Finding the Area of Composite Figures

A composite figure combines two or more basic shapes. The calculation is usually straightforward once the figure is partitioned without gaps or overlap.

Split the figure into familiar pieces

An L-shaped floor can be split into two rectangles. Find each rectangle's area with length × width, then add the results. If a triangle is part of the shape, use one-half × base × height, with the height perpendicular to the selected base.

Subtract a missing region

Sometimes it is easier to find the area of the outer rectangle and subtract a cutout. If a 12-by-8 rectangle has a 4-by-3 corner removed, its area is 96 − 12 = 84 square units. The dimensions of the removed region must match the actual cutout.

Use dimensions carefully

• Mark every length on the diagram and solve for missing lengths before finding area.
• Do not confuse perimeter lengths with perpendicular heights.
• Use each region once; overlapping partitions double-count area.
• Keep units consistent and square them in the final answer.
• Estimate the area to check that the result is plausible.

Example with a rectangle and triangle

A sign has a 10-by-6 rectangular base and a triangular top with base 10 and height 3. The total area is 10 × 6 + 1/2 × 10 × 3 = 60 + 15 = 75 square units. The triangle's height must be perpendicular to its base.

Find missing lengths before finding areas

A diagram may show only the outside dimensions, while an interior cutout needs a length found by subtraction. If a large rectangle is 12 units wide and an adjoining section is 7 units wide, the remaining width is 5 units. Mark that result on the sketch before applying a formula. For a triangle, use the perpendicular height rather than a slanted side; for a semicircle, use half the diameter to find its radius. Unlabeled internal measurements can often be recovered from aligned edges.

Choose a partition that avoids overlap

A composite region can often be divided in several ways. Pick pieces whose dimensions are known and that cover the whole region exactly once. If two rectangles overlap, adding their full areas counts the shared region twice; subtract the overlap or choose a different partition. A careful sketch with dividing lines helps reveal gaps and double-counting before any calculation begins.

Combine straight and curved boundaries

For a rectangle with a semicircle attached, total area is the rectangle's area plus one-half the circle area, provided the shapes meet along an edge rather than overlap. If the semicircle has diameter 10, its radius is 5 and its area is (1/2)π(5²) = 12.5π square units. Do not add the diameter as an area, and do not use the semicircle's curved arc when calculating a perimeter unless the question asks for the outside boundary.

A circular cutout in a larger shape is handled by subtraction. Subtract πr² from the enclosing shape's area, using the cutout's radius. If exact form is acceptable, retain π; if a decimal is required, use the requested approximation and round at the end. The units of all component areas must match before adding or subtracting.

Read the diagram's scale and labels

Do not assume a drawing is to scale unless the prompt says it is. Use the given dimensions and geometric relationships. A missing side may be equal to another because the figure is a rectangle, but visual similarity alone is not proof. When a shape is composed of rectangles, parallel edges and right angles can help determine missing lengths; when a triangle appears, the altitude must meet its chosen base at 90 degrees.

Estimate and check

1. Estimate the enclosing rectangle or other simple shape as an upper bound if regions are removed.
2. Estimate each piece before calculating to catch misplaced decimals.
3. Check that the total is larger than each included piece and smaller than the enclosing area when appropriate.
4. Use square units, and convert units before combining measurements.
5. Re-read whether the question asks for area, perimeter, or only a shaded or exposed region.

Break the figure into familiar pieces

A composite figure is made from two or more basic shapes. To find its area, divide it into rectangles, triangles, trapezoids, or circles whose dimensions can be identified. Add nonoverlapping pieces or subtract a missing part from a larger enclosing shape. Draw internal guide lines and label dimensions before calculating.

For an L-shaped region that fits inside a 10 by 8 rectangle with a 4 by 3 corner removed, calculate 10×8 − 4×3 = 80 − 12 = 68 square units. The subtraction method can be simpler than splitting the L into two rectangles. Check that the removed section is truly outside the region and that no area is counted twice.

Use only dimensions that match each subshape

A dimension may apply to the whole figure rather than one piece. If a composite shape is split, derive missing segment lengths from the total: a full width of 12 with a cutout width of 5 leaves 7 for the remaining section. Align boundaries in a sketch and label the subtraction. For a triangle, use the perpendicular height, not a sloping side, in A = ½bh.

For composite shapes with circles, distinguish radius from diameter and determine whether the curved part is a semicircle, quarter circle, or full circle. A semicircle’s area is half of πr². If a circular part is removed, subtract its area. Use the same units for every measurement and retain π until rounding is requested.

Check boundaries, units, and reasonableness

Area measures the interior, so shared internal edges do not add or subtract area on their own. Perimeter is a different task: trace only the outside boundary of the combined figure. A line used to split a shape for area is not part of the external perimeter unless it is also an exposed edge.

The composite area should not exceed the area of a known enclosing rectangle, and it should be positive. Use square units. If dimensions are mixed, convert them before multiplying. Estimating the rough fraction of the bounding rectangle that is filled can reveal a misplaced cutout or arithmetic error.

• Sketch and divide the figure into nonoverlapping basic shapes.
• Add parts or subtract a missing region from a larger shape.
• Derive missing lengths from total dimensions.
• Use perpendicular heights and correct circle fractions.
• Report square units and compare with a bounding-shape estimate.

Exam takeaway

Partition into basic shapes or subtract cutouts, use the correct formula and perpendicular dimensions, and keep units in square measure.