Grade 6 Math: Area of Composite Figures

Grade 6 focus: A composite figure (or compound shape) is made from simpler figures—often rectangles and triangles. You find total area by adding areas of non-overlapping parts, or by subtracting a “cut-out” from a larger rectangle.

Video lesson: Watch this Math with Mr. J example of area for composite figures made from rectangles.

Strategy: Decompose

1. Draw dashed lines to split the shape into rectangles/triangles you recognize.
2. Label unknown lengths using properties of rectangles (opposite sides equal).
3. Compute each area, then add (or subtract for holes).

Example (add regions)

An L-shape can split into two rectangles. Add their areas.

Example (subtract)

A rectangle with a rectangular hole: area = big rectangle − hole.

Explain your reasoning

Grade-level tasks often ask you to justify decompositions. Sketch, label, and show each area calculation.

Common mistakes

• Double-counting overlapping regions.
• Missing a hidden segment length needed for a sub-figure.
• Using perimeter where area is required.

Composite Figures: Complete Decomposition Guide

Master the systematic approach: identify basic shapes within the composite figure, separate them visually with dividing lines, calculate each area using standard formulas, and combine results. This method works for any complexity level. Examples include rectangle plus triangle (house shape), L-shapes created by subtracting one rectangle from another, and more complex multi-part figures.

Worked Examples with Full Reasoning

Rectangle plus triangle: 8×5=40 plus (1/2)x8x3=12 equals 52 cm². L-shaped figure: 12×10=120 minus 4×3=12 equals 108 cm². Trapezoid plus rectangle: 6×4=24 plus (1/2)(6+4)x3=15 equals 39 cm². Each example shows identifying shapes, calculating areas, and combining.

Review triangles, trapezoids, and compound figures for additional practice and related concepts.

Mastering Composite Figures: Decomposition Strategy

Composite figures are shapes constructed by combining two or more basic geometric forms like rectangles, triangles, circles, and trapezoids. Rather than seeking an obscure formula for an unusual shape, you employ decomposition: identify the basic shapes, calculate the area of each, and combine the results. This systematic approach works for any complexity level and teaches fundamental geometric reasoning.

The Decomposition Process

Begin by examining the composite figure carefully. Identify which basic geometric shapes comprise the overall figure. Visualize dividing lines that separate the composite figure into component shapes. Assign dimensions to each component based on the given measurements. Calculate the area of each basic shape using standard formulas. Add all areas together if combining shapes. Subtract removed areas if a piece has been cut out from a larger shape. Finally, verify that your units are correct (square centimeters, square inches, etc.) and that the final answer is reasonable given the figure’s approximate size.

Worked Example: Rectangle and Triangle Composition

A house-shaped figure consists of a rectangular base and a triangular roof. The rectangle measures 8 cm wide by 5 cm tall. The triangle sits on the rectangle’s top edge, with a base of 8 cm and height of 3 cm. Calculate the rectangle area: length times width equals 8 times 5 equals 40 cm². Calculate the triangle area: one-half times base times height equals one-half times 8 times 3 equals 12 cm². Add the areas: 40 + 12 equals 52 cm² total. This is the area of the entire house-shaped composite figure.

Worked Example: L-Shaped Figure Using Subtraction

An L-shaped courtyard can be calculated as a large rectangle with a smaller rectangular section removed from one corner. The outer dimensions are 12 meters by 10 meters, giving a large area of 120 m². The removed rectangular section measures 4 meters by 3 meters, with area 12 m². The L-shaped area equals 120 minus 12 equals 108 m². This method is often simpler than trying to separate the L into two rectangles when one section’s dimensions aren’t explicitly given.

Worked Example: Trapezoid and Rectangle Combination

A composite figure consists of a rectangle (6 cm by 4 cm) with a trapezoid attached to its top. The trapezoid has parallel sides of 6 cm (bottom, touching the rectangle) and 4 cm (top), with a height of 3 cm. Rectangle area: 6 times 4 equals 24 cm². Trapezoid area: one-half times (6 + 4) times 3 equals one-half times 10 times 3 equals 15 cm². Combined area: 24 + 15 equals 39 cm².

Common Composite Figure Patterns

House shapes combining rectangles and triangles appear frequently in geometry. L-shaped or stair-step figures use rectangular decomposition. Shapes with cut-out sections require subtraction of the removed area from a larger shape. Circular sections might be added or subtracted from rectangular bases.

Review our guides on triangles, trapezoids, and compound figures for additional practice and related concepts.

Mastering Composite Figures: Decomposition Strategy

Composite figures are shapes constructed by combining two or more basic geometric forms like rectangles, triangles, circles, and trapezoids. Rather than seeking an obscure formula for an unusual shape, you employ decomposition: identify the basic shapes, calculate the area of each, and combine the results. This systematic approach works for any complexity level and teaches fundamental geometric reasoning.

The Decomposition Process

Begin by examining the composite figure carefully. Identify which basic geometric shapes comprise the overall figure. Visualize dividing lines that separate the composite figure into component shapes. Assign dimensions to each component based on the given measurements. Calculate the area of each basic shape using standard formulas. Add all areas together if combining shapes. Subtract removed areas if a piece has been cut out from a larger shape. Finally, verify that your units are correct (square centimeters, square inches, etc.) and that the final answer is reasonable given the figure’s approximate size.

Worked Example: Rectangle and Triangle Composition

A house-shaped figure consists of a rectangular base and a triangular roof. The rectangle measures 8 cm wide by 5 cm tall. The triangle sits on the rectangle’s top edge, with a base of 8 cm and height of 3 cm. Calculate the rectangle area: length times width equals 8 times 5 equals 40 cm². Calculate the triangle area: one-half times base times height equals one-half times 8 times 3 equals 12 cm². Add the areas: 40 + 12 equals 52 cm² total. This is the area of the entire house-shaped composite figure.

Worked Example: L-Shaped Figure Using Subtraction

An L-shaped courtyard can be calculated as a large rectangle with a smaller rectangular section removed from one corner. The outer dimensions are 12 meters by 10 meters, giving a large area of 120 m². The removed rectangular section measures 4 meters by 3 meters, with area 12 m². The L-shaped area equals 120 minus 12 equals 108 m². This method is often simpler than trying to separate the L into two rectangles when one section’s dimensions aren’t explicitly given.

Worked Example: Trapezoid and Rectangle Combination

A composite figure consists of a rectangle (6 cm by 4 cm) with a trapezoid attached to its top. The trapezoid has parallel sides of 6 cm (bottom, touching the rectangle) and 4 cm (top), with a height of 3 cm. Rectangle area: 6 times 4 equals 24 cm². Trapezoid area: one-half times (6 + 4) times 3 equals one-half times 10 times 3 equals 15 cm². Combined area: 24 + 15 equals 39 cm².

Common Composite Figure Patterns

House shapes combining rectangles and triangles appear frequently in geometry. L-shaped or stair-step figures use rectangular decomposition. Shapes with cut-out sections require subtraction of the removed area from a larger shape. Circular sections might be added or subtracted from rectangular bases.

Review our guides on triangles, trapezoids, and compound figures for additional practice and related concepts.