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7.6 Arc Length, Sector Area & 3D Geometry / Volume

Key Takeaways

  • Inscribed angles have their vertex on the circle and measure exactly half of their intercepted arc and central angle ($m\angle \text{inscribed} = \frac{1}{2} m\angle \text{central}$).
  • Arc length represents a fraction of circumference: $s = \frac{\theta^\circ}{360^\circ}(2\pi r)$ in degrees, or the simplified $s = r\theta$ when $\theta$ is in radians.
  • Sector area represents a fraction of total circle area: $A = \frac{\theta^\circ}{360^\circ}(\pi r^2)$ in degrees, or $A = \frac{1}{2}r^2\theta$ when $\theta$ is in radians.
  • Volume formulas for 3D solids (provided on the SAT reference sheet) include cylinders ($V = \pi r^2 h$), cones ($V = \frac{1}{3}\pi r^2 h$), spheres ($V = \frac{4}{3}\pi r^3$), and pyramids ($V = \frac{1}{3}Bh$).
  • Linear scaling of 3D solids expands surface area by $k^2$ and volume by $k^3$; when only a single dimension is altered (e.g., doubling radius), evaluate the formula powers directly ($r^2 \implies 2^2 = 4\times$ volume).
Last updated: August 2026

7.6 Arc Length, Sector Area & 3D Geometry / Volume

The final section of Geometry and Trigonometry synthesizes circular geometry (arcs, central angles, inscribed angles, and sectors) with three-dimensional solid geometry (volume and surface area). On the Digital SAT, these questions evaluate fractional proportionality, radian fluency, and dimensional scaling laws.

1. Central Angles vs. Inscribed Angles

+-----------------------------------------------------------------------------+
|                     CENTRAL VS. INSCRIBED ANGLE THEOREMS                    |
|                                                                             |
|   [ CENTRAL ANGLE ]                     [ INSCRIBED ANGLE ]                 |
|   Vertex is at the CENTER O             Vertex is ON the CIRCUMFERENCE      |
|                                                                             |
|                  A                                     A                    |
|                 /                                     /                     |
|                /                                     /                      |
|               O---θ                                 /                       |
|                \                                   B--- 2θ                  |
|                 \                                   \                       |
|                  C                                   \                      |
|                                                       C                     |
|   Central Angle = Arc Measure           Inscribed Angle = 1/2 * Arc Measure |
|   Angle AOC = Arc AC                    Angle ABC = 1/2 * Arc AC            |
|                                         Angle ABC = 1/2 * Angle AOC         |
+-----------------------------------------------------------------------------+

Core Inscribed Angle Corollaries

  1. Angles Subtending the Same Arc: Inscribed angles that intercept the exact same arc are equal in measure.
  2. Thales's Theorem (Inscribed Right Angle): An angle inscribed in a semicircle (subtending a diameter) is always a right angle ($90^\circ$).
+-----------------------------------------------------------------------------+
|                        THALES'S SEMICIRCLE THEOREM                          |
|                                                                             |
|                                    C (90 deg)                               |
|                                   / \                                       |
|                                  /   \                                      |
|                                 /     \                                     |
|                                /       \                                    |
|                             A +----O----+ B                                 |
|                                  Diameter                                   |
|                                                                             |
|   Whenever a triangle has a diameter as one side and its third vertex on    |
|   the circle, it is GUARANTEED to be a RIGHT TRIANGLE (Angle C = 90 deg).   |
+-----------------------------------------------------------------------------+

2. Arc Length & Sector Area Formulas

An arc is a fraction of the circle's circumference, and a sector (pie slice) is a fraction of the circle's total area.

+-----------------------------------------------------------------------------+
|                     ARC LENGTH & SECTOR AREA FORMULAS                       |
|                                                                             |
|   Measurement   Degrees Formula (θ in degrees)   Radians Formula (θ in rad) |
|   -----------   ------------------------------   -------------------------- |
|   Fraction      θ / 360°                         θ / 2π                     |
|   Arc Length    s = (θ / 360°) * 2πr             s = r * θ                  |
|   Sector Area   A = (θ / 360°) * πr²             A = 1/2 * r² * θ           |
+-----------------------------------------------------------------------------+

Worked Example 1: Arc Length and Sector Area in Radians

Problem: A circle has a radius of $8\text{ cm}$. A central angle of $\theta = \frac{3\pi}{4}\text{ radians}$ intercepts an arc of length $s$ and encloses a sector of area $A$. What are the values of $s$ and $A$?

Step-by-Step Solution:

  1. Compute arc length using $s = r\theta$: s=8×(3π4)=2×3π=6π cms = 8 \times \left(\frac{3\pi}{4}\right) = 2 \times 3\pi = 6\pi\text{ cm}
  2. Compute sector area using $A = \frac{1}{2}r^2\theta$: A=12(82)(3π4)=12(64)(3π4)=32×3π4=8×3π=24π cm2A = \frac{1}{2}(8^2)\left(\frac{3\pi}{4}\right) = \frac{1}{2}(64)\left(\frac{3\pi}{4}\right) = 32 \times \frac{3\pi}{4} = 8 \times 3\pi = 24\pi\text{ cm}^2

3. 3D Solid Geometry Formulas (What the Reference Sheet Gives You)

The Bluebook Reference Sheet is reachable from every question in both Math modules, not just at the start, so you never have to gamble on recalling a volume formula. It supplies all five volume formulas below. It supplies no surface-area formula at all — every SA formula here is one you must memorize.

