Geometry Foundations

Area of Composite Figures

Finding the area of two-dimensional figures composed of triangles, quadrilaterals, and circles.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Circumference & Area of Circles

Breaking a Complicated Shape into Familiar Pieces

An oddly shaped room combines a rectangular section with a semicircular reading nook on one end. Before reading on, think about a strategy for finding its total floor area — one that doesn't require a brand-new formula for this one-of-a-kind shape.

Definition — Composite Figure

A shape made up of two or more simpler, familiar shapes joined together, such as a rectangle combined with a triangle or a circle.

The strategy doesn't need a new formula at all: decompose the complicated shape into simpler pieces you already know how to measure — rectangles, triangles, circles — find each piece's area separately, and add them together. If part of the shape is a cutout, like a hole, subtract that piece's area instead.

Worked Example — Adding the Areas of a Composite Figure

The room from the opening question: a 12-by-10-foot rectangle, plus a semicircular nook with a 6-foot diameter attached to one 10-foot wall. Rectangle area: 12 × 10 = 120 square feet. Semicircle area (half of a full circle with radius 3): (π × 3²) ÷ 2 ≈ 14.1 square feet. Total area: 120 + 14.1 ≈ 134.1 square feet.

Worked Example — Subtracting a Cutout Area

A square patio measures 10 feet on each side, with a circular fountain of radius 2 feet built into the middle (not part of the walkable area). Find the walkable patio area. Square area: 10 × 10 = 100 square feet. Circle area: π × 2² ≈ 12.57 square feet. Walkable area: 100 − 12.57 ≈ 87.4 square feet.

Geometry Canvas

Point mode (click to add a point)LabelsMeasurementsSnap to grid

Construct

Tool

Objects

1.
Color
P1
P1: a free point, draggable on the plane
2.
Color
P2
P2: a free point, draggable on the plane
3.
Color
P3
P3: a free point, draggable on the plane
4.
Color
poly1
poly1: the polygon through P1, P2, P3

Measurements

poly1area = 15perimeter = 17.66

Tip

Sketch dashed lines showing exactly how the composite shape splits into simpler pieces before calculating anything — it's easy to miscount or double-count a region when working from the final combined shape alone.

Common Mistakes

Adding a cutout region's area instead of subtracting it, treating a hole as if it were extra floor space.
A region that's removed from the overall shape, like a fountain or hole, subtracts from the total area rather than adding to it.
Using the same measurement, like a diameter, for two different pieces of a composite figure without checking whether each piece actually shares that exact dimension.
Check each simpler shape's own dimensions carefully — a composite figure's pieces don't automatically share matching measurements just because they're joined together.

Key Takeaways

A composite figure's area is found by decomposing it into simpler shapes with known area formulas.
Add the areas of shapes that make up the figure; subtract the areas of any cutout regions.
Sketching how a composite figure splits apart before calculating helps avoid missed or double-counted regions.

Summary

Composite figures show that a small set of area formulas can measure almost any shape, once it's broken into familiar pieces. The final unit shifts from geometry to statistics and probability, starting with three ways to describe the center of a data set.

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