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Spatial Geometry

CSCA Spatial Geometry study guide organized around the publicly available CSCA syllabus. Practice Mathematics questions on aicsca.com.

Before planning this topic, check the CSCA Exam Guide 2026 for exam dates, registration, fees, and subject requirements.

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This study guide is organized around the publicly available CSCA syllabus for international undergraduate applicants.

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International students preparing for CSCA Math, Physics, Chemistry, or Chinese exams.

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Spatial Geometry

Spatial geometry extends mathematics from the 2D plane to 3D space. In the CSCA exam, the focus is on **basic calculations in the spatial coordinate system** (distance, midpoint), **formulas for solids** (prism, cone, sphere), and the simple **equation of a sphere**.

1. Spatial Rectangular Coordinate System

To determine positions in space, we establish a system with three mutually perpendicular axes ($x, y, z$). The origin is $O$.

* **Coordinates of Point P**: $(x, y, z)$.

* **Basic Formulas**:

Let $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$.

1. **Distance Formula**: An extension of the planar distance formula.

$$|AB| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$

2. **Midpoint Formula**: The midpoint $M$ of segment $AB$ is $(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2})$.

2. Solids and Formulas

This is a high-frequency topic for calculation problems. Memorize the Volume ($V$) and Surface Area ($S$) formulas.

| Type | Solid | Volume Formula ($V$) | Lateral/Surface Area ($S$) |

| :--- | :--- | :--- | :--- |

| **Prism/Cylinder** | Cylinder, Prism | $$V = S_{\text{base}} \cdot h$$ | $S_{\text{cyl_lateral}} = 2\pi rh$ |

| **Pyramid/Cone** | Cone, Pyramid | $$V = \frac{1}{3} S_{\text{base}} \cdot h$$ | $S_{\text{cone_lateral}} = \pi rl$ ($l$ is slant height) |

| **Sphere** | Sphere | $$V = \frac{4}{3}\pi R^3$$ | $$S = 4\pi R^2$$ |

3. Equation of a Sphere

The set of points in space at a fixed distance (Radius) from a fixed point (Center).

* **Standard Equation**:

Center $(a, b, c)$, Radius $R$:

$$(x-a)^2 + (y-b)^2 + (z-c)^2 = R^2$$

* **Special Case**: If the center is at the origin $(0,0,0)$, the equation is $x^2 + y^2 + z^2 = R^2$.

4. Spatial Vector Basics

Operations for spatial vectors $\vec{a} = (x, y, z)$ are analogous to plane vectors:

* **Magnitude**: $|\vec{a}| = \sqrt{x^2+y^2+z^2}$.

* **Add/Sub**: Add or subtract corresponding coordinates.

* **Dot Product**: $\vec{a} \cdot \vec{b} = x_1x_2 + y_1y_2 + z_1z_2$.

* **Perpendicularity**: $\vec{a} \perp \vec{b} \iff \vec{a} \cdot \vec{b} = 0$.

5. Practice Examples

**Example 1**: Find the distance from point $A(1, 2, 3)$ to the origin $O$.

**Solution**: $|AO| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14}$.

**Example 2**: A sphere has its center at $(1, -2, 0)$ and radius 3. Find its equation and check if $P(3, 0, 1)$ lies on it.

**Solution**:

Equation: $(x-1)^2 + (y+2)^2 + z^2 = 9$.

Substitute $P(3, 0, 1)$: $(3-1)^2 + (0+2)^2 + 1^2 = 4 + 4 + 1 = 9$.

Since $9 = 9$, point $P$ is on the sphere.

**Example 3**: A cone has a base radius of 3 and a height of 4. Find its volume.

**Solution**:

Base Area $S_{\text{base}} = \pi r^2 = 9\pi$.

Volume $V = \frac{1}{3} S_{\text{base}} h = \frac{1}{3} \cdot 9\pi \cdot 4 = 12\pi$.