Solve Perimeter and Area Problems - 3rd Grade Math

1. Choose the rectangle with the same perimeter as $3$ by $7$: $2$ by $8$ or $4$ by $6$.

   $2$ by $8$. Both have $P = 20$; $3$ by $7$ gives $P = 20$, $2$ by $8$ gives $P = 20$.

2. Find the missing side $x$ if a triangle has perimeter $P = 17$ and sides $6$, $5$, and $x$.

   $x = 6$. From $17 = 6 + 5 + x$, solve: $17 = 11 + x$, so $x = 6$.

3. Find the missing side length if $P = 18$ and the other sides are $4$, $5$, and $6$.

   $3$. Subtract known sides from perimeter: $18 - 4 - 5 - 6 = 3$.

4. Which value is the perimeter of a rectangle with $l = 6$ and $w = 2$: $16$ or $12$?

   $16$. Calculate $P = 2(6) + 2(2) = 12 + 4 = 16$.

5. What is the perimeter of a rectangle with $l = 9$ and $w = 4$?

   $26$. Use $P = 2l + 2w = 2(9) + 2(4) = 18 + 8$.

6. What is the perimeter of a rectangle with $l = 8$ and $w = 3$?

   $22$. Using $P = 2(8) + 2(3) = 16 + 6 = 22$.

7. Find the missing length $l$ if a rectangle has perimeter $24$ and width $4$.

   $l = 8$. Use $24 = 2l + 2(4)$, so $2l = 16$, thus $l = 8$.

8. Two rectangles both have area $12$: $3$ by $4$ and $2$ by $6$. What are their perimeters, in order?

   $14$ and $16$. $3$ by $4$ has $P = 14$; $2$ by $6$ has $P = 16$.

9. What is the perimeter of a triangle with side lengths $3$, $4$, and $5$?

   $12$. Add all three sides: $3 + 4 + 5 = 12$.

10. Identify the area of a $5 \times 1$ rectangle and a $3 \times 3$ rectangle with the same perimeter $12$.

    $5$ and $9$. $5 × 1 = 5$ and $3 × 3 = 9$; both rectangles have perimeter 12.

11. Identify two rectangles with the same area $24$ but different perimeters: $6 \times 4$ and $8 \times 3$.

    Perimeters: $20$ and $22$ units. $6 \times 4$ has $P = 20$; $8 \times 3$ has $P = 22$.

12. Identify the perimeter of a shape with sides $8$ m, $3$ m, $8$ m, and $3$ m.

    $22$ m. Add all sides: $8 + 3 + 8 + 3 = 22$.

13. What is the perimeter of a square with side length $7$ inches?

    $28$ inches. Use $P = 4(7) = 28$ for a square.

14. What is the perimeter of a square with side length $s = 6$?

    $24$. Using $P = 4(6) = 24$ for a square.

15. What is the perimeter of a square with side length $7$?

    $28$. Multiply side length by 4: $7 × 4 = 28$.

16. What is the perimeter of a pentagon with side lengths $2$, $2$, $3$, $3$, and $4$?

    $14$ units. Add all five sides: $2 + 2 + 3 + 3 + 4 = 14$.

17. Which statement is true: Same perimeter always means same area, or it can mean different areas?

    Same perimeter can mean different areas. Rectangles can have equal perimeters but different shapes/areas.

18. State the formula for the perimeter of a rectangle with length $l$ and width $w$.

    $P = 2l + 2w$. Add length and width twice each to get the distance around.

19. Identify the area of a rectangle with $l = 6$ and $w = 2$.

    $12$. Area equals length times width: $6 × 2 = 12$.

20. State the formula for the perimeter of a polygon with side lengths $s_1, s_2, \dots, s_n$.

    $P = s_1 + s_2 + \cdots + s_n$. Add all side lengths to find the perimeter.

21. Identify the perimeter of a pentagon with side lengths $2$, $3$, $2$, $4$, and $3$.

    $14$. Sum all five sides: $2 + 3 + 2 + 4 + 3 = 14$.

22. Find the side length $s$ of a square with perimeter $P = 28$.

    $s = 7$. From $28 = 4s$, divide both sides by 4 to get $s = 7$.

23. Identify the area of a rectangle with $l = 4$ and $w = 3$.

    $12$. Multiply length by width: $4 × 3 = 12$.

24. State the formula for the perimeter of a square with side length $s$.

    $P = 4s$. Multiply side length by 4 since all sides are equal.

25. What is the perimeter of a rectangle with $l = 9$ and $w = 2$?

    $22$. Using $P = 2(9) + 2(2) = 18 + 4 = 22$.

26. What is the perimeter of a triangle with side lengths $4$, $5$, and $6$?

    $15$. Sum all three sides: $4 + 5 + 6 = 15$.

27. What is the perimeter of a $6 \times 2$ rectangle (length $6$, width $2$)?

    $16$. Calculate $P = 2(6) + 2(2) = 12 + 4 = 16$.

28. Find the unknown length $l$ if a rectangle has $P = 22$ and $w = 4$.

    $l = 7$. From $22 = 2l + 2(4)$, solve: $22 = 2l + 8$, so $2l = 14$.

