Area and Volume Relationships - ISEE Lower Level Quantitative Reasoning

1. What is the volume of a rectangular prism with $l=5$, $w=3$, and $h=2$?

   $30$. Product of dimensions: $5 \times 3 \times 2 = 30$.

2. What is the area of a triangle with $b=12$ and $h=5$?

   $30$. Half the product of base and height gives the area: $\frac{1}{2} \times 12 \times 5 = 30$.

3. State the formula for the circumference of a circle with radius $r$.

   $C=2\pi r$. The circumference of a circle is twice pi times its radius.

4. A cylinder has volume $45\pi$ and radius $r=3$. What is its height $h$?

   $5$. Divide volume by $\pi r^2$ to find height: $45\pi \div (\pi \times 9) = 5$.

5. What is the volume of a cube with side length $s=4$?

   $64$. Cube the side length: $4^3 = 64$.

6. State the formula for the volume of a cylinder with radius $r$ and height $h$.

   $V=\pi r^2h$. The volume of a cylinder is pi times the square of its radius times its height.

7. A cube's side length is doubled. By what factor does its volume change?

   Volume is multiplied by $8$. Doubling side length scales volume by $2^3 = 8$.

8. A rectangle has area $48$ and width $6$. What is its length?

   $8$. Divide area by width to solve for length: $48 \div 6 = 8$.

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

   $A=\frac{1}{2}(b_1+b_2)h$. The area of a trapezoid is half the sum of its parallel bases multiplied by the height.

10. What is the volume of a cylinder with $r=2$ and $h=7$ in terms of $\pi$?

    $28\pi$. Pi times radius squared times height: $\pi \times 2^2 \times 7 = 28\pi$.

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

    $A=bh$. The area of a parallelogram is the product of its base and corresponding height.

12. What is the area of a circle with radius $r=3$ in terms of $\pi$?

    $9\pi$. Pi times radius squared: $\pi \times 3^2 = 9\pi$.

13. State the formula for the area of a triangle with base $b$ and height $h$.

    $A=\frac{1}{2}bh$. The area of a triangle is half the product of its base and height.

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

    $A=lw$. The area of a rectangle is calculated as the product of its length and width.

15. What is the area of a trapezoid with $b_1=6$, $b_2=10$, and $h=4$?

    $32$. Half the sum of bases times height: $\frac{1}{2} (6 + 10) \times 4 = 32$.

16. A rectangle's length and width are each tripled. By what factor does its area change?

    Area is multiplied by $9$. Tripling dimensions scales area by $3^2 = 9$.

17. Identify the scale factor for area when all lengths are multiplied by $k$.

    Area is multiplied by $k^2$. When linear dimensions are scaled by $k$, areas scale by the square of $k$.

18. Identify the scale factor for volume when all lengths are multiplied by $k$.

    Volume is multiplied by $k^3$. When linear dimensions are scaled by $k$, volumes scale by the cube of $k$.

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

    $36$. Multiply length by width to find the area: $9 \times 4 = 36$.

20. State the formula for the volume of a rectangular prism with dimensions $l,w,h$.

    $V=lwh$. The volume of a rectangular prism is the product of its length, width, and height.

21. State the formula for the area of a circle with radius $r$.

    $A=\pi r^2$. The area of a circle is pi times the square of its radius.

22. State the formula for the volume of a cube with side length $s$.

    $V=s^3$. The volume of a cube is the cube of its side length.

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