Calculadora de Figuras Compuestas
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A composite figure (also called a compound shape or compound figure) is any 2D shape built by combining two or more standard geometric shapes — rectangles, triangles, circles, trapezoids, or sectors. Real-world examples are everywhere: an L-shaped room floor plan, a sheet of metal with a circular hole, a stadium track outline, an arched window, a sign that combines a triangle on top of a rectangle.
The trick to solving composite figures is decomposition: split the complex shape into non-overlapping simpler shapes you already know how to solve, compute each sub-area, then add them together. For shapes with holes or cut-outs (like the metal sheet with a hole), you subtract the cut-out area from the bounding shape. The AI calculator on this page does the decomposition automatically — describe the figure in plain English, or upload a photo, and it returns the total area, total perimeter, and a step-by-step breakdown showing each sub-shape it identified.
If you are working on homework, you can also use the worked examples below as templates. Each example shows a different composition pattern (additive, subtractive, with curves) so you can match your figure to the closest example and adapt the approach.
Ejemplos resueltos
Ejemplo 1: Forma de L (aditiva — dos rectángulos)
An L-shaped office floor plan: the long arm is 12 m × 4 m, and the short arm is 5 m × 3 m attached to the bottom-right.
Decomposition: two non-overlapping rectangles.
Long arm area = 12 × 4 = 48 m²
Short arm area = 5 × 3 = 15 m²
Total area = 48 + 15 = 63 m²
Perimeter: walk the outline once. From bottom-left corner: right 12, up 4, left 7 (12 − 5), up 3, left 5, down 7 (4 + 3) = total 38 m. (Don't double-count the shared internal edge — it isn't part of the outer perimeter.)
Ejemplo 2: Dona (sustractiva — círculo menos círculo)
A washer (donut shape) with outer radius 5 cm and inner radius 2 cm.
Decomposition: outer circle minus inner circle.
Outer area = π × 5² = 25π ≈ 78.54 cm²
Inner hole = π × 2² = 4π ≈ 12.57 cm²
Net area = 25π − 4π = 21π ≈ 65.97 cm²
Perimeter: outer circumference + inner circumference (both edges visible) = 2π(5) + 2π(2) = 14π ≈ 43.98 cm.
Ejemplo 3: Estadio (rectángulo + dos semicírculos)
A running track outline: a rectangle 100 m × 40 m with a semicircle of diameter 40 m on each short end.
Decomposition: central rectangle + two semicircles (which together form one full circle).
Rectangle area = 100 × 40 = 4,000 m²
Two semicircles = π × 20² = 400π ≈ 1,256.64 m²
Total area = 4,000 + 1,256.64 ≈ 5,256.64 m²
Perimeter: two long sides + circumference of one full circle = 2(100) + 2π(20) = 200 + 40π ≈ 325.66 m.
Ejemplo 4: Ventana con arco superior (rectángulo + semicírculo)
A church window: a rectangle 1.5 m wide × 2 m tall topped by a semicircle of diameter 1.5 m.
Decomposition: rectangle + half a circle.
Rectangle area = 1.5 × 2 = 3 m²
Semicircle area = ½ × π × 0.75² = 0.28125π ≈ 0.88 m²
Total area ≈ 3.88 m²
Perimeter (the outer outline only — exclude the shared straight edge between rectangle and semicircle): 2(2) + 1.5 + π(0.75) ≈ 7.86 m.
In-Depth Tutorial: Calculadora de Figuras Compuestas
Composite figures (a.k.a. compound shapes) are 2D figures built by combining two or more standard geometric shapes. The defining skill is decomposition — recognising which simpler shapes you can split the composite into, computing each sub-area, and then adding (or subtracting) them. This tutorial walks the four most common decomposition patterns + the perimeter trick, so you can both describe figures to the AI accurately and double-check its work.
