Find Area of Composite Figures - 3rd Grade Math

Refer to the figure. Two friends measured the same rectilinear shape. Alex broke it into a $5$ ft by $8$ ft rectangle and a $3$ ft by $2$ ft rectangle. What is the total area of the shape?
Answer: $46$ square feet $5 \times 8 = 40$ and $3 \times 2 = 6$ . Total: $40 + 6 = 46$ sq ft. B ignores the small rectangle. A adds side lengths. D multiplies wrong numbers.
What is the total area of the concrete pad?
Answer: $156$ square yards The large rectangle has an area of 12 times 15, or 180 square yards. The removed section has an area of 4 times 6, or 24 square yards. Subtracting the removed section, 180 minus 24 equals 156 square yards, so C is correct. Choice A (132) subtracts too much area. Choice B (144) does not account for the full removed section. Choice D (168) does not subtract the removed section at all.
Look at the shaded region in the figure below. What is the area of the shaded part of this rectangle?
Answer: $39$ square inches The large rectangle is 9 in × 7 in = 63 square inches. The unshaded rectangle in the top right is 4 in × 6 in = 24 square inches. The shaded area = 63 - 24 = 39 square inches.
Refer to the L-shaped figure. The figure can be split into two rectangles. One rectangle is $8$ feet by $3$ feet, and the other rectangle is $4$ feet by $5$ feet. What is the total area of the figure?
Answer: $44$ square feet Area of first rectangle: $8 \times 3 = 24$ sq ft. Area of second rectangle: $4 \times 5 = 20$ sq ft. Total: $24 + 20 = 44$ sq ft. A only counts the second rectangle. B only counts the first. D adds all side lengths incorrectly.
Use the figure. The shape is divided into three non-overlapping rectangles with areas of $14$ sq in, $9$ sq in, and $6$ sq in. What is the total area of the shape?
Answer: $29$ square inches Add all three: $14 + 9 + 6 = 29$ sq in. A, B, and C each leave out one of the three parts.
Maria's bedroom floor is made of two connected rectangles. One rectangle is $6$ meters long and $4$ meters wide. The other rectangle is $3$ meters long and $2$ meters wide. What is the total area of Maria's bedroom floor?
Answer: $30$ square meters When a shape is made of two connected rectangles, you can find the total area by calculating each rectangle's area separately and then adding them together. Remember that the area of a rectangle is found by multiplying length by width, and area is always measured in square units. Start with the larger rectangle: $6 \times 4 = 24$ square meters. Then the smaller rectangle: $3 \times 2 = 6$ square meters. Adding them gives $24 + 6 = 30$ square meters, which matches choice C. Choice A ( $15$ ) comes from adding length and width instead of multiplying — that's the formula for perimeter thinking, not area. Choice B ( $24$ ) is only the area of the larger rectangle; it forgets to include the smaller one. Choice D ( $48$ ) doubles the larger rectangle's area, which might happen if you accidentally multiplied $6 \times 4 \times 2$ , or confused area with something else entirely. A helpful strategy for composite shape problems: break the figure into simple rectangles, find each area one at a time, and then add. Draw a quick sketch if you can — it helps you avoid missing a piece or double-counting. And always double-check that you multiplied (for area) rather than added (for perimeter).
A school playground is made of two rectangles. The first section measures $5$ yards by $5$ yards. The second section measures $7$ yards by $4$ yards. What is the total area of the playground?
Answer: $53$ square yards $5 \times 5 = 25$ and $7 \times 4 = 28$ . Total: $25 + 28 = 53$ sq yd. A only counts the second section. B adds side lengths. D multiplies incorrect values.
The figure shows a garden bed. To find its area, Jin decomposed it into rectangles in two different ways. Method 1 gives him rectangles with areas $8$ , $6$ , and $4$ square units. Method 2 gives him rectangles with areas $10$ and $8$ square units. Which statement is true?
Answer: Both methods are correct and give the same total area Both methods correctly decompose the figure into non-overlapping rectangles. Method 1: $8 + 6 + 4 = 18$ square units. Method 2: $10 + 8 = 18$ square units. Since area is additive, different valid decompositions must yield the same total area. Choice A incorrectly suggests more rectangles means more accuracy. Choice B incorrectly suggests fewer rectangles is better. Choice D fails to recognize that $18 = 18$ .
Use the figure to answer the question. The rectilinear figure is made from two non-overlapping rectangles. Rectangle A measures $7$ cm by $2$ cm and Rectangle B measures $3$ cm by $6$ cm. What is the area of the whole figure?