29. State the rule for finding the perimeter of any polygon from its side lengths.

    Add all side lengths: $P = s_1 + s_2 + \cdots$. Perimeter is the sum of all side lengths.

30. Which rectangle has perimeter $12$ and larger area: $5 \times 1$ or $3 \times 3$?

    $3 \times 3$. $3 × 3$ has area 9, which is larger than $5 × 1$'s area of 5.

31. Which rectangle has the larger perimeter: $2$ by $9$ or $3$ by $6$?

    $2$ by $9$. $2$ by $9$ has $P = 22$; $3$ by $6$ has $P = 18$.

32. What is the area of a rectangle with $l = 6$ and $w = 4$ (for comparing area vs. perimeter)?

    $24$ square units. Area equals length times width: $A = 6 \times 4 = 24$.

33. What is the perimeter of a rectangle with area $16$ and side lengths $2$ and $8$?

    $20$ units. Use $P = 2(2) + 2(8) = 4 + 16 = 20$.

34. Find the missing width $w$ if a rectangle has perimeter $30$ and length $10$.

    $w = 5$. Use $30 = 2(10) + 2w$, so $2w = 10$, thus $w = 5$.

35. What is the perimeter of a pentagon with side lengths $3$, $3$, $4$, $4$, and $6$?

    $20$. Sum all five sides: $3 + 3 + 4 + 4 + 6 = 20$.

36. Find the missing side length $x$ if a triangle has perimeter $20$ and sides $6$, $7$, and $x$.

    $x = 7$. Subtract known sides from perimeter: $20 - 6 - 7 = 7$.

37. Find and correct the mistake: A student says $P = l + w$ for a rectangle.

    Correct: $P = 2l + 2w$. Student forgot to multiply by 2; must count all 4 sides.

38. What is the perimeter of a rectangle with side lengths $8$, $3$, $8$, and $3$?

    $22$. Add all four sides: $8 + 3 + 8 + 3 = 22$.

39. Find the unknown width $w$ if a rectangle has $P = 30$ and $l = 10$.

    $w = 5$. From $30 = 2(10) + 2w$, solve: $30 = 20 + 2w$, so $2w = 10$.

40. What is the perimeter of a rectangle with $l = 9$ cm and $w = 4$ cm?

    $26$ cm. Use $P = 2(9) + 2(4) = 18 + 8 = 26$.

41. Which statement is true: same perimeter means same area, or same perimeter can have different areas?

    Same perimeter can have different areas. Rectangles can share perimeter but have different dimensions and areas.

42. Find the unknown side $x$ if a rectangle has perimeter $P = 18$, $l = 5$, and $w = x$.

    $x = 4$. From $18 = 2(5) + 2x$, solve: $18 = 10 + 2x$, so $x = 4$.

43. What is the perimeter of a $4 \times 3$ rectangle (length $4$, width $3$)?

    $14$. Calculate $P = 2(4) + 2(3) = 8 + 6 = 14$.

44. Which rectangle has area $12$ but a different perimeter than $4 \times 3$: $6 \times 2$ or $3 \times 4$?

    $6 \times 2$. $6 × 2$ has area 12 and perimeter 16, different from $4 × 3$'s perimeter 14.

45. Which statement is true: same area means same perimeter, or same area can have different perimeters?

    Same area can have different perimeters. Rectangles can share area but have different dimensions and perimeters.

46. Identify two rectangles with the same perimeter $20$ but different areas: $6 \times 4$ and $7 \times 3$.

    Areas: $24$ and $21$ square units. $6 \times 4 = 24$ and $7 \times 3 = 21$ with same $P = 20$.

47. Find the missing side $x$ if a quadrilateral has perimeter $25$ and sides $6$, $8$, $5$, and $x$.

    $x = 6$. Subtract known sides: $25 - 6 - 8 - 5 = 6$.

48. Which expression represents the perimeter of a rectangle with sides $a$ and $b$?

    $2a + 2b$. Rectangle perimeter formula with different variable names.

49. What is the perimeter of a polygon with side lengths $3$, $5$, $4$, and $6$ units?

    $18$ units. Add all sides: $3 + 5 + 4 + 6 = 18$.

50. What is the perimeter of a square with side length $s$?

    $P = 4s$. A square has 4 equal sides, so multiply side length by 4.

51. Find the unknown side $s$ of a square if the perimeter is $P = 36$.

    $s = 9$. Divide perimeter by 4 to find side: $36 ÷ 4 = 9$.

52. Which statement is true: Same area always means same perimeter, or it can mean different perimeters?

    Same area can mean different perimeters. Rectangles can have equal areas but different shapes/perimeters.

53. Find the missing side $x$ if a quadrilateral has perimeter $P = 24$ and sides $7$, $5$, $6$, and $x$.

    $x = 6$. From $24 = 7 + 5 + 6 + x$, solve: $24 = 18 + x$, so $x = 6$.

54. What are the areas of rectangles $2$ by $8$ and $4$ by $6$ (same perimeter), in order?

    $16$ and $24$. $2 \times 8 = 16$ and $4 \times 6 = 24$ show different areas.

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