Mensuration 3D

3D Mensuration deals with calculating:

• Volume: Space occupied by a solid figure.

• Surface Area:

  • Curved Surface Area (CSA): The area around the sides (excluding top and bottom).
  • Total Surface Area (TSA): CSA + area of bases (top and bottom).

If 2D tells you how much paper to cover a shape, 3D tells you how much water can fill it — and how much paint is needed to coat it entirely.

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Key Solids and Their Formulas

We’ll cover:

• Cube
• Cuboid
• Cylinder
• Cone
• Sphere & Hemisphere
• Frustum of Cone
• Prism & Pyramid
• Composite Solids

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1. Cube

All sides = $a$

• Volume: $a^3$
• TSA: $6a^2$
• CSA: $4a^2$
• Diagonal: $\sqrt{3}a$

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2. Cuboid

Length = $l$, Breadth = $b$, Height = $h$

• Volume: $l \times b \times h$
• TSA: $2(lb + bh + hl)$
• CSA: $2h(l + b)$
• Diagonal: $\sqrt{l^2 + b^2 + h^2}$

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3. Cylinder

Radius = $r$, Height = $h$

• Volume:

  $$\pi r^2 h$$

• CSA:

  $$2\pi r h$$

• TSA:

  $$2\pi r(h + r)$$

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4. Cone

Radius = $r$, Height = $h$, Slant height = $l = \sqrt{r^2 + h^2}$

• Volume:

  $$\frac{1}{3}\pi r^2 h$$

• CSA:

  $$\pi r l$$

• TSA:

  $$\pi r(l + r)$$

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5. Sphere

Radius = $r$

• Volume:

  $$\frac{4}{3} \pi r^3$$

• Surface Area:

  $$4\pi r^2$$

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6. Hemisphere

Radius = $r$

• Volume:

  $$\frac{2}{3} \pi r^3$$

• CSA:

  $$2\pi r^2$$

• TSA:

  $$3\pi r^2$$

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7. Frustum of a Cone

Top radius = $r$, Bottom radius = $R$, Height = $h$, Slant height = $l = \sqrt{h^2 + (R - r)^2}$

• Volume:

  $$\frac{1}{3}\pi h(R^2 + r^2 + Rr)$$

• CSA:

  $$\pi (R + r)l$$

• TSA:

  $$\pi(R + r)l + \pi R^2 + \pi r^2$$

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8. Prism (Right)

Base area = $A$, Height = $h$

• Volume: $A \times h$
• TSA: $2A + \text{perimeter of base} \times h$

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9. Pyramid (Right)

Base area = $A$, Height = $h$, Slant height = $l$

• Volume:

  $$\frac{1}{3} A h$$

• TSA:

  $$A + \frac{1}{2} \times \text{Perimeter of base} \times l$$

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10. Composite Solids

• Volume = Sum of constituent volumes
• TSA = Add outer surface areas, subtract internal contacts
• Use only visible outer boundaries for surface area

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Visual Derivations and Insights

• Cylinder: Imagine rolling a rectangle around a circle.
• Cone: Think of slicing a pizza slice and rolling it.
• Sphere: Like revolving a semicircle about its diameter.
• Frustum: Cut a cone horizontally — apply subtraction.

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Conceptual Tips and Common Mistakes

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Examples

Example 1

A cube has edge 5 cm. Find TSA and volume.

• TSA = $6a^2 = 6 \times 25 = 150$ cm²
• Volume = $a^3 = 125$ cm³

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Example 2

A cylinder has radius 3 cm, height 7 cm. Find CSA and volume.

• CSA = $2\pi r h = 2 \times 3.14 \times 3 \times 7 = 131.88$ cm²
• Volume = $\pi r^2 h = 3.14 \times 9 \times 7 = 197.82$ cm³

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Example 3

A cone has radius 6 cm and height 8 cm. Find its CSA.

• Slant height $l = \sqrt{36 + 64} = 10$
• CSA = $\pi r l = 3.14 \times 6 \times 10 = 188.4$ cm²

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Example 4

Find the volume of a sphere with radius 5 cm.

$$V = \frac{4}{3} \pi r^3 = \frac{4}{3} \times 3.14 \times 125 = 523.33 \text{ cm}^3$$

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Example 5

A cone of height 16 cm is cut parallel to its base forming a frustum. Top and bottom radii are 3 cm and 5 cm. Find volume.

• Use:

  $$V = \frac{1}{3} \pi h (R^2 + r^2 + Rr) = \frac{1}{3} \pi \times 16 (25 + 9 + 15) = \frac{1}{3} \pi \times 16 \times 49 = \frac{784}{3} \pi \approx 821.9 \text{ cm}^3$$