Composite & Shaded-Region Area

A composite figure is built from familiar shapes — a rectangle with a semicircle stuck on, an L-shape, a square with a circle punched out. There's no single formula for these, and that's the point: the skill is breaking the figure into pieces you already have formulas for.

Every composite problem is one of two moves. Addition: the figure is made of pieces glued together, so add their areas. Subtraction: the figure is a big shape with a piece removed, so subtract the hole from the whole.

Adding pieces

Slice the figure along natural lines into rectangles, triangles, trapezoids, and half or quarter circles. Find each piece's area separately, then add. An L-shape, for instance, splits into two rectangles with one straight cut — and there are usually two valid ways to cut it, both giving the same total.

The only tricky part is finding missing side lengths. Opposite sides of the figure must account for the same total distance: if the whole bottom is $8$ and a top piece covers $4$ of it, the remaining width is $8 - 4 = 4$ . In the L-shape below, one straight dashed cut splits the figure into two rectangles whose areas you can add.

Subtracting holes

A shaded region is often what's left after cutting a shape out of a bigger one — a square with a circular hole, a ring between two circles. Compute the outer area, compute the removed area, and subtract: shaded $=$ whole $-$ hole.

Keep exact forms as long as possible. If a square of side $10$ loses a circle of radius $3$ , write the answer as $100 - 9\pi$ and only convert to a decimal at the end if asked.

Semicircles and quarter circles

Half and quarter circles show up constantly in composite figures. A semicircle's area is $\displaystyle \frac{1}{2}\pi r^2$ and a quarter circle's is $\displaystyle \frac{1}{4}\pi r^2$ . Watch the given measurement: if the flat side of a semicircle sits on a side of length $6$ , that $6$ is the diameter, so the radius is $3$ .

Worked examples

Example 1: an L-shape (addition)

An L-shaped figure is $8$ units across the bottom and $6$ units tall on the left. The bottom strip is $2$ units tall, and the left column is $4$ units wide. Find the total area.

Cut along the inside corner into two rectangles

Bottom strip:

8

wide,

2

tall

A_1 = (8)(2) = 16

Left column above the strip:

4

wide,

6 - 2 = 4

tall

A_2 = (4)(4) = 16

Add the pieces

A = 16 + 16 = 32

Answer: $A = 32$ square units

Example 2: square with a circle removed (subtraction)

A square has a side of $10$ , and a circle of radius $3$ is cut out of it. Find the area of the shaded region that remains.

Area of the square

A_{\text{square}} = 10^2 = 100

Area of the removed circle

A_{\text{circle}} = \pi (3)^2 = 9\pi

Subtract the hole from the whole

A = 100 - 9\pi

Approximate if a decimal is needed

A \approx 100 - 28.3 = 71.7

Answer: $A = 100 - 9\pi \approx 71.7$ square units

Example 3: rectangle plus semicircle

A figure is a $12$ by $8$ rectangle with a semicircle attached to one $8$ -unit side. Find the total area to the nearest tenth.

Area of the rectangle

A_1 = (12)(8) = 96

The semicircle's diameter is

8

, so its radius is

4

r = \dfrac{8}{2} = 4

Area of the semicircle

A_2 = \dfrac{1}{2}\pi (4)^2 \approx 25.1

Add the pieces

A \approx 96 + 25.1 = 121.1

Answer: $A \approx 121.1$ square units

Try one yourself

Common questions

How do I decide whether to add or subtract?

Look at how the figure is built. If it's pieces joined together, add. If it's a larger shape with a region cut away — a hole, a notch, an unshaded part — subtract. Some figures need both: add two pieces, then subtract a hole.

Does it matter how I split the figure?

No — any split into familiar shapes gives the same total. Pick the cut that makes the side lengths easiest to find. If your two ways of splitting give different answers, a side length was read wrong somewhere.

How do I find the sides the diagram doesn't label?

Use the sides it does label. Parallel edges across the figure must add up to the same total — if the full height is $6$ and one piece takes $2$ of it, the rest is $4$ . Every missing length is a sum or difference of given ones.

What radius do I use for an attached semicircle?

Half of the side it's attached to. The flat edge of the semicircle is a diameter, so a semicircle sitting on a side of length $6$ has radius $3$ . Using $6$ as the radius is the classic mistake.

Want the video version?

Allday Everyday Math has video lessons, practice, and an AI tutor for every topic, Pre-Algebra through Algebra 2.

Try it for $1

Composite & Shaded-Region Area

Adding pieces

Subtracting holes

Semicircles and quarter circles

Worked examples

Example 1: an L-shape (addition)

Example 2: square with a circle removed (subtraction)

Example 3: rectangle plus semicircle

Try one yourself

Common questions

Keep learning

Area & Perimeter of Triangles & Quadrilaterals

Sectors & Segment Area

Area of Regular Polygons

Perimeter & Area on the Coordinate Plane