Area and Volume Pattern - Formula Calculation

This pattern asks you to calculate an area, perimeter, or volume by plugging given dimensions into the appropriate formula. Know the key formulas, substitute carefully, and simplify.

 

Perimeter of Rectangles and Squares

The perimeter of a rectangle is $P = 2l + 2w$. For a square with side $s$, it's $P = 4s$.

A rectangular community garden has a length of 12 meters and a width of 5 meters. What is the perimeter, in meters, of the garden?

A) $17$
B) $22$
C) $29$
D) $34$

$P = 2(12) + 2(5) = 24 + 10 = 34$. The answer is D. Option A is $l + w$ (half the perimeter). Option B adds only three sides.

A square floor tile has a side length of 14 inches. What is the perimeter, in inches, of the tile?

A) $14$
B) $28$
C) $56$
D) $196$

$P = 4(14) = 56$. The answer is C. Option D ($196$) is the area $14^2$, not the perimeter.

Common trap: Confusing perimeter with area. Perimeter is the total distance around; area is the space inside.

 

Area of Rectangles, Triangles, and Parallelograms

Rectangle: $A = lw$. Triangle: $A = \dfrac{1}{2}bh$. Parallelogram: $A = bh$.

What is the area, in square meters, of a triangle with a base of 10 meters and a height of 8 meters?

A) $18$
B) $36$
C) $40$
D) $80$

$A = \dfrac{1}{2}(10)(8) = 40$. The answer is C. Option D ($80$) forgets the $\dfrac{1}{2}$. Option A adds the dimensions instead of multiplying.

A rectangular garden plot has a length of 8 meters and a width of 5 meters. What is the area, in square meters?

A) $13$
B) $26$
C) $32$
D) $40$

$A = 8 \times 5 = 40$. The answer is D. Option A is $l + w$, and option B is $2(l + w)$ — that's the perimeter, not the area.

 

Area of Trapezoids

A trapezoid with parallel bases $b_1$ and $b_2$ and height $h$ has area $A = \dfrac{1}{2}(b_1 + b_2)h$.

A trapezoid has parallel bases of 8 inches and 12 inches and a height of 5 inches. What is the area, in square inches?

A) $25$
B) $50$
C) $100$
D) $240$

$A = \dfrac{1}{2}(8 + 12)(5) = \dfrac{1}{2}(20)(5) = 50$. The answer is B.

 

Area and Circumference of Circles

Circle area: $A = \pi r^2$. Circumference: $C = 2\pi r$. Remember that the radius is half the diameter.

A circle has a diameter of 38 inches. What is the area, in square inches?

A) $19\pi$
B) $38\pi$
C) $361\pi$
D) $1{,}444\pi$

The radius is $\dfrac{38}{2} = 19$. So $A = \pi(19)^2 = 361\pi$. The answer is C. Option D uses the diameter instead of the radius: $38^2 = 1{,}444$.

A circular garden has a radius of 31 feet. Which expression gives the circumference, in feet?

A) $\pi \cdot 31^2$
B) $2 \cdot 31$
C) $2\pi \cdot 31$
D) $\pi \cdot 31$

$C = 2\pi r = 2\pi(31)$. The answer is C. Option A is the area formula, not circumference.

Common trap: Using diameter in $\pi r^2$ instead of radius, giving an answer 4 times too large.

 

Volume of Rectangular Prisms and Cubes

Rectangular prism: $V = lwh$. Cube with edge $s$: $V = s^3$.

What is the volume, in cubic centimeters, of a rectangular prism with length 5, width 4, and height 3?

A) $12$
B) $20$
C) $120$
D) $60$

$V = 5 \times 4 \times 3 = 60$. The answer is D.

The edge length of a cube is 12 inches. What is the volume, in cubic inches?

A) $36$
B) $144$
C) $864$
D) $1{,}728$

$V = 12^3 = 1{,}728$. The answer is D. Option B ($144 = 12^2$) is the area of one face. Option A ($36 = 3 \times 12$) confuses the formula entirely.

 

Volume of Cylinders

A right circular cylinder with radius $r$ and height $h$ has volume $V = \pi r^2 h$.

A grain silo has an interior height of 10 feet and a radius of 6 feet. What is the volume, in cubic feet?

