Find Area by Composing and Decomposing - 6th Grade Math

1. A floor section is a trapezoid with bases $8\text{ m}$ and $12\text{ m}$ and height $5\text{ m}$. What area is covered?

   $50\text{ m}^2$. Trapezoid: $\frac{1}{2}(8+12) \times 5 = 50$.

2. What is the area of an $L$-shape made from a $10\times 8$ rectangle with a $4\times 3$ corner removed?

   $68\text{ square units}$. Large rectangle minus removed corner: $80-12=68$.

3. What is the area of a rhombus with diagonals $d_1=10$ and $d_2=6$ units?

   $30\text{ square units}$. Apply $A=\frac{1}{2}d_1d_2$: $\frac{1}{2}(10)(6)=30$.

4. What is the area of a right triangle with legs $6$ and $8$?

   $24$ square units. Right triangle area: $\frac{1}{2} \times 6 \times 8 = 24$.

5. What is the area of a trapezoid with $b_1=6$, $b_2=10$, and $h=5$ square units?

   $40$. Apply $A = \frac{1}{2}(6+10) \times 5 = \frac{1}{2} \times 16 \times 5$.

6. What is the area of a rectangle with $l=11$ and $w=3$ square units?

   $33$. Multiply length by width: $11 \times 3 = 33$.

7. What is the height $h$ of a trapezoid with area $45$ and bases $b_1=5$, $b_2=10$?

   $6$. Solve $45 = \frac{1}{2}(5+10) \times h$ for $h$.

8. State the formula for the area of a triangle using base $b$ and height $h$.

   $A=\frac{1}{2}bh$. Half the base times height gives triangle area.

9. Identify the height to use if a triangle has base $10$ and perpendicular height $7$.

   $h=7$. The perpendicular height is always used in area calculations.

10. What is the height of a parallelogram with area $56$ and base $7$?

    $8$. Solve $56 = 7 \times h$ for $h$.

11. What is the area of a triangle with base $12$ units and height $5$ units?

    $30\text{ square units}$. Apply $A=\frac{1}{2}bh$: $\frac{1}{2}(12)(5)=30$.

12. Which measurement must be perpendicular to the base when finding area with $A=bh$?

    $\text{The height (altitude)}$. Height must form a $90°$ angle with the base.

13. What is the base of a triangle with area $30$ and height $6$?

    $10$. Solve $30 = \frac{1}{2} \times b \times 6$ for $b$.

14. What is the area of a composite shape: a $9\times 6$ rectangle plus a triangle with base $9$ and height $4$?

    $72$. Rectangle $54$ plus triangle $\frac{1}{2} \times 9 \times 4 = 18$.

15. A triangular sign has base $14\text{ cm}$ and height $9\text{ cm}$. What is its area?

    $63\text{ cm}^2$. Apply triangle formula: $\frac{1}{2} \times 14 \times 9$.

16. What is the area of a rhombus with diagonals $12$ and $5$?

    $30$ square units. Apply $\frac{1}{2} \times 12 \times 5 = 30$.

17. What is the area of a composite shape: rectangle $8\times 5$ plus right triangle with legs $4$ and $6$?

    $52\text{ square units}$. Rectangle: $8×5=40$; triangle: $\frac{1}{2}(4)(6)=12$; total: $52$.

18. What is the area of a square with side length $7$ units?

    $49\text{ square units}$. Square area is side squared: $7^2=49$.

19. Identify the correct base-height pair for $A=\frac{1}{2}bh$ in any triangle.

    A base and its perpendicular height. Height must be perpendicular to the chosen base.

20. Find the missing height $h$ of a parallelogram with area $45$ and base $9$.

    $5$. Solve $45 = 9h$ to find height.

21. What is the area of a trapezoid with bases $6$ and $10$ and height $5$?

    $40$ square units. Average bases $(6+10)/2 = 8$, times height $5$.

22. What is the area of a polygon split into two triangles with areas $18$ and $25$ square units?

    $43\text{ square units}$. Add the areas of component shapes: $18+25=43$.

23. What is the area of a right triangle with legs $6$ and $8$ units?

    $24\text{ square units}$. Use $A=\frac{1}{2}bh$ with legs as base and height: $\frac{1}{2}(6)(8)=24$.

24. What is the area of an L-shape formed from a $10\times 8$ rectangle with a $4\times 3$ rectangle removed?

    $68$. Large rectangle $80$ minus small rectangle $12$.

25. State the formula for the area of a kite or rhombus using diagonals $d_1$ and $d_2$.

    $A=\frac{1}{2}d_1d_2$. Half the product of the two diagonals.

