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Properties of Simple Solid Figures

CSCA Properties of Simple Solid Figures study guide organized around the publicly available CSCA syllabus. Practice Mathematics questions on aicsca.com.

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Properties of Simple Solid Figures

In the CSCA exam, the focus of spatial geometry is on the **Equation of a Sphere** and the calculation of **Volume and Surface Area** for common solids (Cuboids, Cylinders, Cones).

1. Sphere

The sphere is the most frequently tested solid in analytic geometry. Definition: The set of points in space at a fixed distance (Radius) from a fixed point (Center).

#### 1.1 Equation of a Sphere

* **Standard Equation**:

Center $C(a, b, c)$, Radius $R$ ($R>0$):

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

*Special Case*: Center at Origin $O(0,0,0)$, then $x^2 + y^2 + z^2 = R^2$.

* **General Equation**:

$$x^2 + y^2 + z^2 + Dx + Ey + Fz + G = 0$$

* **Condition**: $D^2 + E^2 + F^2 - 4G > 0$.

* **Center**: $(-\frac{D}{2}, -\frac{E}{2}, -\frac{F}{2})$.

* **Radius**: $R = \frac{1}{2}\sqrt{D^2 + E^2 + F^2 - 4G}$.

#### 1.2 Geometric Properties

* **Volume**: $V = \frac{4}{3}\pi R^3$

* **Surface Area**: $S = 4\pi R^2$

2. Cuboid & Cube

Let the length, width, and height be $a, b, c$ (or $l, w, h$).

* **Space Diagonal**: The segment connecting opposite vertices.

$$d = \sqrt{a^2 + b^2 + c^2}$$

* **Total Surface Area**: $S = 2(ab + bc + ca)$

* **Volume**: $V = abc$

* **Cube**: When $a=b=c$, Diagonal $d = \sqrt{3}a$, Surface Area $S=6a^2$, Volume $V=a^3$.

3. Cylinder & Cone

These are often formed by rotation (Solids of Revolution). The exam focuses on their lateral surface nets (unrolled shapes) and formulas.

#### 3.1 Right Circular Cylinder

Base radius $r$, Height $h$.

* **Lateral Area**: $S_{\text{lat}} = 2\pi rh$ (Unrolls to a rectangle)

* **Total Area**: $S_{\text{total}} = 2\pi rh + 2\pi r^2$

* **Volume**: $V = \pi r^2 h$ (Base Area $\times$ Height)

#### 3.2 Right Circular Cone

Base radius $r$, Height $h$, **Slant Height** $l$.

* **Basic Relation**: $l^2 = h^2 + r^2$ (Pythagorean Theorem)

* **Lateral Area**: $S_{\text{lat}} = \pi rl$ (Unrolls to a sector)

* **Total Area**: $S_{\text{total}} = \pi rl + \pi r^2$

* **Volume**: $V = \frac{1}{3}\pi r^2 h$ ($\frac{1}{3} \times$ Base Area $\times$ Height)

4. Typical Examples

**Example 1**: Find the equation of the sphere passing through $P(4, 0, -1)$ with center $C(1, -2, 3)$.

**Solution**:

Radius $R = |CP| = \sqrt{(4-1)^2 + (0-(-2))^2 + (-1-3)^2} = \sqrt{9 + 4 + 16} = \sqrt{29}$.

Equation: $(x-1)^2 + (y+2)^2 + (z-3)^2 = 29$.

**Example 2**: A cone has a base diameter of 6 and height 4. Find its lateral area and volume.

**Solution**:

Radius $r = 3$, Height $h = 4$.

Slant height $l = \sqrt{3^2 + 4^2} = 5$.

Lateral Area $S_{\text{lat}} = \pi rl = 15\pi$.

Volume $V = \frac{1}{3}\pi r^2 h = 12\pi$.

**Example 3**: The sum of all edges of a cuboid is 48, and its space diagonal is $5\sqrt{2}$. Find its total surface area.

**Solution**:

Let dimensions be $a, b, c$. $4(a+b+c) = 48 \Rightarrow a+b+c = 12$.

$d = \sqrt{a^2+b^2+c^2} = 5\sqrt{2} \Rightarrow a^2+b^2+c^2 = 50$.

Using $(a+b+c)^2 = (a^2+b^2+c^2) + S_{\text{total}}$:

$144 = 50 + S_{\text{total}} \Rightarrow S_{\text{total}} = 94$.

5. Study Tips

* **Formulas**: Cone has $\frac{1}{3}$, Cylinder does not. Sphere Volume is $\frac{4}{3}$, Area factor is 4.

* **Geometry**: The right triangle formed by $h, r, l$ in a cone is crucial.

* **Completing the Square**: Always convert general sphere equations to standard form to find the center and radius.