Area, Perimeter, and Volume - Quantitative Reasoning

1. What is the volume formula for a right circular 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 due to the cone's tapering shape.

2. What is the volume formula for a right circular cylinder with radius $r$ and height $h$?

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

3. What is the area formula for a rhombus (or kite) with diagonals $d_1$ and $d_2$?

   $A=\frac{1}{2}d_1d_2$. Uses half the product of the diagonals, as they bisect the rhombus or kite into four right triangles.

4. What is the area of a square with side length $s$?

   $A=s^2$. Squares the side length to compute the area enclosed by the square.

5. What is the area formula for a circle with radius $r$?

   $A=\pi r^2$. Calculates the space inside the circle using pi times the square of the radius.

6. What is the area formula for a triangle with base $b$ and height $h$?

   $A=\frac{1}{2}bh$. Takes half the product of base and height since the area is half of a parallelogram with the same base and height.

7. Identify the scale factor for area if all linear dimensions are multiplied by $k$.

   Area is multiplied by $k^2$. Scales with the square of linear dimensions, multiplying area by $k^2$.

8. What is the surface area formula for a rectangular solid with dimensions $l,w,h$?

   $SA=2(lw+lh+wh)$. Sums the areas of all six faces by doubling the products of each pair of dimensions.

9. What is the volume formula for a pyramid in terms of base area $B$ and height $h$?

   $V=\frac{1}{3}Bh$. Uses one-third of the prism's volume with the same base and height due to the pyramid's apex convergence.

10. What is the volume formula for a sphere with radius $r$?

    $V=\frac{4}{3}\pi r^3$. Derives from integrating cross-sectional areas, resulting in four-thirds pi times radius cubed.

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

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

12. What is the area of a rectangle with $l=7$ and $w=3$?

    $A=21$. Applies the rectangle area formula by multiplying length 7 by width 3.

13. What is the volume of a cylinder with radius $r=2$ and height $h=5$?

    $V=20\pi$. Computes using the cylinder volume formula with base area $\pi (2)^2$ times height 5.

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

    $P=4s$. Multiplies the side length by four since all sides of the square are equal.

15. What is the surface area formula for a sphere with radius $r$?

    $SA=4\pi r^2$. Represents the total surface as four times the area of a great circle of the sphere.

16. What is the circumference formula for a circle with radius $r$?

    $C=2\pi r$. Represents the total length around the circle as twice pi times the radius.

17. Identify the scale factor for perimeter if all linear dimensions are multiplied by $k$.

    Perimeter is multiplied by $k$. Scales linearly with the dimensions, so perimeter increases by the factor $k$.

18. What is the volume formula for a rectangular solid with dimensions $l,w,h$?

    $V=lwh$. Multiplies the three dimensions to determine the space occupied by the rectangular solid.

19. What is the area formula for 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 account for the trapezoid's shape.

20. What is the volume formula for a prism in terms of base area $B$ and height $h$?

    $V=Bh$. Multiplies the base area by the height to fill the prism's cross-sectional shape along its length.

21. What is the area formula for a parallelogram with base $b$ and height $h$?

    $A=bh$. Multiplies base by perpendicular height to find the enclosed area, regardless of side angles.

22. What is the area of a semicircle with radius $r$ (excluding the diameter as boundary detail)?

    $A=\frac{1}{2}\pi r^2$. Halves the full circle's area since a semicircle covers half the space without including the diameter's contribution.

23. Identify the scale factor for volume if all linear dimensions are multiplied by $k$.

    Volume is multiplied by $k^3$. Scales with the cube of linear dimensions, multiplying volume by $k^3$.

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

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

25. What is the total surface area formula for a right circular cylinder with radius $r$ and height $h$?

    $SA=2\pi r^2+2\pi rh$. Adds the areas of two bases and the lateral surface, which unwraps to a rectangle of width $2\pi r$ and height $h$.

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