26. State the area formula for a triangle using base $b$ and height $h$.

    $A=\frac{1}{2}bh$. Area equals half the product of base and height.

27. What is the area of a right triangle with legs $6$ and $8$ square units?

    $24$. Right triangle area is $\frac{1}{2} \times 6 \times 8 = 24$.

28. What is the area of a trapezoid with $b_1=6$, $b_2=10$, and $h=5$?

    $40\text{ square units}$. Apply formula: $\frac{1}{2}(6+10)(5)=\frac{1}{2}(16)(5)=40$.

29. A triangular sign has base $14\text{ in}$ and height $9\text{ in}$. What is its area?

    $63\text{ in}^2$. Apply $A=\frac{1}{2}bh$: $\frac{1}{2}(14)(9)=63$ square inches.

30. What is the area of an L-shape formed by a $10\times 6$ rectangle with a $4\times 3$ rectangle removed?

    $48$ square units. Large rectangle $(60)$ minus small rectangle $(12)$.

31. What is the area of a parallelogram with base $9$ and height $4$ units?

    $36\text{ square units}$. Apply $A=bh$: $(9)(4)=36$.

32. State the formula for the area of a rectangle with length $l$ and width $w$.

    $A=lw$. Length times width for rectangles.

33. Identify the correct method to find a polygon area when no single formula fits the whole shape.

    $\text{Decompose into known shapes and add areas}$. Break complex shapes into triangles/rectangles to calculate.

34. What is the height of a triangle with area $48$ and base $12$?

    $8$. Solve $48 = \frac{1}{2} \times 12 \times h$ for $h$.

35. State the area formula for a parallelogram with base $b$ and height $h$.

    $A=bh$. Parallelogram area is base times perpendicular height.

36. A rectangle is split by a diagonal into two congruent triangles. If the rectangle area is $50$, what is one triangle's area?

    $25$ square units. Diagonal splits rectangle into two equal triangles.

37. Identify the height to use if a triangle has base $10$ and a perpendicular height of $7$.

    $7$. Use the perpendicular distance from base to opposite vertex.

38. State the formula for the area of a trapezoid with bases $b_1,b_2$ and height $h$.

    $A=\frac{1}{2}(b_1+b_2)h$. Average the parallel bases, then multiply by height.

39. What is the area of a square with side length $9$ square units?

    $81$. Square area is side squared: $9^2 = 81$.

40. What is the area of a parallelogram with base $9$ and height $4$ square units?

    $36$. Multiply base by height: $9 \times 4 = 36$.

41. Find the missing height $h$ of a trapezoid with area $48$ and bases $10$ and $6$.

    $6$. Solve $48 = \frac{1}{2}(10+6)h = 8h$ for $h$.

42. A floor is a rectangle $12\text{ ft}\times 9\text{ ft}$. What is its area in $\text{ft}^2$?

    $108\text{ ft}^2$. Apply $A=lw$: $(12)(9)=108$ square feet.

43. What is the area of a triangle with base $10$ and height $7$ square units?

    $35$. Apply $A = \frac{1}{2}bh = \frac{1}{2} \times 10 \times 7$.

44. State the area formula for a trapezoid with bases $b_1,b_2$ and height $h$.

    $A=\frac{1}{2}(b_1+b_2)h$. Average the parallel bases, then multiply by height.

45. State the formula for the area of a parallelogram with base $b$ and height $h$.

    $A=bh$. Base times perpendicular height for parallelograms.

46. State the area formula for a kite or rhombus using diagonals $d_1$ and $d_2$.

    $A=\frac{1}{2}d_1d_2$. Half the product of the two diagonals gives the area.

47. What is the area of a composite shape made of a $6\times 4$ rectangle plus a right triangle with legs $4$ and $3$?

    $30$ square units. Rectangle $(24)$ plus triangle $(6)$ equals $30$.

48. What is the area of a rhombus with diagonals $d_1=12$ and $d_2=5$ square units?

    $30$. Apply $A = \frac{1}{2} \times 12 \times 5 = 30$.

49. What is the area of a triangle with base $12$ and height $5$?

    $30$ square units. Apply $A = \frac{1}{2}bh = \frac{1}{2} \times 12 \times 5$.

50. What is the area of a parallelogram with base $9$ and height $4$?

    $36$ square units. Multiply base by perpendicular height: $9 \times 4$.

51. Find the missing height $h$ of a triangle with area $36$ and base $12$.

    $6$. Solve $36 = \frac{1}{2} \times 12 \times h$ for $h$.

Study these flashcards