Geometry III — 3D Figures and Coordinate Geometry

What This Module Covers

This module covers three-dimensional figures — boxes, cylinders, cones, and spheres — plus the coordinate geometry toolkit the GRE tests repeatedly: slope, distance, midpoint, and the equation of a circle. These two topic areas share a common thread: they both require you to apply the right formula to the right shape, and the GRE loves to set traps right at the formula level.

Why It Matters on the GRE

3D geometry and coordinate geometry together show up on nearly every GRE Quant section. The questions aren't trying to test advanced math — they're testing whether you have the right formulas memorized and whether you can apply them without slipping on a detail (like that sneaky 1/3 in the cone volume formula). A student who knows these cold can pick up several reliable points.

Core Concepts

Rectangular Prism (Box) and Cube

A rectangular prism — the GRE usually just calls it a "box" — has three dimensions: length, width, and height.

Must Know: Volume = l × w × h. Surface area = 2(lw + lh + wh). Space diagonal (the line from one corner through the interior to the opposite corner) = √(l² + w² + h²).

The space diagonal is one of the GRE's favorite 3D traps. It looks like a Pythagorean theorem problem, and it is — but you need all three dimensions, not just two. Think of it as applying the Pythagorean theorem twice in sequence.

A cube is just a box where all three sides are equal (side = s). That gives you: Volume = s³, Surface area = 6s², and Space diagonal = s√3.

Example — Space diagonal of a box:

1. A box has dimensions 3 × 4 × 12. Find the space diagonal.
2. Apply the formula: d = √(3² + 4² + 12²)
3. Calculate: √(9 + 16 + 144) = √169
4. d = 13

Verify: 9 + 16 = 25, and 25 + 144 = 169. √169 = 13. ✓

Cylinder

A cylinder has a circular base with radius r and a height h.

Must Know: Volume = πr²h. Lateral (curved) surface area = 2πrh. Total surface area = 2πrh + 2πr² (lateral surface plus both circular ends).

The total surface area formula trips students up because they forget to add the two circular caps. Picture unrolling the cylinder: you get a rectangle (the lateral surface) plus two circles at top and bottom.

Example — Cylinder volume:

1. A cylinder has radius 3 and height 5. Find the volume.
2. Volume = πr²h = π(3²)(5)
3. Volume = π(9)(5) = 45π

Cone

A cone has the same base as a cylinder — radius r — and height h.

Must Know: Volume = (1/3)πr²h. A cone holds exactly one-third the volume of a cylinder with the same base and height. This fraction is non-negotiable and the GRE tests it directly.

You can think of it this way: if you filled a cone with water and poured it into a matching cylinder, you'd need to do that exactly three times to fill the cylinder. That image is worth remembering on test day.

Sphere

A sphere has one measurement: radius r.

Must Know: Volume = (4/3)πr³. Surface area = 4πr².

Both formulas involve r² or r³ and a coefficient with 4 in it — don't mix them up. Volume has the (4/3) and the cube; surface area has 4 and the square.

Slope and Lines

Slope measures how steeply a line rises or falls. Given two points (x₁, y₁) and (x₂, y₂):

Must Know: Slope m = (y₂ − y₁) / (x₂ − x₁). Parallel lines have equal slopes. Perpendicular lines have slopes that are negative reciprocals: m₁ × m₂ = −1.

A positive slope rises from left to right; a negative slope falls. Horizontal lines have slope 0 (no rise). Vertical lines have undefined slope (division by zero — the run is 0).

The perpendicular slope rule catches students constantly. If one line has slope 2, the perpendicular line has slope −1/2. You flip the fraction AND change the sign. "Just make it negative" is wrong — that gives you −2, which is a different line entirely.

Example — Finding a perpendicular slope:

1. Line A has slope 3/4. What is the slope of a perpendicular line?
2. Flip the fraction: 4/3. Change the sign: −4/3.
3. Perpendicular slope = −4/3
4. Verify: (3/4) × (−4/3) = −12/12 = −1. ✓

Distance and Midpoint

Must Know: Distance = √[(x₂ − x₁)² + (y₂ − y₁)²]. Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

The distance formula is just the Pythagorean theorem in disguise — the horizontal and vertical gaps between the two points are the legs, and the distance is the hypotenuse. Always square the differences before adding; taking the square root of the sum of the unsquared differences is a common error.

