Mathematics · Euclidean Geometry

Area and Perimeter of Composite 2D Shapes

Reference entry · last updated September 20, 2026

A composite shape is a plane region built by joining simpler shapes such as rectangles, triangles, circles, and trapezoids. Its area equals the sum of the component areas when the components do not overlap, or the difference when one region is removed from another. Its perimeter is the total length of the boundary of the final region: every exposed edge counts once, including the boundary of any hole, and it is not the sum of the component perimeters.

1. First principles

Area is additive over disjoint regions. If a region \(R\) is partitioned into pieces \(R_1, R_2, \dots, R_k\) whose interiors do not overlap, then the total area is:

\( A(R) = A(R_1) + A(R_2) + \cdots + A(R_k) \)

Subtraction is the same principle applied to a hole: if \(R_2\) lies entirely inside \(R_1\), the remaining region has area \(A(R_1) - A(R_2)\). Perimeter is different. It measures the length of the complete boundary of the final region: the outer boundary plus the boundary of every hole. Shared interior edges introduced only by the decomposition are not part of it [1, 2].

2. Component area formulas

Shape	Area	Perimeter
Rectangle	\( A = lw \)	\( P = 2l + 2w \)
Parallelogram	\( A = bh \)	Sum of four sides
Triangle	\( A = \tfrac{1}{2}bh \)	\( P = a + b + c \)
Trapezoid	\( A = \tfrac{1}{2}(b_1 + b_2)h \)	Sum of four sides
Circle	\( A = \pi r^2 \)	\( C = 2\pi r \)
Semicircle	\( A = \tfrac{1}{2}\pi r^2 \)	\( \pi r + 2r \)

The perimeter of a circular arc is the arc length alone. A semicircle's boundary consists of the curved half plus the straight diameter, so its perimeter is \( \pi r + 2r \), not \( \pi r \).

3. Additive and subtractive composition

Decomposition is a choice, and more than one valid decomposition exists. An L-shaped region can be split into two rectangles or treated as a large rectangle with a rectangular corner removed. Both routes give the same area, which is a useful check on the work.

Additive method: split the region into non-overlapping pieces, compute each area, and add.
Subtractive method: enclose the region in a larger simple shape and subtract the areas of the missing parts.

Choose the method that produces fewer, simpler pieces. The total is invariant under the choice.

4. Perimeter of a composite shape

Trace every exposed edge of the final region once and add those segment lengths and arc lengths. The boundary includes the outer contour and the contour of every hole. Interior cut lines introduced only by the decomposition do not belong to the boundary and are not counted. In the additive example below, the shared edge between the two rectangles appears in both component perimeters but contributes zero to the boundary of the union [1].

Removing an interior \(2 \times 2\) square hole from a \(6 \times 4\) rectangle adds the hole's perimeter \(8\) to the rectangle's \(20\), for a total of \(28\). Counting only the outer contour would report \(20\) and miss the hole.

A common error is to sum component perimeters. For two rectangles sharing a side of length \(s\), that sum overcounts the true perimeter by \(2s\), the shared edge counted on both sides.

5. Worked example

Take an L-shape formed by a \(6 \times 4\) rectangle with a \(2 \times 2\) square removed from one corner.

Area: \( 6 \cdot 4 - 2 \cdot 2 = 24 - 4 = 20 \) square units.
Perimeter: tracing the remaining boundary gives segments \(6\), \(4\), \(4\), \(2\), \(2\), \(2\), totaling \(20\) units.

Decompose the L-shape into a \(6 \times 2\) rectangle and a \(4 \times 2\) rectangle. Their areas sum to \(12 + 8 = 20\), as required. Their perimeters sum to \(16 + 12 = 28\), which exceeds the true perimeter by \(8\). The excess is twice the shared edge of length \(4\), counted once in each component perimeter. Tracing every exposed edge is the reliable method.

References

^H.S.M. Coxeter, Introduction to Geometry, 2nd ed., John Wiley & Sons, 1969.
^Euclid, Elements, Books I and VI (Area of rectilinear figures and the equality of figures on equal bases).