A) $60\pi$
B) $120\pi$
C) $360\pi$
D) $600\pi$

$V = \pi(6)^2(10) = 360\pi$. The answer is C. Option A uses $r$ instead of $r^2$.

A cylinder has a diameter of 14 cm and a height of 10 cm. What is the volume, in cubic centimeters?

A) $70\pi$
B) $140\pi$
C) $490\pi$
D) $1{,}960\pi$

Radius $= 7$. $V = \pi(7)^2(10) = 490\pi$. The answer is C. Option D ($1{,}960\pi$) uses the diameter: $\pi(14)^2(10)$.

 

Volume of Cones

A right circular cone has volume $V = \dfrac{1}{3}\pi r^2 h$.

A cone has a height of 9 inches and a base radius of 5 inches. What is the volume, in cubic inches?

A) $15\pi$
B) $75\pi$
C) $135\pi$
D) $225\pi$

$V = \dfrac{1}{3}\pi(5)^2(9) = \dfrac{1}{3}\pi(225) = 75\pi$. The answer is B. Option D forgets the $\dfrac{1}{3}$.

Watch for diameter vs. radius. If the problem says "base diameter of 48 meters," the radius is 24.

 

Volume of Spheres

A sphere with radius $r$ has volume $V = \dfrac{4}{3}\pi r^3$.

A sphere has a diameter of 12 inches. What is the volume, in cubic inches?

A) $144\pi$
B) $288\pi$
C) $576\pi$
D) $2{,}304\pi$

Radius $= 6$. $V = \dfrac{4}{3}\pi(6)^3 = \dfrac{4}{3}\pi(216) = 288\pi$. The answer is B.

A common SPR variant asks: "The volume is $k\pi$ cubic inches. What is $k$?" Just compute the volume and factor out $\pi$.

A sphere has a diameter of 6 inches. The volume is $k\pi$ cubic inches. What is $k$?

Radius $= 3$. $V = \dfrac{4}{3}\pi(3)^3 = \dfrac{4}{3}\pi(27) = 36\pi$. So $k = 36$.

 

Surface Area of Rectangular Prisms

The total surface area is $SA = 2lw + 2lh + 2wh$.

A wooden chest has length 25 in, width 12 in, and height 18 in. What is the exterior surface area?

$SA = 2(25)(12) + 2(25)(18) + 2(12)(18) = 600 + 900 + 432 = 1{,}932$ square inches.

A crate has length 30 in, width 18 in, and height 12 in. What is the surface area?

$SA = 2(30)(18) + 2(30)(12) + 2(18)(12) = 1{,}080 + 720 + 432 = 2{,}232$ square inches.

Tip: Surface area questions ask you to find the total area of all six rectangular faces. Organize your work by computing the area of each pair of opposite faces separately, then adding.

 

What to Do on Test Day

• Key formulas to memorize:
• Rectangle: $A = lw$, $P = 2l + 2w$
• Triangle: $A = \dfrac{1}{2}bh$
• Trapezoid: $A = \dfrac{1}{2}(b_1 + b_2)h$
• Circle: $A = \pi r^2$, $C = 2\pi r$
• Rectangular prism: $V = lwh$, $SA = 2lw + 2lh + 2wh$
• Cylinder: $V = \pi r^2 h$
• Cone: $V = \dfrac{1}{3}\pi r^2 h$
• Sphere: $V = \dfrac{4}{3}\pi r^3$
• Diameter vs. radius: If the problem gives a diameter, divide by 2 before using any formula with $r$. Using the diameter instead of the radius in $\pi r^2$ gives an answer 4 times too large — and that wrong answer will be one of the choices.
• Perimeter vs. area: Perimeter is the distance around; area is the space inside. The SAT always includes the other one as a trap.
• MCQ questions: If the answer choices contain $\pi$ (like $361\pi$), leave your answer in terms of $\pi$ — don't compute a decimal.
• SPR questions: You cannot type $\pi$ into the answer box. If the question asks for an exact numerical answer (like "What is $k$?" where the volume is $k\pi$), factor out $\pi$ and enter just the number. If the question asks for the actual area or volume, you must multiply by $\pi \approx 3.14159$ and give a decimal.
• "The volume is $k\pi$. What is $k$?" — This is the SAT's way of keeping $\pi$ out of your final answer. Compute the volume, factor out $\pi$, and enter the coefficient.