Area, Perimeter, and Volume - ISEE Upper Level Quantitative Reasoning

1. State the formula for the surface area of a rectangular prism with length $l$, width $w$, height $h$.

   $SA=2(lw+lh+wh)$. Adds twice the areas of each pair of opposite faces to compute total surface area.

2. State the formula for the surface area of a cylinder with radius $r$ and height $h$.

   $SA=2\pi r^2+2\pi rh$. Sums the areas of two bases and the lateral surface to find total surface area.

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

   $A=lw$. Multiplies length by width to determine the total enclosed space within the rectangle.

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

   $A=\pi r^2$. Squares the radius and multiplies by $\pi$ to compute the circle's enclosed area.

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

   $P=2l+2w$. Sums twice the length and twice the width to account for opposite sides of the rectangle.

6. State the formula for the area of a sector with radius $r$ and central angle $\theta$ degrees.

   $A=\frac{\theta}{360}\pi r^2$. Takes the fraction of the full circle's area based on the central angle in degrees.

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

   $8$. Divides the area by the width to solve for length using $A=lw$.

8. State the formula for the volume of a rectangular prism with length $l$, width $w$, height $h$.

   $V=lwh$. Multiplies length, width, and height to find the space occupied by the rectangular prism.

9. Find the volume of a cylinder with $r=2$ and $h=9$ in terms of $\pi$.

   $36\pi$. Multiplies base area $\pi r^2$ by height for the cylinder volume.

10. Find the area of a triangle with base $b=10$ and height $h=6$.

    $30$. Uses the triangle area formula $A=\frac{1}{2}bh$ with the provided base and height.

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

    $A=\frac{1}{2}bh$. Computes half the product of base and height, as a triangle is half of a parallelogram with the same base and height.

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

    $C=2\pi r$. Multiplies the diameter by $\pi$ to find the total length around the circle.

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

    $V=\pi r^2h$. Multiplies the base area by height to determine the cylinder's volume.

14. Find the perimeter of a rectangle with $l=9$ and $w=4$.

    $26$. Applies the perimeter formula $P=2l+2w$ by substituting the given length and width.

15. Find the area of a trapezoid with $b_1=8$, $b_2=14$, and $h=3$.

    $33$. Averages the bases and multiplies by height per the trapezoid area formula.

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

    $A=\frac{1}{2}(b_1+b_2)h$. Averages the two bases and multiplies by height to calculate the trapezoid's area.

17. State the formula for the surface area of a sphere with radius $r$.

    $SA=4\pi r^2$. Multiplies four times the area of a great circle to cover the entire surface.

18. Find the circumference of a circle with diameter $d=12$ in terms of $\pi$.

    $12\pi$. Converts diameter to radius and applies the circumference formula $C=2\pi r$.

19. Find the volume of a rectangular prism with $l=3$, $w=4$, and $h=5$.

    $60$. Multiplies length, width, and height according to the rectangular prism volume formula.

20. State the formula for the volume of a cone with radius $r$ and height $h$.

    $V=\frac{1}{3}\pi r^2h$. Takes one-third of the cylinder's volume with the same base and height.

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

    $A=bh$. Multiplies base by perpendicular height to find the area of the parallelogram.

22. State the formula for the arc length of a sector with radius $r$ and central angle $\theta$ degrees.

    $s=\frac{\theta}{360}\cdot 2\pi r$. Calculates the fraction of the full circumference corresponding to the central angle in degrees.

23. Find the area of a circle with radius $r=5$ in terms of $\pi$.

    $25\pi$. Squares the radius and multiplies by $\pi$ using the circle area formula.

24. State the formula for the area of a rhombus with diagonals $d_1$ and $d_2$.

    $A=\frac{1}{2}d_1d_2$. Multiplies half the diagonals, as they bisect the rhombus into four right triangles of equal area.

25. State the formula for the volume of a sphere with radius $r$.

    $V=\frac{4}{3}\pi r^3$. Integrates the volumes of infinitesimal shells or uses the formula derived from calculus for the sphere's volume.

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