The Circular Portions Of The Following Figures Are Semicircles

You're staring at a diagram. That's why a rectangle with a curved top. Day to day, a square with two half-circles cut out of opposite sides. Plus, a triangle with a semicircle sitting on each leg. The problem statement reads: The circular portions of the following figures are semicircles.

And your brain does that little freeze thing. You know the circumference formula. Because you know the formula for the area of a circle. But suddenly they're halved, combined, overlapping, or subtracted — and the clean formulas you memorized don't slot in neatly anymore.

This is where most students (and honestly, plenty of adults) get stuck. Not because the math is hard. And because the visual parsing* is hard. The moment a semicircle stops being a standalone shape and starts being a component* of something else, the rules feel like they shift.

They don't. But the way you have to think about them does.

What This Actually Means

When a problem tells you "the circular portions are semicircles," it's giving you a constraint. Every curve you see that looks like a half-circle is exactly a half-circle. On top of that, not a quarter-circle. Not a 60-degree arc. In practice, not "approximately half. " Exactly half.

That means:

• The diameter of each semicircle lies on a straight line segment in the figure
• The radius is half that segment's length
• The curved boundary is exactly 180 degrees of a full circle
• The area of that curved region is exactly half the area of the full circle with that radius

Sounds obvious. Even so, the difference between two parallel lines. But the trap is assuming the diameter* is always given directly. Often it's not. The width of a rectangle minus two cutouts. The diameter might be the side of a square. The hypotenuse of a right triangle. You have to find* the diameter before you can use the semicircle formulas.

And that's the real skill here: not memorizing πr²/2, but recognizing where the radius lives in the diagram.

Why These Problems Show Up Everywhere

Standardized tests love semicircle composites. SAT, ACT, GRE, GMAT, math competitions from AMC to AIME to national olympiads — they all use them. Why?

Because they test three things at once:

1. Consider this: 3. Geometric reasoning — can you deduce missing lengths from the figure's constraints? Now, 2. Also, Formula fluency — do you actually know the area and arc length formulas cold? Synthesis — can you combine pieces (add areas, subtract overlaps, split perimeters) without double-counting or missing a boundary?

It's efficient testing. One diagram, five minutes, hits half the geometry curriculum.

But beyond tests, this shows up in real design. Practically speaking, architecture (arched windows, doorways), engineering (stress distribution in curved beams), manufacturing (sheet metal cutouts), even logo design. The math is the same whether you're scoring points or specifying a waterjet cut.

How to Read a Composite Semicircle Figure

Step 1: Inventory Every Curve

Before you calculate anything, trace every curved boundary with your finger (or mentally). Still, label each one: Semicircle A, Semicircle B... Six? Are there two? Four? Day to day, * Count them. Are any inside* another shape (cutouts) versus on the boundary?

This sounds trivial. On top of that, it's not. Even so, under time pressure, people miss curves. Especially small ones tucked into corners or overlapping larger ones.

Step 2: Identify Every Diameter

For each semicircle, find the line segment that forms its diameter. Sometimes it's labeled. Sometimes it's a side of a polygon. Sometimes it's shared* — two semicircles back-to-back on the same line segment, forming a full circle together. That's a classic pattern: two semicircles on the same diameter = one circle. Area = πr². Perimeter contribution = 2πr (the two arcs together).

If the diameter isn't directly given, what geometric relationship determines it? Is it:

• A side of a square/rectangle?
• The hypotenuse of a right triangle (Thales' theorem: angle inscribed in a semicircle is 90°)?
• The sum/difference of other labeled segments?
• A radius of a larger circle?

Write the diameter in terms of known quantities* for each semicircle. This is your variable map.

Step 3: Decide What the Question Actually Wants

Total area of the shaded region? Perimeter of the whole figure? Also, area between* two semicircles? On the flip side, ratio of two curved areas? The target determines which formulas you combine and whether you add or subtract.

Don't start calculating until you know the destination.

Common Configurations (And How to Handle Them)

The Arbelos (Shoemaker's Knife)

Three semicircles on the same line, all on the same side. Two small ones inside a large one, diameters adding up to the large diameter. The region between* them — the curved triangular shape — is the arbelos.

Archimedes studied this. But the area of the arbelos equals the area of a circle with diameter equal to the distance between the two small semicircles' centers*. Wild, right? But you don't need to memorize that. That's why you just need to see: large semicircle area minus two small semicircle areas. That's it. The magic is that the result simplifies to something clean.

Semicircles on the Sides of a Right Triangle

Classic. In real terms, semicircles drawn outward on each leg and the hypotenuse. Because of that, the sum of the areas of the two smaller semicircles equals the area of the largest one. Practically speaking, why? On top of that, because area scales with radius squared, and the Pythagorean theorem gives you a² + b² = c². The π/8 factor cancels out.

For more on this topic, read our article on how to divide a small number by a big number or check out replace with an expression that will make the equation valid.