Answer: $32$ square centimeters $7 \times 2 = 14$ and $3 \times 6 = 18$ . Total: $14 + 18 = 32$ sq cm. A is only Rectangle B. B is the perimeter-like sum. D adds a small extra piece.
Look at the floor plan below. Carlos calculated the area by adding $3 \times 4 = 12$ and $2 \times 5 = 10$ to get $22$ square units. His friend Emma calculated it by adding $3 \times 7 = 21$ and $2 \times 1 = 2$ to get $23$ square units. Who is correct and what is the actual area?
Answer: Both made errors; the actual area is $21$ square units The correct decomposition is into two rectangles: $3 \times 4 = 12$ and $3 \times 3 = 9$ , giving $12 + 9 = 21$ square units. Carlos used incorrect dimensions for his second rectangle ( $2 \times 5$ instead of $3 \times 3$ ). Emma's calculation of $3 \times 7$ creates overlap since there is no $7$ -unit continuous dimension. Choice A accepts Carlos's error. Choice B accepts Emma's error. Choice D uses yet another incorrect decomposition.
Maya's L-shaped garden can be split into two rectangles: one measuring 6 feet by 5 feet, and another measuring 3 feet by 6 feet. What is the total area of the garden?
Answer: $48$ square feet The first rectangle has an area of 6 times 5, or 30 square feet, and the second has an area of 3 times 6, or 18 square feet. Adding these gives 30 plus 18, which is 48 square feet total. Choice B, 52 square feet, Choice C, 44 square feet, and Choice D, 40 square feet, all come from errors in multiplying or adding the two rectangle areas.
A garden is shaped like the letter T. The top rectangle is $9$ feet by $2$ feet. The bottom rectangle is $3$ feet by $5$ feet. What is the total area of the garden?
Answer: $33$ square feet When a shape is made of two rectangles joined together (like a T, L, or plus sign), you can find its total area by finding the area of each rectangle separately and then adding them together. Remember, the area of a rectangle is length × width. Start with the top rectangle: $9 \times 2 = 18$ square feet. Then the bottom rectangle: $3 \times 5 = 15$ square feet. Add them: $18 + 15 = 33$ square feet. That matches choice D. Choice A ( $19$ ) comes from adding the dimensions instead of multiplying — someone did $9+2+3+5=19$ , which gives a perimeter-like number, not area. Choice B ( $27$ ) comes from finding only one rectangle's area correctly and mixing up the other; for example, $9+2+3+5+...$ or adding $18+9$ . Choice C ( $45$ ) is the trap answer — it comes from multiplying $9 \times 5 = 45$ , treating the whole T as one big rectangle. But the T shape has empty corners, so it isn't a full $9 \times 5$ rectangle. When you see a composite shape (a shape made of more than one rectangle), always break it into separate rectangles, find each area with length × width, then add. Watch out for the trap of multiplying the largest dimensions together as if the shape were one rectangle — that only works for actual rectangles, not for T, L, or plus shapes.
Use the figure shown to answer this question. Anna wants to carpet this T-shaped room. What is the total area she needs to cover?
Answer: $88$ square feet The T-shaped room can be split into two rectangles. The horizontal top section is 12 ft × 4 ft = 48 square feet. The vertical bottom section is 5 ft × 8 ft = 40 square feet. Total area = 48 + 40 = 88 square feet.
A rectangular pool deck is made of two parts. The swimming area is $10$ m by $5$ m, and the wading area is $4$ m by $4$ m. If both parts are covered with tile, how many square meters of tile are needed?
Answer: $66$ square meters When a shape is made up of two separate rectangles, you can find the total area by calculating the area of each rectangle and then adding them together. The area of a rectangle is length × width, measured in square units. Start with the swimming area: $10 \text{ m} \times 5 \text{ m} = 50$ square meters. Then the wading area: $4 \text{ m} \times 4 \text{ m} = 16$ square meters. Add the two areas together: $50 + 16 = 66$ square meters. That matches choice A. Choice B ( $50$ ) is only the swimming area — it forgets to include the wading area entirely. Choice C ( $23$ ) comes from adding the side lengths instead of multiplying them ( $10 + 5 + 4 + 4 = 23$ ), which confuses perimeter-style addition with area. Choice D ( $80$ ) comes from multiplying the two areas' dimensions incorrectly, like doing $10 \times 4 \times 2 = 80$ or a similar mix-up — a trap when you rush and combine numbers without thinking about what each represents. A helpful strategy: when a problem describes two shapes, underline each one separately and calculate its area before combining. Remember, area uses multiplication (length × width) and is always in square units, while perimeter uses addition. Getting these two ideas mixed up is the most common trap on area questions.