The four decomposition patterns
- Additive (L-shape, T-shape, signs): the composite is several non-overlapping standard shapes touching edge-to-edge. Sum the areas; for perimeter, sum the outer edges only.
- Subtractive (donut, perforated plate): a smaller shape is cut out of a larger one. Subtract the cut-out area from the bounding area; for perimeter, BOTH the outer edge and the cut-out edge count.
- Curved (stadium track, arched window): straight + curved boundaries (semicircles, sectors). Compute the straight parts as rectangles/triangles and the curved parts as appropriate circular fragments.
- Coordinate-defined (irregular polygon from vertex list): if the composite is a single closed polygon with 5+ vertices, the Shoelace formula gives the area directly without explicit decomposition.
Worked Example 1 — Additive (L-shape)
Figure: An L-shaped office floor plan. Long arm 12 m × 4 m, short arm 5 m × 3 m attached to the bottom-right of the long arm.
Decomposition: 2 rectangles. Long arm area = 48 m². Short arm = 15 m². Total = 63 m².
Perimeter: walk the outer outline once and don't double-count internal edges. Starting bottom-left, going clockwise: right 12, up 4, left 7 (= 12 − 5), up 3, left 5, down 7 (= 4 + 3). Total = 38 m.
Worked Example 2 — Subtractive (donut/washer)
Figure: A washer (donut) with outer radius 5 cm and inner radius 2 cm.
Decomposition: outer circle MINUS inner circle. Outer area = π × 5² = 25π ≈ 78.54 cm². Inner hole = π × 2² = 4π ≈ 12.57 cm². Net area = 25π − 4π = 21π ≈ 65.97 cm².
Perimeter: both the outer circumference AND the inner hole's circumference count, because both are visible boundaries. Total = 2π(5) + 2π(2) = 14π ≈ 43.98 cm. (Naive intuition says "perimeter is just the outside" — wrong for figures with enclosed holes.)
Worked Example 3 — Curved (stadium track)
Figure: Athletics track: a 100 m × 40 m rectangle with a semicircle (diameter 40 m) on each short end.
Decomposition: rectangle + two semicircles. The two semicircles together form one full circle of radius 20.
Rectangle = 100 × 40 = 4,000 m².
Full circle = π × 20² = 400π ≈ 1,256.64 m².
Total = ≈ 5,256.64 m².
Perimeter: two long sides + one full circle's circumference (the two semicircle arcs together) = 2(100) + 2π(20) = 200 + 40π ≈ 325.66 m. The short rectangle sides are NOT counted — they're shared with the semicircles' diameters and lie inside the outline.
The perimeter trick
Perimeter is the most error-prone part of composite figures because students reflexively want to "add all the edges" — including shared internal ones. The rule is simple:
Perimeter = sum of EXPOSED boundary segments only.
Walk the outline once with a finger. Every segment your finger touches counts. Edges where two sub-shapes meet (and the boundary is "inside" the figure) DON'T count. For figures with enclosed holes (donuts, ring-shaped frames), both the outer outline AND the hole edges count.
How to describe a figure to the AI
The clearer your description, the better the AI's decomposition. Include:
- Sub-shape names (rectangle, triangle, semicircle) and their dimensions in consistent units.
- How they connect (e.g. "L-shape: 12×4 horizontal rectangle, with a 5×3 rectangle attached at the bottom-right").
- Any cut-outs (e.g. "with a circular hole of radius 2 in the centre").
- Orientation if it matters (e.g. "with the semicircle facing up").
You can also upload a photo (sketch, screenshot, textbook scan). The AI Vision model reads the figure, identifies labels, and runs the same decomposition.
Common mistakes
- Double-counting shared edges in perimeter. Walk the outer outline; internal seams between sub-shapes don't contribute.
- Forgetting the hole's edge contributes to perimeter for ring shapes. If the hole is fully enclosed, both edges show.
- Mixing units within one figure (e.g. rectangle in m, hole radius in cm). Convert everything to one unit before computing.
