - 25\pi\) square feet.
   • Approximately \(1200 - 78.5 = 1121.5\) square feet.

This is one of the best examples of examples of finding the area of irregular shapes by subtracting curved cut-outs from a simple base shape.

Using grids: sketchy, hand-drawn irregular shapes

Not every irregular shape comes with nice, clean measurements. Sometimes you just have a sketch of a pond, a garden bed, or a blob-like region on graph paper.

Teachers, especially in middle school geometry, often use these grid-based examples of finding the area of irregular shapes. The idea is simple:

• Draw the shape on grid paper.
• Count full squares inside.
• Estimate partial squares.

A pond on grid paper

Imagine a hand-drawn pond outline on graph paper, where each square is 1 ft by 1 ft.

• You count about 40 full squares completely inside the shape.
• You count about 30 half-squares around the edges.

You can estimate area by combining halves into wholes:

• 30 half-squares ≈ 15 full squares.
• Total ≈ \(40 + 15 = 55\) square feet.

This method is approximate, but it’s surprisingly useful for organic shapes. Agencies that map land and water areas, like the U.S. Geological Survey, use more advanced versions of this idea with high-resolution grids and digital data.

Coordinate geometry examples of finding the area of irregular shapes

When a shape is drawn on a coordinate plane, you can use the vertices’ coordinates to find the area, even if the boundary looks irregular.

A land plot on a map

Suppose a land plot has corners at these coordinates (in feet):

• A(0, 0)
• B(10, 0)
• C(14, 6)
• D(4, 8)

This quadrilateral is an irregular shape. One simple strategy:

• Split it into two triangles: ABC and ACD.
• Use the formula for the area of a triangle using coordinates, or break each triangle further into right triangles and rectangles.

For triangle ABC:

• Base AB = 10 (from (0,0) to (10,0)).
• You can think of C(14,6) and create a right triangle with base from x = 10 to x = 14 and height 6.

A cleaner approach is to use the “shoelace formula,” which is taught in many high school and college geometry courses and appears in open resources like those linked through OER Commons. While the algebra can look a bit heavy, the idea is the same: use coordinates to break down an irregular polygon into manageable pieces.

The key point: if you know the coordinates of the corners, you can find the area of the irregular polygon, even if the sides aren’t aligned with the axes.

2024–2025: digital tools as modern examples of finding the area of irregular shapes

In 2024–2025, you’re not limited to paper and pencil. There are now plenty of apps and online tools that provide real examples of finding the area of irregular shapes from photos, maps, or drawn outlines.

Some ways people are doing this today:

• Online map tools: You can trace the outline of a property or field directly on a satellite map, and the tool reports the area. Farmers, surveyors, and city planners use this routinely.
• Home design apps: Many home remodeling apps let you sketch an irregular room, and they automatically calculate area for flooring or paint. This is essentially a digital version of the breakdown methods we’ve been using in our examples of irregular shapes.
• Tablet or phone stylus tools: Some apps let you draw the shape with your finger or stylus on a grid, then count squares or approximate area.

Even though the tools are high-tech, the underlying math is the same as in our best examples of finding the area of irregular shapes: divide the shape into simpler parts, approximate where needed, and add or subtract.

For classroom-friendly digital activities and geometric problem-solving tasks, websites supported by organizations like NSF.gov and university education departments (for example, Harvard Graduate School of Education) often highlight technology-rich geometry lessons.

A repeatable strategy you can reuse

Looking back at all these examples of examples of finding the area of irregular shapes, there’s a clear pattern:

• See the simple shapes hiding inside. Rectangles, triangles, and circles are your go-to building blocks.
• Decide whether to add or subtract. If pieces are sticking out, you add. If pieces are missing, you subtract.
• Use grids or coordinates when outlines are messy. They turn blobs into countable or calculable shapes.
• Use tools, but understand the method. Apps and online tools are great, but they’re using the same logic you’re practicing.

Once you start seeing irregular shapes as combinations of familiar ones, the fear factor drops fast. Whether you’re planning a garden, estimating flooring, or solving a geometry problem, these real examples of finding the area of irregular shapes give you a template: break, calculate, combine.

FAQ: examples of finding the area of irregular shapes

Q: What are some everyday examples of irregular shapes where I need to find the area?
You’ll see them in yards, oddly shaped rooms, driveways that curve, decks with cut-out corners, garden beds, swimming pools, and playgrounds with rounded edges. Any time the boundary isn’t a simple rectangle or circle, you’re dealing with an irregular shape.

Q: Can you give a quick example of finding the area of an irregular shape made of rectangles?
Yes. Suppose a hallway is shaped like a “T”: a 20 ft by 4 ft corridor with a 10 ft by 6 ft section across one end. Treat it as two rectangles. Corridor area: \(20 \times 4 = 80\) square feet. Cross-section area: \(10 \times 6 = 60\) square feet. Total: \(80 + 60 = 140\) square feet.

Q: How do I handle an irregular shape if I only have a sketch and no measurements?
Put it on grid paper or overlay a digital grid. Count full squares, estimate partial squares, and add them. This gives an approximate area, which is often good enough for rough planning. This grid method appears in many middle school curricula and is supported by research on spatial reasoning from universities like those listed on nsf.gov.

Q: Are there real examples where professionals need to find the area of irregular shapes?
Yes. Surveyors measure land parcels that almost never form perfect rectangles. Architects design creative floor plans. Landscape designers shape ponds and gardens. City planners map irregular districts. In all these jobs, people use the same strategies shown in our best examples of finding the area of irregular shapes.

Q: What if the irregular shape has curved sides instead of straight lines?
If the curved parts are parts of circles (like semicircles or quarter-circles), you can use circle formulas and add or subtract those pieces. If the curves are more complicated, you can approximate them with many small rectangles (Riemann sums) or use software that integrates under the curve. In school, you’ll usually see circle-based curves, like rounded patios or arches.

Q: Is it better to add pieces or subtract pieces when working with irregular shapes?
Use whichever makes the outline easier to describe. If the shape fits neatly inside a rectangle with a few chunks missing, subtraction is easier. If it’s made of several obvious pieces stuck together, addition is better. Many of the most efficient examples of examples of finding the area of irregular shapes use a mix of both approaches.

Q: How accurate are area estimates from grids or digital sketches?
They can be very close if the grid is fine enough or the tool is high-resolution. For rough planning (like estimating mulch or grass), a small error is usually acceptable. For legal or engineering work, professionals use precise measuring tools, GPS, and advanced software, as described in resources from agencies like the U.S. Geological Survey.