Areas of irregular shapes

Real-life shapes are rarely tidy rectangles or triangles. Houses have odd-shaped rooms; states on a map have curvy borders; ponds and gardens have irregular outlines. To find the area of these irregular shapes, we use a few clever strategies.

Concept

Strategy 1: Split into known shapes

If a shape is made of straight edges, you can almost always split it into rectangles and triangles. Find the area of each piece, then add them up.

For example, an L-shaped room can be split into two rectangles. A T-shaped path can be split into three rectangles. The trick is to look for natural lines that cut the shape into the simplest possible pieces.

Strategy 2: Subtract from a bigger shape

Sometimes it is easier to surround the shape with a big rectangle, find the area of the big rectangle, and subtract the pieces that aren't part of the shape. This works especially well for shapes with "notches" cut out of them.

For example, a rectangle with a square hole in the middle: find the rectangle area, subtract the square area.

Strategy 3: Count unit squares (for curved or irregular figures)

For shapes whose boundary is curved (like a leaf, a country, or a pond), exact formulas are not available. Place the shape on graph paper and:

• Count the fully inside squares as $1$ each.
• Count the partly inside squares (any cell whose more than half is inside the shape) as $1$, those with less than half as $0$. Alternatively count all partial squares as $\frac{1}{2}$ each.
• Ignore squares barely touching the boundary.

This gives an estimate, not an exact answer , but it works for any shape, and gets more accurate as you use finer grid paper.

Pick's theorem (peek ahead)

There is a beautiful exact formula for the area of a polygon whose corners sit on grid points (called a lattice polygon):

$\text{Area} = i + \frac{b}{2} - 1$

where $i$ is the number of grid points strictly inside the polygon, and $b$ is the number of grid points on its boundary. This is Pick's theorem. You will see it again in higher classes, but it is fun to test it on small examples now.

Worked examples

Example 1. An L-shaped room is made of a $5$ m by $4$ m rectangle and a $3$ m by $2$ m rectangle joined along one $2$-m edge. Find its area.

• Piece $1$: $5 \times 4 = 20$ m².
• Piece $2$: $3 \times 2 = 6$ m².
• Total area $= 20 + 6 = 26$ m².

Example 2. A rectangle is $12$ m by $9$ m. A square swimming pool of side $3$ m is dug out of the middle. Find the remaining area.

• Rectangle area $= 12 \times 9 = 108$ m².
• Pool area $= 3^2 = 9$ m².
• Remaining $= 108 - 9 = 99$ m².

Example 3. A path of width $1$ m is built around the inside of a rectangular plot of $10$ m by $8$ m. Find the area of the path.

• Outer rectangle: $10 \times 8 = 80$ m².
• Inner rectangle (inside the path): $(10 - 2) \times (8 - 2) = 8 \times 6 = 48$ m².
• Path area $= 80 - 48 = 32$ m².

Example 4. A leaf is placed on $1$-cm graph paper. Counting: $14$ full squares inside, and $18$ partial squares. Estimate its area.

• Estimate $= 14 \times 1 + 18 \times \frac{1}{2} = 14 + 9 = 23$ cm².

Example 5. Use Pick's theorem to verify the area of a triangle with vertices at $(0,0)$, $(4,0)$, $(0,3)$.

• Boundary lattice points: on $(0,0)$-$(4,0)$ we have $5$ points (including ends). On $(4,0)$-$(0,3)$ , the gcd of $4$ and $3$ is $1$, so $1+1 = 2$ end points. On $(0,3)$-$(0,0)$ we have $4$ points. Avoid double-counting corners: $b = 5 + 2 + 4 - 3 = 8$. Hmm , careful: gcd of distances determines interior points on each segment. Let me redo: segment $(0,0)$ to $(4,0)$ has gcd $= 4$, giving $4+1 = 5$ lattice points. Segment $(4,0)$ to $(0,3)$ has gcd $= 1$, giving $2$ lattice points (just the endpoints). Segment $(0,3)$ to $(0,0)$ has gcd $= 3$, giving $4$ lattice points. Total boundary $b = 5 + 2 + 4 - 3 = 8$.
• Interior lattice points: triangle is small; checking shows $i = 3$.
• Pick's area $= 3 + \frac{8}{2} - 1 = 3 + 4 - 1 = 6$.
• Formula area $= \frac{1}{2} \times 4 \times 3 = 6$. ✓

Try it yourself

1. An L-shape is formed by a $7 \times 3$ rectangle and a $4 \times 2$ rectangle. Find its area (assume they fit cleanly).
2. A rectangular room is $12$ m by $9$ m. A square pillar of side $2$ m takes some floor space. What is the floor area you can walk on?
3. A path of $2$ m wide runs along the inside of a $20$ m by $15$ m garden. Find the path area.
4. A T-shape is formed by stacking a $6 \times 2$ rectangle on top of a $4 \times 2$ rectangle (centred). Find the total area.
5. A leaf covers $20$ full grid squares and $12$ partial squares on $1$-cm graph paper. Estimate its area.
6. A picture frame has outer size $30$ cm by $20$ cm and inner picture $24$ cm by $14$ cm. Find the frame area.
7. Pick's theorem: a triangle has vertices at $(0,0)$, $(5,0)$, $(0,5)$. Find its area (a) by formula and (b) using Pick's theorem.
8. Find the area of a rectangle $4 \times 6$ with a corner triangle of legs $2$ and $3$ cut out.

Activity

Map area estimate. Pick a small state of India on a map , say Goa or Kerala. Place a transparent grid over it (or trace the state onto graph paper). Count the unit squares to estimate the area in "map units". If the map's scale tells you how many real km each grid square stands for, convert your estimate to km². Compare with the official area.