- Using the slanted side as height in triangle sub-shapes. Triangle area requires the perpendicular height, not the slanted side.
When to use a different calculator
- For 3D composite solids (cylinder on top of cube, sphere with cylindrical hole), use the AI Geometry Problem Solver — it handles volume and surface-area composition using the same additive/subtractive principles.
- For simple regular polygons (one shape, not composite), the dedicated polygon / triangle / circle calculators are faster and cheaper (no AI credits).
- For irregular polygons defined by vertex coordinates, the Polygon Coordinates Calculator uses the Shoelace formula directly, no AI needed.
Preguntas frecuentes – Calculadora de Figuras Compuestas
A composite figure (also called a compound shape) is a 2D figure built by combining two or more standard geometric shapes such as rectangles, triangles, circles, semicircles, or trapezoids. Examples include L-shaped rooms, stadium tracks, donut washers, arched windows, and sheets of material with cut-out holes.
Use the decomposition method: (1) split the figure into non-overlapping simpler shapes you can solve individually; (2) compute each sub-area using its standard formula; (3) sum the parts for the total area. If the figure has holes or cut-outs, subtract those areas from the bounding shape. The 4 worked examples above show this method applied to L-shape, donut, stadium, and arched window.
They mean the same thing — both terms describe a 2D figure built from two or more simpler shapes. "Composite figure" is the more common term in US geometry textbooks; "compound shape" is more common in UK and AU curricula. This calculator handles both terms identically.
Walk the outer outline once and sum the visible edge lengths. Do NOT include internal shared edges between the sub-shapes — those are inside the figure and aren't part of its perimeter. For figures with curved sides (semicircles, arcs), use the relevant fraction of the circle's circumference (e.g. πr for a semicircle).
Yes — describe the cut-out clearly (e.g. "minus a 4×3 rectangle from the lower-left corner" or "with a circular hole of radius 2 in the centre") and the AI subtracts the hole area from the bounding total. For perimeter when a hole is fully enclosed, both the outer outline AND the hole's edge contribute to the total perimeter (since both are visible boundaries).
Curved boundaries are handled — describe arcs by their radius and central angle (or as a fraction of a full circle, e.g. "quarter circle of radius 5"), and the AI applies the correct sector or segment formula. Photo upload also captures curves directly. For full circles in cut-outs, area = πr² and circumference = 2πr.
If your composite is an irregular polygon defined by its vertices (rather than rectangles + circles), list the vertex coordinates clockwise starting from any vertex (e.g. "vertices at (0,0), (10,0), (10,4), (6,4), (6,8), (0,8)"). The AI applies the Shoelace Formula to compute the area directly. For more than ~8 vertices, the dedicated Polygon with Coordinates Calculator is faster.
This calculator is optimised for 2D composite figures (area and perimeter). For 3D composite solids — such as a cylinder on top of a cube, or a sphere with a cylindrical hole drilled through it — use the AI Geometry Problem Solver, which handles volume and surface-area composition for solids using the same additive/subtractive decomposition principles.
Any consistent unit. Describe your figure in cm / m / inches / feet / etc. — the AI returns results in the same unit you described. Mixing units within one figure (e.g. a rectangle in metres and a hole in centimetres) will give wrong answers; convert to one unit first.
Yes — every AI Solve request returns a full breakdown: each sub-shape identified, the formula applied, the intermediate area, and how they combine into the total. This is the same format used in the worked examples above, so you can copy the structure into your homework.
The AI computes to standard floating-point precision (~15 significant digits) and rounds to 4 decimal places by default. For irrational results involving π, you receive both the exact symbolic form (e.g. 21π) and the decimal approximation (≈ 65.97). For homework requiring exact answers, use the symbolic form.
Each calculation uses 3 credits, including photo-based requests. New accounts receive 30 free credits — enough for 10 composite-figure solves.