Area and Volume Pattern - Inverse Calculation

This pattern gives you a measurement (like area, volume, or perimeter) and asks you to work backward to find a missing dimension. The key is to set up the formula, substitute what you know, and solve for the unknown.

 

Finding a Missing Side from Perimeter

To find a missing side of a polygon, subtract the known sides from the total perimeter.

The perimeter of a quadrilateral is 30 inches. Three sides measure 5, 7, and 10 inches. What is the fourth side?

A) $2$
B) $8$
C) $22$
D) $52$

Fourth side $= 30 - 5 - 7 - 10 = 8$. The answer is B.

A rectangular garden has a perimeter of 28 meters. One side is 5 meters. What is the adjacent side?

A) $5$
B) $9$
C) $10$
D) $18$

$P = 2l + 2w$, so $28 = 2(5) + 2w$, giving $2w = 18$ and $w = 9$. The answer is B. Option D ($18$) forgets to divide by 2.

 

Finding a Missing Angle

The interior angles of any quadrilateral sum to $360°$.

In quadrilateral $PQRS$, the angles are $70°$, $110°$, and $85°$. What is the fourth angle?

A) $85$
B) $95$
C) $155$
D) $265$

$360 - 70 - 110 - 85 = 95°$. The answer is B.

 

Finding the Height of a Triangle from Its Area

Use $A = \dfrac{1}{2}bh$ and solve for $h$: $h = \dfrac{2A}{b}$.

A triangular banner has an area of 60 square feet and a base of 12 feet. What is the height?

A) $5$
B) $6$
C) $10$
D) $48$

$h = \dfrac{2(60)}{12} = \dfrac{120}{12} = 10$. The answer is C. Option A divides area by base without the factor of 2.

A triangle has area 108 square meters and base 12 meters. What is the height?

A) $9$
B) $12$
C) $96$
D) $18$

$h = \dfrac{2(108)}{12} = 18$. The answer is D. Option A ($9$) divides $108 \div 12$ without doubling first.

 

Finding the Edge Length of a Cube from Its Volume

Since $V = s^3$, the edge length is $s = \sqrt[3]{V}$.

The volume of a cube is 27 cubic centimeters. What is the edge length?

A) $9$
B) $3$
C) $54$
D) $27$

$s = \sqrt[3]{27} = 3$. The answer is B.

 

Finding the Height of a Prism from Its Volume

For any prism, $V = (\text{base area}) \times h$, so $h = \dfrac{V}{\text{base area}}$.

A rectangular prism has volume 390 cubic inches and base area 26 square inches. What is the height?

A) $26$
B) $195$
C) $15$
D) $10{,}140$

$h = \dfrac{390}{26} = 15$. The answer is C. Option D multiplies instead of dividing.

A right triangular prism has volume 1,200 cubic centimeters. The base is a right triangle with legs 10 cm and 12 cm. What is the height of the prism?

A) $10$
B) $20$
C) $22$
D) $60$

Base area $= \dfrac{1}{2}(10)(12) = 60$. Height $= \dfrac{1{,}200}{60} = 20$. The answer is B.

 

Finding a Cylinder's Height or Radius from Its Volume

$V = \pi r^2 h$. Solve for the missing dimension.

A cylinder has volume $300\pi$ cubic centimeters and radius 5 cm. What is the height?

A) $25$
B) $60$
C) $12$
D) $30$

$300\pi = \pi(5)^2 h = 25\pi h$, so $h = \dfrac{300}{25} = 12$. The answer is C.

A cylinder has volume $891\pi$ cubic inches and height 11 inches. What is the radius?

$891\pi = \pi r^2(11)$, so $r^2 = \dfrac{891}{11} = 81$, giving $r = 9$.

A cylinder has volume $576\pi$ cubic centimeters and height 9 cm. What is the radius?

$r^2 = \dfrac{576}{9} = 64$, so $r = 8$.

 

Finding a Cone's Radius from Its Volume

$V = \dfrac{1}{3}\pi r^2 h$. Solve for $r$: $r^2 = \dfrac{3V}{\pi h}$.

A cone has height 12 cm and volume $324\pi$ cubic centimeters. What is the radius?

$324\pi = \dfrac{1}{3}\pi r^2(12) = 4\pi r^2$, so $r^2 = \dfrac{324}{4} = 81$ and $r = 9$.

 

Finding Volume from Surface Area (Spheres)

Sphere surface area: $SA = 4\pi r^2$. Sphere volume: $V = \dfrac{4}{3}\pi r^3$.

Given the surface area, first find $r$ from $SA = 4\pi r^2$, then compute the volume.

A sphere has surface area $324\pi$ square inches. The volume is $k\pi$ cubic inches. What is $k$?

$4\pi r^2 = 324\pi$, so $r^2 = 81$ and $r = 9$. Then $V = \dfrac{4}{3}\pi(9)^3 = \dfrac{4}{3}\pi(729) = 972\pi$. So $k = 972$.

A sphere has surface area $900\pi$ square feet. The volume is $k\pi$ cubic feet. What is $k$?

$r^2 = \dfrac{900}{4} = 225$, so $r = 15$. $V = \dfrac{4}{3}\pi(15)^3 = \dfrac{4}{3}\pi(3{,}375) = 4{,}500\pi$. So $k = 4{,}500$.

 

Multi-Step Inverse: Height Equals Diameter, Then Find Lateral Surface Area

Some hard problems chain two inversions: find a dimension from volume, then use it to compute a surface area.

A cylinder has volume $432\pi$ cubic inches and its height equals its diameter. The lateral surface area is $k\pi$ square inches. What is $k$?

Let $r$ be the radius. Then height $= 2r$. $V = \pi r^2(2r) = 2\pi r^3 = 432\pi$, so $r^3 = 216$ and $r = 6$. Height $= 12$. Lateral surface area $= 2\pi r h = 2\pi(6)(12) = 144\pi$. So $k = 144$.

Strategy: When you see "the volume is $k\pi$" or "the surface area is $k\pi$," factor out $\pi$ early. It simplifies the arithmetic and you just need to find $k$.

 

What to Do on Test Day

• The process is always the same: plug known values into the formula, then solve for the unknown. The only question is which formula to use.
• Key inverse formulas:
• Height of a triangle: $h = \dfrac{2A}{b}$ — don't forget the factor of 2
• Side from perimeter: subtract the known sides from the total
• Edge of a cube: $s = \sqrt[3]{V}$
• Height of a prism: $h = \dfrac{V}{\text{base area}}$
• Height of a cylinder: $h = \dfrac{V}{\pi r^2}$
• Radius of a cylinder: $r = \sqrt{\dfrac{V}{\pi h}}$
• The most common trap is forgetting a factor of 2 or $\dfrac{1}{3}$. For triangles, $A = \dfrac{1}{2}bh$, so $h = \dfrac{2A}{b}$, not $\dfrac{A}{b}$. For cones, $V = \dfrac{1}{3}\pi r^2 h$, so solving for $r$ requires multiplying by 3.
• "What is $k$?" — Factor out $\pi$ early to simplify your arithmetic. The answer is just the number in front of $\pi$.
• SPR vs. MCQ: On SPR questions you cannot type $\pi$, so if the problem asks for an actual measurement (not $k$), compute the decimal. On MCQ, match the form of the answer choices.
• Two-step inversions (find a dimension from volume, then use it in another formula) are common on harder questions. Label each intermediate result clearly.