Area and Volume Relationships - ISEE Lower Level: Quantitative Reasoning

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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