Example — Distance between two points:

1. Find the distance between (1, 2) and (4, 6).
2. d = √[(4 − 1)² + (6 − 2)²]
3. d = √[3² + 4²] = √[9 + 16] = √25
4. d = 5

Equation of a Circle

Must Know: A circle centered at (h, k) with radius r has equation (x − h)² + (y − k)² = r². A point is ON the circle if it satisfies the equation exactly, INSIDE if the left side is less than r², and OUTSIDE if greater.

If the center is at the origin, the equation simplifies to x² + y² = r². Notice the equation uses r², so if the equation says x² + y² = 25, the radius is 5, not 25.

Graphing Lines

The standard form y = mx + b tells you everything. The slope is m, and b is the y-intercept (where the line crosses the y-axis). To find the x-intercept, set y = 0 and solve for x. Two lines intersect at the point (x, y) that satisfies both equations simultaneously — solve the system by substitution or elimination.

Common Traps

• The Cone's Missing Third: Students write πr²h for cone volume, forgetting the 1/3 factor. The cone formula is always (1/3)πr²h — burn that fraction into memory.
• Perpendicular vs. Opposite: Saying "perpendicular slopes are opposites" gets you −2 when the answer is −1/2. Perpendicular slopes are negative reciprocals. Flip AND negate.
• 2D Diagonal vs. 3D Diagonal: For a box, the space diagonal needs all three dimensions: √(l² + w² + h²). Using only two dimensions gives you the face diagonal, which is a different thing entirely.
• Distance Formula Shortcut Gone Wrong: Some students try to subtract the coordinates without squaring first. The formula requires squaring each difference before summing — skipping that step produces a meaningless number.
• Circle Radius vs. r²: If the circle equation reads (x − 2)² + (y + 3)² = 49, the radius is 7, not 49. Always take the square root of the right-hand side to get r.
• Surface Area vs. Volume Confusion: Both sphere formulas involve r and a multiple of 4 and π — but volume has (4/3)r³ and surface area has 4r². When in doubt, remember: volume is always one dimension "bigger" (cubic), surface area is always squared.

GRE Strategy

• Memorize the 3D formulas in a single sitting and group them: box (lwh), cylinder (πr²h), cone (1/3 × cylinder), sphere (4/3 πr³). The cone and sphere both have that fraction in front — that's your cue.
• On coordinate geometry problems, sketch the points. A 10-second sketch almost always reveals the answer pattern before you touch a formula.
• When you see "perpendicular," immediately write: m₁ × m₂ = −1. Then plug in the known slope and solve for the unknown. No guessing.
• For circle problems, rewrite the equation in standard form first. If you see x² + y² + 4x − 6y = 12, complete the square before doing anything else.
• On Quantitative Comparison problems with 3D figures, look for a ratio. If two shapes share a dimension, factor it out — the ratio often simplifies to a clean number like 3 (cone vs. cylinder) without requiring you to compute actual volumes.

Worked Example

Question: A cylinder and a cone have the same radius and the same height. The volume of the cylinder is how many times the volume of the cone?

(A) 1/3 (B) 1/2 (C) 2 (D) 3 (E) 4

Solution:

Step 1 — Write the formulas. Volume of cylinder = πr²h Volume of cone = (1/3)πr²h

Step 2 — Set up the ratio. Since the radius and height are the same for both, let's call them r and h and divide:

Cylinder ÷ Cone = πr²h ÷ (1/3)πr²h

Step 3 — Simplify. The πr²h terms cancel completely, leaving: 1 ÷ (1/3) = 1 × 3 = 3

Step 4 — State the answer. The cylinder's volume is 3 times the cone's volume. The answer is (D) 3.

Verify: Plug in numbers. Let r = 1, h = 3.

• Cylinder: π(1)²(3) = 3π
• Cone: (1/3)π(1)²(3) = π
• Ratio: 3π / π = 3 ✓

Why the other choices are wrong:

• (A) 1/3 — This is the inverted relationship. If you accidentally compared cone ÷ cylinder, you'd get 1/3. The question asks for cylinder ÷ cone.
• (B) 1/2 — There's no factor of 2 anywhere in these formulas. This answer has no basis; it's there to catch students who confuse this with a halving relationship.
• (C) 2 — Again, no factor of 2 in the relevant formulas. A student guessing "roughly double" without using the formulas would land here.
• (E) 4 — This would only appear if you thought the cone formula had a 1/4 factor. It doesn't. The cone formula has 1/3, full stop.

The key insight: you never needed actual numbers. The moment you write both formulas side by side, the ratio is obvious. That's the GRE Quant mindset — set up the structure, let the algebra do the work.