Worked Examples: SAT Strategy
Example 1: Easy (Working Backward from Volume)
Question: A right circular cylinder has a volume of 432 cubic centimeters and the area of its base is 24 square centimeters. What is the height of the cylinder, in centimeters?
SAT Strategy: Do not get intimidated by the $\pi$ in the cylinder formula ($V = \pi r^2 h$). Remember that the area of the circular base is exactly $\pi r^2$.
- The volume of any prism or cylinder can be written as $V = B \times h$, where $B$ is the area of the base.
- Substitute the known values: $432 = 24 \times h$.
- Divide both sides by 24: $h = \frac{432}{24} = 18$.
- Answer: 18.
Example 2: Medium (Algebraic Dimensions)
Question: A rectangle has an area of 155 square inches. The length of the rectangle is 4 inches less than 7 times the width of the rectangle. What is the width of the rectangle, in inches?
SAT Strategy: Translate the text into an algebraic expression and set up a quadratic equation using the area formula $A = l \times w$.
- Let the width be $w$. The length is $7w - 4$.
- The area is $w(7w - 4) = 155$.
- Expand: $7w^2 - 4w = 155$.
- Move everything to one side: $7w^2 - 4w - 155 = 0$.
- You can use the Desmos calculator to find the roots, or factor. The positive root is $w = 5$. (Check: $7(5) - 4 = 31$, and $5 \times 31 = 155$).
- Answer: 5.
Example 3: Hard (Scaling Area)
Question: Triangle $ABC$ is similar to triangle $XYZ$. The area of triangle $ABC$ is $45$ square inches. If the lengths of the sides of triangle $XYZ$ are $3$ times the lengths of the corresponding sides of triangle $ABC$, what is the area of triangle $XYZ$ in square inches?
SAT Strategy: When a 2D shape's side lengths are scaled by a factor of $k$, its AREA is scaled by $k^2$. (If it were a 3D shape, its VOLUME would scale by $k^3$).
- The scale factor of the side lengths is $k = 3$.
- The scale factor for the area is $k^2 = 3^2 = 9$.
- Multiply the original area by 9: $45 \times 9 = 405$.
- Answer: 405.
Practice Questions
Level 1: Easy (Foundational SAT)
- A rectangle has a length of $12$ inches and a width of $5$ inches. What is the area of the rectangle, in square inches? (Grid-in)
- What is the volume of a rectangular prism with a length of $4$ cm, a width of $3$ cm, and a height of $10$ cm?
- A) $17$
- B) $60$
- C) $120$
- D) $144$
- A circle has a radius of $6$ meters. Which of the following is the area of the circle, in square meters?
- A) $6\pi$
- B) $12\pi$
- C) $36\pi$
- D) $144\pi$
- The area of a triangle is $40$ square centimeters. If the base of the triangle is $10$ centimeters, what is the height of the triangle in centimeters? (Grid-in)
- A right circular cylinder has a radius of $3$ and a height of $5$. What is its volume?
- A) $15\pi$
- B) $30\pi$
- C) $45\pi$
- D) $75\pi$
Level 2: Medium (Standard SAT)
- A square has an area of $81$ square inches. What is the perimeter of the square, in inches?
- A) $9$
- B) $18$
- C) $36$
- D) $81$
- The volume of a sphere is $\frac{256\pi}{3}$ cubic centimeters. What is the radius of the sphere, in centimeters? ($V = \frac{4}{3}\pi r^3$) (Grid-in)
- A rectangle has an area of $48$ square meters. The length of the rectangle is $2$ meters longer than its width. What is the length of the rectangle, in meters?
- A) $6$
- B) $8$
- C) $10$
- D) $12$
- The surface area of a cube is $150$ square centimeters. What is the volume of the cube, in cubic centimeters? (Grid-in)
- A right circular cone has a base area of $16\pi$ and a volume of $48\pi$. What is the height of the cone? ($V = \frac{1}{3}\pi r^2 h$)
- A) $3$
- B) $4$
- C) $9$
- D) $12$
Level 3: Hard (Advanced SAT)
- The circumference of a circle is $10\pi$ inches. What is the area of the circle, in square inches?
- A) $10\pi$
- B) $20\pi$
- C) $25\pi$
- D) $100\pi$
- Cylinder A and Cylinder B are similar right circular cylinders. The radius of Cylinder B is $2$ times the radius of Cylinder A. How many times greater is the volume of Cylinder B than the volume of Cylinder A? (Grid-in)
- A right rectangular prism has a square base with a side length of $x$ inches and a height of $5$ inches. If the total surface area of the prism is $112$ square inches, what is the value of $x$? (Grid-in)
- A rectangular pool is $10$ meters long and $5$ meters wide. The pool is filled with water to a depth of $2$ meters. A solid cube with a side length of $1$ meter is dropped into the pool and sinks completely to the bottom. Assuming no water spills out, by how many meters does the water level in the pool rise?
- A) $0.02$
- B) $0.10$
- C) $0.20$
- D) $0.50$
- A triangle has an area of $T$. If the base of the triangle is increased by $20%$ and the height is decreased by $10%$, what is the new area of the triangle in terms of $T$?
- A) $0.90T$
- B) $1.08T$
- C) $1.10T$
- D) $1.20T$
Level 4: Very Hard (Slightly beyond SAT)
- A right circular cone and a right circular cylinder have the same radius and the same volume. What is the ratio of the height of the cone to the height of the cylinder? (Grid-in)
- The diagonal of a square measures $10\sqrt{2}$ units. What is the area of the square, in square units? (Grid-in)
- A right circular cylinder has a radius of $r$ and a height of $h$. If the radius is increased by $10%$ and the height is decreased by $10%$, by what percentage does the volume of the cylinder increase?
- A) It decreases by $1%$
- B) It remains the same
- C) It increases by $8.9%$
- D) It increases by $10%$
- The volume of a solid right pyramid with a square base is $400$ cubic units. If the height of the pyramid is $12$ units, what is the perimeter of the square base?
- A) $10$
- B) $20$
- C) $40$
- D) $100$
- A sphere is inscribed perfectly inside a cube such that the sphere touches all six faces of the cube. What is the ratio of the volume of the sphere to the volume of the cube?
- A) $\frac{\pi}{6}$
- B) $\frac{\pi}{4}$
- C) $\frac{\pi}{3}$
- D) $\frac{2\pi}{3}$
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