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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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