4.4 Quadrilaterals & 3-D Geometry

4.4 Quadrilaterals & 3-D Geometry

Quadrilaterals (four-sided polygons) and three-dimensional solid figures (rectangular prisms, cubes, and cylinders) represent the final major geometry topic on the GRE Quantitative Reasoning section. Test questions in this domain evaluate your understanding of polygon classifications, area and perimeter calculations, spatial reasoning, volume, total surface area, and 3-D space diagonals.

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1. Classification & Properties of Quadrilaterals

A quadrilateral is any four-sided polygon. The sum of the interior angles of any convex quadrilateral is always:

$$S = (4 - 2) \times 180^\circ = 360^\circ$$

Quadrilaterals form a hierarchical structure based on side parallelity, side length equality, and angle measures:

Quadrilateral Family Hierarchy & Formulas

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2. Deep Dive: Rhombuses and Trapezoids

• Rhombus Diagonals Property: The diagonals of a rhombus intersect at right angles ($90^\circ$) and bisect each other into four smaller congruent right triangles.
  • If a rhombus has side $s$ and diagonals $d_1, d_2$, then each right triangle has legs $\frac{d_1}{2}$ and $\frac{d_2}{2}$ and hypotenuse $s$: $$\left(\frac{d_1}{2}\right)^2 + \left(\frac{d_2}{2}\right)^2 = s^2$$
• Trapezoid Average Base: The area of a trapezoid can also be thought of as the average base multiplied by the height: $$\text{Area} = \text{Average Base} \times h = \left(\frac{b_1 + b_2}{2}\right) h$$

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3. Three-Dimensional (3-D) Geometry Solids

The GRE tests three main 3-D geometric figures: rectangular prisms, cubes, and right circular cylinders.

1. Rectangular Prisms (Box)

A 3-D solid bounded by six rectangular faces.

• Volume ($V$): Total space inside the prism. $$V = l \times w \times h$$
• Total Surface Area ($SA$): Total outer surface area of all six faces. $$SA = 2(lw + lh + wh)$$
• 3-D Space Diagonal ($d$): The straight line distance connecting two opposite vertices inside the prism. $$d = \sqrt{l^2 + w^2 + h^2}$$

2. Cubes

A special rectangular prism where length, width, and height are all equal to edge length $s$ ($l = w = h = s$).

• Volume ($V$): $$V = s^3$$
• Total Surface Area ($SA$): Six square faces of area $s^2$. $$SA = 6s^2$$
• 3-D Space Diagonal ($d$): $$d = \sqrt{s^2 + s^2 + s^2} = \sqrt{3s^2} = s\sqrt{3}$$

3. Right Circular Cylinders

A 3-D solid with two parallel circular bases of radius $r$ separated by height $h$.

• Volume ($V$): $$V = \pi r^2 h$$
• Lateral Surface Area: Area of the rolled side wall (excluding the top and bottom circles). $$\text{Lateral SA} = 2\pi r h$$
• Total Surface Area: Lateral surface area plus the area of the two circular caps. $$\text{Total SA} = 2\pi r h + 2\pi r^2 = 2\pi r(h + r)$$

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4. 3-D Solids Formulas Summary Table

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5. Worked GRE Step-by-Step Examples

Example 1: Rhombus Area and Perimeter from Diagonals

Problem: A rhombus has diagonals of lengths 12 and 16. Find the area and perimeter of the rhombus.

Solution:

1. Calculate Area using Diagonals Formula: $$\text{Area} = \frac{1}{2} d_1 d_2 = \frac{1}{2} \times 12 \times 16 = 96$$
2. Find Side Length $s$ using Diagonals:
   • The diagonals bisect each other at $90^\circ$, forming four right triangles with legs equal to half the diagonals: $$\text{Leg}_1 = \frac{12}{2} = 6, \quad \text{Leg}_2 = \frac{16}{2} = 8$$
   • Applying the Pythagorean Theorem to find side $s$ (the hypotenuse): $$s^2 = 6^2 + 8^2 = 36 + 64 = 100 \implies s = 10$$ (Note: This is a 6-8-10 multiple of the 3-4-5 Pythagorean triple!)
3. Calculate Perimeter: $$\text{Perimeter} = 4s = 4 \times 10 = 40$$

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Example 2: Cylinder Dimension Changes

Problem: A cylinder has radius $r$ and height $h$. If the radius is doubled and the height is reduced by half, how does the new volume compare to the original volume?

Solution:

1. Write Original Volume Formula: $$V_1 = \pi r^2 h$$
2. Substitute New Dimensions:
   • New radius $r' = 2r$
   • New height $h' = \frac{1}{2}h$
3. Calculate New Volume $V_2$: $$V_2 = \pi (r')^2 h' = \pi (2r)^2 \left(\frac{1}{2}h\right) = \pi (4r^2) \left(\frac{1}{2}h\right) = 2\pi r^2 h$$
4. Compare Volumes: $$V_2 = 2 V_1$$
5. Conclusion: The new volume is twice (2 times) the original volume.

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6. GRE Exam Strategies & Common Pitfalls

• Do Not Confuse Height with Slant Side in Trapezoids and Parallelograms: Always ensure height is measured perpendicular ($90^\circ$) to the base. If given a non-perpendicular side, drop a perpendicular line to create a right triangle and solve for the true height!
• Total vs. Lateral Surface Area: Read cylinder and box questions carefully! If a container is described as an "open-top box" or "open cylinder," omit the top cap area from your surface area sum.
• Space Diagonal 3D Pythagorean Theorem: The 3-D diagonal formula $d = \sqrt{l^2 + w^2 + h^2}$ is just applying the standard 2D Pythagorean Theorem twice. Memorizing the direct 3D formula saves multiple computational steps on test day.