Refer to the diagram below. Jake is painting this wall that has a rectangular window cut out. What is the area of the wall he needs to paint?
Answer: $54$ square feet The total wall area is 8 ft × 9 ft = 72 square feet. The window opening is 3 ft × 6 ft = 18 square feet. The paintable area = 72 - 18 = 54 square feet.
Refer to the figure. The floor plan of a hallway is made from two rectangles. The long rectangle is $12$ ft by $3$ ft. The shorter rectangle is $4$ ft by $5$ ft. What is the total floor area of the hallway?
Answer: $56$ square feet $12 \times 3 = 36$ and $4 \times 5 = 20$ . Total: $36 + 20 = 56$ sq ft. B only counts the long rectangle. A adds the wrong pieces. D multiplies incorrectly.
A rectangular rug is $8$ feet by $6$ feet. A smaller rectangular rug that is $4$ feet by $3$ feet is placed next to it, not overlapping. What is the combined area of both rugs?
Answer: $60$ square feet When you're asked to find the combined area of two shapes, the strategy is simple: find each area separately, then add them together. The area of a rectangle is found by multiplying length × width. Start with the larger rug: $8 \times 6 = 48$ square feet. Then the smaller rug: $4 \times 3 = 12$ square feet. Adding these together gives $48 + 12 = 60$ square feet, which matches choice B. Choice A ( $21$ ) comes from adding the dimensions instead of multiplying them: $8 + 6 + 4 + 3 = 21$ . That's a perimeter-style mistake, not an area calculation. Choice C ( $48$ ) is only the area of the larger rug — it forgets to include the smaller rug entirely. Choice D ( $72$ ) comes from multiplying all four numbers together in a mixed-up way (for example, $8 \times 3 \times 3 = 72$ or similar errors), which doesn't represent either rug's area correctly. A helpful tip: whenever a problem says "combined," "total," or "altogether" with shapes, treat it as a two-step problem — calculate each area first, then add. Also, remember the difference between area (multiply length × width, answer in square units) and perimeter (add all sides, answer in regular units). Mixing these up is one of the most common traps in geometry questions.
Refer to the figure. The rectilinear shape is split by a dashed line into two rectangles. The larger rectangle is $10$ in by $4$ in. The smaller rectangle is $2$ in by $3$ in. What is the area of the whole shape?
Answer: $46$ square inches $10 \times 4 = 40$ and $2 \times 3 = 6$ . Total: $40 + 6 = 46$ sq in. B ignores the small rectangle. A subtracts instead of adding. D multiplies incorrect dimensions.
Use the figure. A stage is made of two rectangles. The main rectangle is $8$ ft by $4$ ft. A step in front is $6$ ft by $2$ ft. What is the total floor area of the stage and step together?
Answer: $44$ square feet $8 \times 4 = 32$ and $6 \times 2 = 12$ . Total: $32 + 12 = 44$ sq ft. B ignores the step. A adds side lengths. D multiplies wrong values.
Ms. Lee's classroom carpet is shaped like a rectangle with a smaller rectangle attached. The main part is $9$ ft by $6$ ft, and the attached part is $3$ ft by $3$ ft. What is the total area of the carpet?
Answer: $63$ square feet When a shape is made from two rectangles joined together, you can find the total area by finding the area of each rectangle separately and then adding them together. Remember, the area of a rectangle is length × width, measured in square units. Start with the main part of the carpet: $9 \times 6 = 54$ square feet. Then find the area of the attached part: $3 \times 3 = 9$ square feet. Add them together: $54 + 9 = 63$ square feet. That matches choice A. Choice B ( $54$ square feet) is only the area of the main rectangle — it forgets to add the smaller attached piece. Choice C ( $27$ square feet) comes from adding the side lengths instead of multiplying, or from other addition mistakes with the numbers given; it's far too small for a carpet this size. Choice D ( $72$ square feet) likely comes from multiplying $9 \times 6$ and then adding $3 + 3 + 3 + 3 = 12$ (the perimeter of the small square) instead of its area of $9$ . When you see a compound shape, always break it into simple rectangles, calculate each area with length × width, then add. Double-check that you multiplied (for area) rather than added the sides (which gives perimeter). Labeling each piece on a quick sketch helps you avoid mixing up the two.

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