This shows up as "prove the shaded areas are equal" or "find the area of the lune" (the crescent between the hypotenuse semicircle and the triangle).

Overlapping Semicircles (Venn-Style)

Two semicircles of equal radius, diameters on the same line, overlapping symmetrically. So naturally, the overlap region is lens-shaped. Area = 2 × (sector area − triangle area). You need the central angle of the sector, which comes from the geometry of the overlap (usually 60° or 90° in contest problems).

Semicircles Cut Out of a Rectangle/Square

A rectangle. Also, two semicircles cut out of the top and bottom edges, diameters equal to the width. Still, or cut from the left and right edges. Worth adding: or four quarter-circles at the corners (which is the same as two semicircles). And shaded area = rectangle area − semicircle areas. Perimeter = straight edges + curved edges.

Watch for: are the cutouts removed* (holes) or just boundaries* (the figure includes the curved indent)? The area calculation is the same. The perimeter differs — holes add interior boundary length.

Concentric Semicircles (Rainbow Bands)

Multiple semicircles sharing the same center, different radii. Like a target cut in half. Area of a band = difference of two semicircle areas = (π/2)(R² − r²). Perimeter of a band = outer arc + inner arc + two radial edges.

It looks simple on paper, but it's easy to get wrong.

The Formulas

The Formulas

Here is the toolkit. Keep these in your head, not on a cheat sheet.

Semicircle area: $A = \frac{\pi r^2}{2}$

If you're given the diameter $d$, substitute $r = d/2$ and get $A = \frac{\pi d^2}{8}$.

Semicircle arc length (the curved part only): $\ell = \pi r = \frac{\pi d}{2}$

Semicircle perimeter (the full boundary — arc + diameter): $P = \pi r + 2r = r(\pi + 2)$

This distinction matters constantly. When a problem asks for "perimeter," it almost always means the total boundary length you'd trace if you walked around the figure. So naturally, for a lone semicircle, that includes the straight diameter. For a shaded region inside* a composite figure, the diameter edge might be interior and therefore excluded.

Lune (crescent) area: A lune is the region inside one semicircle but outside another. Its area is typically: $A_{\text{lune}} = A_{\text{semicircle}} - A_{\text{overlap or triangle}}$

In the right-triangle lune configuration, the lune area equals the triangle area. Because of that, that's the elegant result. You get there by writing the semicircle areas in terms of the sides, subtracting the circular segment areas, and watching the Pythagorean theorem collapse everything into $A_{\triangle}$.

Annular band (between two concentric semicircles): $A_{\text{band}} = \frac{\pi}{2}(R^2 - r^2)$

Sector of a semicircle: If the central angle is $\theta$ (in radians), the area of that sector within a semicircle of radius $r$ is: $A_{\text{sector}} = \frac{1}{2}r^2\theta$

We're talking about the same sector formula as for full circles — it doesn't care whether the boundary is a semicircle or a full circle. What matters is the angle you're sweeping.

Overlap of two equal semicircles (lens): $A_{\text{lens}} = 2\left(\frac{r^2}{2}\theta - \frac{r^2}{2}\sin\theta\right) = r^2(\theta - \sin\theta)$

where $\theta$ is the half-angle subtended at each center by the chord that defines the overlap. For the symmetric case where two semicircles of radius $r$ have their diameters on the same line and overlap so that each center lies on the other semicircle's arc, $\theta = \pi/3$ (60°) and the lens area simplifies nicely.

The Problem-Solving Protocol

When you face a shaded-region problem involving semicircles, follow this sequence:

1. Identify the components. Name every semicircle, every straight edge, every intersection point. Label radii and diameters.
2. Determine the operation. Is the shaded region the sum of parts, the difference between a larger shape and cutouts, or a more complex combination (e.g., a semicircle minus a triangle plus another semicircle)?
3. Write the area expression symbolically first. Keep $\pi$ and $r$ as symbols. Don't plug in numbers until the last step. This is where cancellation and simplification happen — and where you avoid arithmetic errors.
4. Check the perimeter separately. Area and perimeter are independent questions. A figure can have a simple area and a complicated perimeter, or vice versa.
5. Look for hidden relationships. Pythagorean triples, tangent lines, inscribed angles, symmetry. The geometry usually hands you the relationships; your job is to see them.

Final Thought

Semicircle problems feel intimidating because the curves introduce $\pi$, and $\pi$ makes everything look messy. When you compose these shapes — nesting them, overlapping them, cutting them from polygons — the operations remain addition and subtraction. The area of a semicircle is just half the area of a circle. The arc length is just half the circumference. But the underlying structure is almost always simple arithmetic dressed in curved clothing. The challenge is purely geometric: seeing which pieces belong to the shaded region and which don't.

Master that vision, and no semicircle configuration — from Archimedes' arbelos to a lune on a right triangle — will resist you.