8.2 Congruent & Similar Figures, 3D Figures, Area/Perimeter/Volume
Key Takeaways
- Congruent figures have identical shape AND size — all corresponding sides and angles equal; similar figures have identical shape with proportional sides and equal corresponding angles.
- If two similar 2D figures have linear scale factor k, their areas scale by k²; for similar 3D figures, surface areas scale by k² and volumes scale by k³.
- Rectangular solid volume = lwh, surface area = 2(lw + lh + wh); cylinder volume = πr²h, surface area = 2πr² + 2πrh.
- Cone volume = ⅓πr²h; sphere volume = (4/3)πr³, sphere surface area = 4πr²; pyramid volume = ⅓ × base area × height.
- Doubling a sphere's radius multiplies its volume by 8 and its surface area by 4 — a classic GRE Quantitative Comparison trap.
Congruent & Similar Figures, 3D Figures, Area/Perimeter/Volume
Quick Answer: Congruent shapes match exactly in size and shape; similar shapes share angles with proportional sides. The scaling rules — area as k², volume as k³ — are the single most tested idea in this section, and the 3D formula set (rectangular solid, cylinder, cone, sphere, pyramid) covers every volume question on the test.
Congruent Figures
Two geometric figures are congruent if they have the same shape and the same size — all corresponding sides equal, all corresponding angles equal. Congruent triangles follow specific tests (SSS, SAS, ASA, AAS, HL for right triangles), but on the GRE you mostly use congruence as a label: when the question states two figures are congruent, treat every corresponding measure as equal without further proof.
Similar Figures
Two figures are similar if they have the same shape but not necessarily the same size — corresponding angles equal and corresponding sides proportional. The constant of proportionality k is the similarity ratio (or linear scale factor).
If figure B is similar to figure A with linear scale factor k (each side of B is k times the corresponding side of A), then:
- Perimeter of B = k × perimeter of A.
- Area of B = k² × area of A.
- (For 3D) Surface area of B = k² × surface area of A.
- (For 3D) Volume of B = k³ × volume of A.
Worked Example — Similar Triangles and Area
Triangle ABC has area 18. Triangle DEF is similar to ABC with each side 3 times the corresponding side of ABC. What is the area of DEF?
- Linear scale factor k = 3.
- Area scale factor = k² = 9.
- Area of DEF = 9 × 18 = 162.
Worked Example — QC on Similar Solids
Quantity A: Volume of a sphere with radius 4. Quantity B: Volume of 8 spheres each of radius 2.
- Volume scales as k³. The radius ratio is 4/2 = 2, so the larger sphere has 2³ = 8 times the volume of one small sphere.
- Quantity A = 8 × (volume of one small sphere).
- Quantity B = 8 × (volume of one small sphere).
- The two quantities are equal — answer (C).
This is a classic GRE shortcut: eight small spheres of half the radius pack exactly into the volume of one large sphere of the original radius.
Three-Dimensional Figures and Their Formulas
| Figure | Volume | Surface Area |
|---|---|---|
| Rectangular solid (l × w × h) | lwh | 2(lw + lh + wh) |
| Cube (side s) | s³ | 6s² |
| Cylinder (radius r, height h) | πr²h | 2πr² + 2πrh |
| Cone (radius r, height h) | ⅓πr²h | πr² + πrℓ (ℓ = slant height) |
| Sphere (radius r) | (4/3)πr³ | 4πr² |
| Pyramid (base area B, height h) | ⅓ × B × h | B + (½ × perimeter × slant height) |
The cylinder is a "stacked circle": its volume is the circle area πr² times the height h. The cone and pyramid each carry the factor ⅓ — the same base-and-height formula as a cylinder or rectangular solid but divided by 3. Forgetting the ⅓ is the most common cone/pyramid error on the GRE.
Worked Example — Cylinder Volume
A cylindrical tank has radius 5 m and height 12 m. What is its volume, in cubic meters?
- V = πr²h = π × 25 × 12 = 300π ≈ 942 cubic meters.
Worked Example — Cube Versus Sphere
A cube has side length 6. What is the radius of a sphere with the same volume as the cube?
- Cube volume = 6³ = 216.
- Sphere: (4/3)πr³ = 216 → r³ = 162/π ≈ 51.57 → r ≈ 3.72.
- This kind of "equal volume" comparison is a frequent Numeric Entry framing.
Scaling Relationships: The Single Most Useful Shortcut
When a GRE question changes a figure's linear dimensions, the area and volume changes follow the square and cube of the linear ratio. Memorize these multipliers:
| Linear scale k | Area multiplier k² | Volume multiplier k³ |
|---|---|---|
| 2 | 4 | 8 |
| 3 | 9 | 27 |
| 4 | 16 | 64 |
| ½ | ¼ | ⅛ |
| ⅓ | 1/9 | 1/27 |
Worked Example — Doubling Dimensions
A rectangular solid has dimensions 3 × 4 × 5 (volume 60). If each dimension is doubled, what is the new volume?
- Linear scale k = 2 → volume multiplier = 8.
- New volume = 8 × 60 = 480. (Direct check: 6 × 8 × 10 = 480 ✓.)
Worked Example — Halving a Sphere
If the radius of a sphere is halved, by what factor does the surface area change?
- Linear scale k = ½ → surface area multiplier = (½)² = ¼.
- New surface area = ¼ × original.
Perimeter, Area, and Circumference — Quick Reference
Keep the 2D formulas at your fingertips since 3D surface areas build on them:
- Square: A = s², P = 4s.
- Rectangle: A = lw, P = 2(l + w).
- Triangle: A = ½bh, P = a + b + c.
- Parallelogram: A = bh.
- Trapezoid: A = ½(b₁ + b₂)h.
- Circle: A = πr², C = 2πr.
Why This Matters for the GRE
Roughly one in four geometry items on the GRE involves either similarity scaling or a 3D volume formula. The scaling shortcuts let you bypass the on-screen calculator entirely — recognizing that doubling a radius multiplies a sphere's volume by 8 turns a multi-step computation into a one-line comparison. The trap is forgetting which dimension scales which way: area always squares, volume always cubes.
Two similar triangles have corresponding sides in the ratio 3 : 5. If the smaller triangle has area 36, what is the area of the larger triangle?
A cylinder has radius 4 and height 10. What is its volume in terms of π?
Quantity A: The volume of a sphere with radius 6. Quantity B: The volume of 27 spheres each with radius 2. Which is larger?
Each dimension of a rectangular solid is doubled. By what factor does the surface area increase?