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Area, Perimeter
& Volume

Master the essential formulas for 2D and 3D shapes tested on the GRE Quantitative section: from rectangles and circles to cylinders and spheres.

GRE Quantitative

What Is Perimeter?

The perimeter of a shape is the total length around its boundary. Think of it as the distance you would walk if you traced the entire outline of the figure.

Rectangle
\(l\)
\(w\)
\( P = 2(l+w) \)
Triangle
\(a\)
\(b\)
\(c\)
\( P = a + b + c \)
Circle
\(r\)
\( C = 2\pi r \)

What Is Area?

Area measures the amount of two-dimensional space enclosed by a shape. It is always expressed in square units (e.g., \(\text{cm}^2\), \(\text{ft}^2\)).

Rectangle
\( A = l \times w \)
Triangle
\( A = \tfrac{1}{2}bh \)
Circle
\( A = \pi r^2 \)
Trapezoid
\( A = \tfrac{1}{2}(b_1 + b_2)h \)

What Is Volume?

Volume measures the three-dimensional space occupied by a solid. It is always expressed in cubic units (e.g., \(\text{cm}^3\), \(\text{m}^3\)).

Box
\( V = lwh \)
Cylinder
\( V = \pi r^2 h \)
Sphere
\( V = \frac{4}{3}\pi r^3 \)

Core 2D Formulas

Shape Perimeter Area
Rectangle \( 2(l + w) \) \( lw \)
Square \( 4s \) \( s^2 \)
Triangle \( a + b + c \) \( \tfrac{1}{2}bh \)
Circle \( 2\pi r \) \( \pi r^2 \)
Trapezoid Sum of sides \( \tfrac{1}{2}(b_1 + b_2)h \)
Parallelogram \( 2(a + b) \) \( bh \)

Core 3D Formulas

Solid Volume Surface Area
Rectangular Box \( lwh \) \( 2(lw + lh + wh) \)
Cube \( s^3 \) \( 6s^2 \)
Cylinder \( \pi r^2 h \) \( 2\pi r(r + h) \)
Sphere \( \tfrac{4}{3}\pi r^3 \) \( 4\pi r^2 \)
Cone \( \tfrac{1}{3}\pi r^2 h \) \( \pi r(r + l) \)

Area vs. Perimeter: Key Distinction

Many GRE problems test whether you can distinguish between these two measurements. They measure fundamentally different things and are not directly proportional.

Area
Measures the space inside a shape. Units are squared: \(\text{m}^2\). Doubling all sides quadruples area.
Perimeter
Measures the boundary length. Units are linear: \(\text{m}\). Doubling all sides doubles perimeter.

Surface Area vs. Volume

Surface area is the total area of all faces of a 3D solid, while volume is the space contained inside. GRE questions may ask for one, the other, or both.

Surface Area
Like wrapping paper: the total outer covering. Units: \(\text{cm}^2\). Scale factor \( k \) multiplies SA by \( k^2 \).
Volume
Like filling with water: the total inner capacity. Units: \(\text{cm}^3\). Scale factor \( k \) multiplies volume by \( k^3 \).

How to Solve These Problems

1
Identify the shape. Read carefully: is it 2D or 3D? Which specific shape (rectangle, cylinder, etc.)?
2
Determine what is being asked. Perimeter? Area? Volume? Surface area? A combination?
3
Write the formula. Select the correct formula from your toolkit and write it before substituting.
4
Substitute and compute. Plug in values carefully. Watch units: keep them consistent throughout.
5
Check units and reasonableness. Area is in \(\text{units}^2\), volume in \(\text{units}^3\). Does the answer make sense?

Example 1: Rectangle

A rectangle has length \(12\) and width \(5\). Find the perimeter and area.

\(12\)
\(5\)
\(12\)
\(5\)

Example 2: Circle

A circle has radius \(7\). Find the circumference and area.

\(r = 7\)

Example 3: Shaded Region

A square with side \(10\) contains an inscribed circle. Find the area of the shaded region (the square minus the circle).

\(10\)
\(10\)
\(r = 5\)
shaded

Example 4: Cylinder

A cylinder has radius \(3\) and height \(10\). Find the volume and total surface area.

\(r = 3\)
\(h = 10\)

Example 5: Scaling Dimensions

If the radius of a sphere is tripled, by what factor do the surface area and volume change?

1
Original SA: \( 4\pi r^2 \). New SA: \( 4\pi(3r)^2 = 4\pi \cdot 9r^2 = 9 \times (4\pi r^2) \). Factor: \( 9 = 3^2 \)
2
Original Volume: \( \tfrac{4}{3}\pi r^3 \). New: \( \tfrac{4}{3}\pi(3r)^3 = 27 \times \tfrac{4}{3}\pi r^3 \). Factor: \( 27 = 3^3 \)

Common Mistakes to Avoid

✕ Confusing \( r \) and \( d \)
The GRE often gives the diameter. Divide by 2 before using a radius formula: \( r = d/2 \).
✕ Wrong height
Using the slant side instead of the perpendicular height in triangle and trapezoid area formulas.
✕ Unit mismatch
Mixing units (e.g., cm and m) in the same formula. Convert before substituting.

Question 1 of 5

A rectangle has a perimeter of 36 and a width of 8. What is its area?

Question 2 of 5

A circle has a diameter of 14. What is its area?

Question 3 of 5

A cube has a volume of 64. What is the surface area of the cube?

Question 4 of 5

If the radius of a cylinder is doubled while the height remains the same, by what factor does the volume increase?

Question 5 of 5

A trapezoid has parallel bases of 6 and 10, and a height of 4. What is its area?