+-----------------------------------------------------------------------------+
|                     3D SOLID GEOMETRY FORMULAS                              |
|                     [SHEET] = on the Reference Sheet                        |
|                     [MEMO]  = you must memorize it                          |
|                                                                             |
|   [ Rectangular Prism ]       [ Right Circular Cylinder ]                   |
|   [SHEET] V = l * w * h       [SHEET] V = π * r² * h                        |
|   [MEMO] SA = 2(lw+lh+wh)     [MEMO] SA = 2πr² + 2πrh (2 bases + wall)      |
|                                                                             |
|   [ Sphere ]                  [ Right Circular Cone ]                       |
|   [SHEET] V = (4/3) * π * r³  [SHEET] V = (1/3) * π * r² * h                |
|   [MEMO] SA = 4 * π * r²      (Note: 1/3 volume of matching cylinder!)      |
|                                                                             |
|   [ General Pyramid ]                                                       |
|   [SHEET] V = (1/3) * B * h   (Base area B = s² for a square pyramid)       |
+-----------------------------------------------------------------------------+

The Reference Sheet also carries the area and circumference of a circle, the area of a rectangle and a triangle, the Pythagorean theorem, the $30^\circ$-$60^\circ$-$90^\circ$ and $45^\circ$-$45^\circ$-$90^\circ$ ratios, and three facts: $360^\circ$ and $2\pi$ radians in a circle, and $180^\circ$ in a triangle. Everything else on this page — arc length, sector area, the inscribed angle theorem, the circle equation $(x-h)^2 + (y-k)^2 = r^2$, and all surface areas — is not provided.

4. 3D Proportional Scaling & Dimensional Exponent Laws

When scaling geometric solids, the exponent on the scale factor matches the dimension of the measurement:

Linear Dimension∝k,Surface Area∝k2,Volume∝k3\text{Linear Dimension} \propto k, \quad \text{Surface Area} \propto k^2, \quad \text{Volume} \propto k^3

+-----------------------------------------------------------------------------+
|                        3D SOLID SCALING RELATIONSHIPS                       |
|                                                                             |
|   Scenario: Every linear dimension of a solid is multiplied by k = 2.        |
|                                                                             |
|   - New Length:        2 * L                (Scale factor k = 2)            |
|   - New Surface Area:  2² * SA = 4 * SA     (Area quadruples)               |
|   - New Volume:        2³ * V  = 8 * V      (Volume octuples)               |
+-----------------------------------------------------------------------------+

Single-Dimension vs. Multi-Dimension Scaling

  • If all dimensions scale by $k$, Volume scales by $k^3$.
  • If only the radius of a cylinder doubles ($r \to 2r$) while height is constant: Vnew=π(2r)2h=4πr2h=4Vold(4× volume)V_{\text{new}} = \pi (2r)^2 h = 4\pi r^2 h = 4 V_{\text{old}} \quad (4\times \text{ volume})
  • If only the height of a cylinder doubles ($h \to 2h$) while radius is constant: Vnew=πr2(2h)=2πr2h=2Vold(2× volume)V_{\text{new}} = \pi r^2 (2h) = 2\pi r^2 h = 2 V_{\text{old}} \quad (2\times \text{ volume})

Worked Example 2: Volume Scaling and Density

Problem: A solid spherical lead ball of radius $3\text{ cm}$ has a mass of $1{,}280\text{ grams}$. A second lead ball made of the same uniform material has a radius of $6\text{ cm}$. What is the mass of the second lead ball in grams?

Step-by-Step Solution:

  1. Determine the linear scale factor $k$: k=r2r1=6 cm3 cm=2k = \frac{r_2}{r_1} = \frac{6\text{ cm}}{3\text{ cm}} = 2
  2. Determine the volume scale factor: Because volume scales as $k^3$: V2V1=k3=23=8\frac{V_2}{V_1} = k^3 = 2^3 = 8
  3. Calculate the mass of the second ball: Because density is constant, $\text{Mass} \propto \text{Volume}$: Mass2=8×Mass1=8×1,280=10,240 grams\text{Mass}_2 = 8 \times \text{Mass}_1 = 8 \times 1{,}280 = 10{,}240\text{ grams}

[!TIP] Desmos Geometry Shortcut: For complicated multi-step solid geometry questions, assign variables directly in Desmos: r = 6, h = 10, V = pi * r^2 * h. Then modify r = 2 * 6 to see the resulting volume change immediately.

Test Your Knowledge

In a circle with center O, points A, B, and C lie on the circumference. Angle ∠ABC is an inscribed angle with measure 42°. What is the measure of central angle ∠AOC subtending the same minor arc AC?

A
B
C
D
Test Your Knowledge

A circle has a radius of 12 centimeters. A sector of this circle is bounded by a central angle measuring 5π/6 radians. What is the area of this sector, in square centimeters?

A
B
C
D
Test Your Knowledge

A right circular cylinder has a base radius of r and a height of h, with a volume of V. If a second cylinder has a base radius of 3r and a height of (1/2)h, what is the volume of the second cylinder in terms of V?

A
B
C
D
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