Find Area of Composite Figures - 3rd Grade Math

1. What is the total area if two non-overlapping rectangles have areas $12$ and $7$ square units?

   $19$ square units. Add non-overlapping areas: $12 + 7 = 19$.

2. Identify the best first step to find area of an L-shaped rectilinear figure.

   Decompose it into non-overlapping rectangles. Breaking into rectangles allows using $l \times w$ for each part.

3. What is a rectilinear figure?

   A shape made only of straight sides meeting at right angles. All corners form 90-degree angles in rectilinear shapes.

4. A floor section is $8 \text{ ft}$ by $3 \text{ ft}$. What is its area in $\text{ft}^2$?

   $24\ \text{ft}^2$. Multiply dimensions: $8 \times 3 = 24$ square feet.

5. Choose the expression for total area of two rectangles with areas $a$ and $b$ (no overlap).

   $a + b$. Non-overlapping areas are simply added together.

6. What is the area of a figure made from rectangles $6\times 2$ and $6\times 1$ (no overlap)?

   $18$ square units. Add areas: $(6 \times 2) + (6 \times 1) = 12 + 6 = 18$.

7. What is the area of a $9\times 5$ rectangle with a $3\times 2$ corner cut out?

   $39$ square units. Subtract cut-out from whole: $(9 \times 5) - (3 \times 2) = 45 - 6 = 39$.

8. A garden is made of two rectangles: $4 \text{ m} \times 5 \text{ m}$ and $2 \text{ m} \times 5 \text{ m}$. Find total area.

   $30\ \text{m}^2$. Add areas: $(4 \times 5) + (2 \times 5) = 20 + 10 = 30$.

9. Find and correct the mistake: total area $= (5\times 3) + (2\times 3) = 21^2$.

   Correct total area is $21$ square units, not $21^2$. Units should be square units, not squared again.

10. What does it mean to say area is additive for non-overlapping rectangles?

    Total area equals the sum of the parts' areas. When rectangles don't overlap, combine their areas by adding.

11. Find the total area: rectangle $A$ is $2 \times 9$ and rectangle $B$ is $3 \times 4$ (non-overlapping).

    $30$ square units. Add areas: $(2 \times 9) + (3 \times 4) = 18 + 12 = 30$.

12. Which equation correctly shows additivity for two rectangles with areas $A_1$ and $A_2$?

    $A_{\text{total}} = A_1 + A_2$. Shows area additivity: total equals sum of parts.

13. What is the area of a rectangle with side lengths $7$ and $4$?

    $28$ square units. Apply area formula: $7 \times 4 = 28$.

14. Find the area of a rectilinear figure made of $2$ rectangles: $3 \times 7$ and $2 \times 7$ (non-overlapping).

    $35$ square units. Add areas: $(3 \times 7) + (2 \times 7) = 21 + 14 = 35$.

15. Identify the best first step to find the area of an L-shaped rectilinear figure.

    Decompose it into non-overlapping rectangles. Breaking complex shapes into rectangles makes area calculation easier.

16. Find the area of a figure split into $3$ rectangles: $2 \times 4$, $1 \times 4$, and $3 \times 2$.

    $18$ square units. Add areas: $(2 \times 4) + (1 \times 4) + (3 \times 2) = 8 + 4 + 6 = 18$.

17. What does it mean that area is additive for non-overlapping rectangles?

    Total area $=$ sum of the parts' areas. Areas of separate parts can be added together to find the whole.

18. Find the area of a rectilinear figure decomposed into $5 \times 2$ and $5 \times 3$ rectangles.

    $25$ square units. Add areas: $(5 \times 2) + (5 \times 3) = 10 + 15 = 25$.

19. What must be true about rectangles when decomposing to add areas correctly?

    They must not overlap and must cover the figure. Overlapping would count some area twice; gaps would miss area.

20. What is the area of a $6 \times 6$ square in square units?

    $36$ square units. Square area equals side squared: $6 \times 6 = 36$.

21. A garden is $9\text{ ft}\times 5\text{ ft}$ with a $3\text{ ft}\times 1\text{ ft}$ section removed. What area?

    $42\text{ ft}^2$. Subtract removed section: $(9 \times 5) - (3 \times 1) = 45 - 3 = 42$.

22. Find the total area of rectangles $3\times 7$, $2\times 7$, and $1\times 7$ (no overlap).

    $42$ square units. Add all parts: $21 + 14 + 7 = 42$.

23. A room floor is $6\text{ m}\times 4\text{ m}$ plus a $2\text{ m}\times 3\text{ m}$ alcove. What area?

    $30\text{ m}^2$. Add main room and alcove: $(6 \times 4) + (2 \times 3) = 24 + 6 = 30$.

24. Which method matches additive area: add the areas of parts or add the perimeters of parts?

    Add the areas of parts. Area is additive; perimeter is not for composite figures.

25. What is the area formula for a rectangle with side lengths $l$ and $w$?

    $A = l \times w$. Multiply length times width to find rectangular area.

26. What is the area of a $8\times 6$ rectangle with a $2\times 6$ strip removed?

    $36$ square units. Subtract removed strip: $(8 \times 6) - (2 \times 6) = 48 - 12 = 36$.

27. What is the area of a $4 \times 6$ rectangle in square units?

    $24$ square units. Multiply length times width: $4 \times 6 = 24$.

28. What is the total area of a $7 \times 4$ rectangle if it is decomposed into $7 \times 1$ and $7 \times 3$?

    $28$ square units. Verify: $(7 \times 1) + (7 \times 3) = 7 + 21 = 28 = 7 \times 4$.

29. What is the area of a figure decomposed into $4\times 4$ and $4\times 2$ rectangles?

    $24$ square units. Add areas: $(4 \times 4) + (4 \times 2) = 16 + 8 = 24$.

30. What is the area of a figure made from $10\times 2$ and $4\times 3$ rectangles (no overlap)?

    $32$ square units. Add areas: $(10 \times 2) + (4 \times 3) = 20 + 12 = 32$.

31. Which unit is correct for area: $\text{cm}$ or $\text{cm}^2$?

    $\text{cm}^2$. Area measures square units, not linear units.

32. What is the area of a rectilinear figure split into rectangles $5\times 3$ and $2\times 3$?

    $21$ square units. Add areas: $(5 \times 3) + (2 \times 3) = 15 + 6 = 21$.

33. Find the missing part area if total area is $35$ and one rectangle part is $20$ square units.

    $15$ square units. Subtract known part from total: $35 - 20 = 15$.

34. Identify the error: A student adds areas of two rectangles that overlap. What is wrong?

    The overlap is counted twice. Overlapping areas violate the non-overlapping requirement.

35. What is the area formula for a rectangle with length $l$ and width $w$?

    $A = l \times w$. Multiply length times width to find rectangular area.

36. Which unit is correct for area: $\text{cm}$, $\text{cm}^2$, or $\text{cm}^3$?

    $\text{cm}^2$. Area is measured in square units, not linear or cubic.

37. Find the missing rectangle area if total area is $35$ and one part is $20$.

    $15$ square units. Subtract known part from total: $35 - 20 = 15$.

38. Find the area of an L-shape split into rectangles $4 \times 3$ and $2 \times 3$ (non-overlapping).

    $18$ square units. Add areas: $(4 \times 3) + (2 \times 3) = 12 + 6 = 18$.

39. What is the total area if two non-overlapping rectangles have areas $18$ and $6$?

    $24$ square units. Area is additive: $18 + 6 = 24$.

40. What is the area of a $3 \times 5$ rectangle in square units?

    $15$ square units. Multiply length times width: $3 \times 5 = 15$.

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