8.2 3D Shapes
Understanding Surface Area
Surface area is the total area of all the faces of a 3D shape. Think of it as the amount of wrapping paper needed to cover the shape completely.
Surface area is the total area of all the faces of a 3D shape. Think of it as the amount of wrapping paper needed to cover the shape completely.
⚡ Surface Area Formulas:
Cuboid (Box)
SA = 2(lw + lh + wh)
Sum of all 6 rectangular faces
Triangular Prism
SA = 2 × triangle area + perimeter × length
2 triangular ends + 3 rectangles
Square Pyramid
SA = base² + 2 × base × slant height
1 square base + 4 triangular faces
Example 1: Cuboid Surface Area
A box has dimensions: length 8 cm, width 5 cm, height 3 cm. Find its surface area.
Formula:
$$\text{SA} = 2(lw + lh + wh)$$
Solution:
$\text{SA} = 2(8 \times 5 + 8 \times 3 + 5 \times 3)$
$\text{SA} = 2(40 + 24 + 15)$
$\text{SA} = 2(79)$
$\text{SA} = 158$ cm²
Answer: 158 cm²
$$\text{SA} = 2(lw + lh + wh)$$
Solution:
$\text{SA} = 2(8 \times 5 + 8 \times 3 + 5 \times 3)$
$\text{SA} = 2(40 + 24 + 15)$
$\text{SA} = 2(79)$
$\text{SA} = 158$ cm²
Answer: 158 cm²
Example 2: Triangular Prism Surface Area
A triangular prism has a triangular face with base 6 cm and height 4 cm. The prism is 10 cm long, and the triangle's sides are 5 cm each. Find the surface area.
Step 1: Calculate the area of the triangular ends
Triangle area $= \frac{1}{2} \times 6 \times 4 = 12$ cm²
Two ends $= 2 \times 12 = 24$ cm²
Step 2: Calculate the three rectangular faces
Base rectangle: $6 \times 10 = 60$ cm²
Two side rectangles: $2 \times (5 \times 10) = 100$ cm²
Total rectangles $= 60 + 100 = 160$ cm²
Step 3: Add all faces
$\text{SA} = 24 + 160 = 184$ cm²
Answer: 184 cm²
Step 1: Calculate the area of the triangular ends
Triangle area $= \frac{1}{2} \times 6 \times 4 = 12$ cm²
Two ends $= 2 \times 12 = 24$ cm²
Step 2: Calculate the three rectangular faces
Base rectangle: $6 \times 10 = 60$ cm²
Two side rectangles: $2 \times (5 \times 10) = 100$ cm²
Total rectangles $= 60 + 100 = 160$ cm²
Step 3: Add all faces
$\text{SA} = 24 + 160 = 184$ cm²
Answer: 184 cm²
Example 3: Square Pyramid Surface Area
A square pyramid has a base of 6 cm and slant height of 5 cm. Find its surface area.
Formula:
$$\text{SA} = \text{base}^2 + 2 \times \text{base} \times \text{slant height}$$
Solution:
Base area $= 6^2 = 36$ cm²
Four triangular faces $= 2 \times 6 \times 5 = 60$ cm²
$\text{SA} = 36 + 60 = 96$ cm²
Answer: 96 cm²
Note: The formula $2 \times \text{base} \times \text{slant}$ calculates all 4 triangular faces at once!
Formula:
$$\text{SA} = \text{base}^2 + 2 \times \text{base} \times \text{slant height}$$
Solution:
Base area $= 6^2 = 36$ cm²
Four triangular faces $= 2 \times 6 \times 5 = 60$ cm²
$\text{SA} = 36 + 60 = 96$ cm²
Answer: 96 cm²
Note: The formula $2 \times \text{base} \times \text{slant}$ calculates all 4 triangular faces at once!
💡 Surface Area Tips:
• Count the faces: Cuboid has 6, triangular prism has 5, square pyramid has 5
• Opposite faces are equal: In cuboids, opposite faces have the same area
• Slant height vs perpendicular height: Pyramids use slant height for surface area
• Units: Always squared (cm², m²) because it's area
• Opposite faces are equal: In cuboids, opposite faces have the same area
• Slant height vs perpendicular height: Pyramids use slant height for surface area
• Units: Always squared (cm², m²) because it's area
🎯 Surface Area Practice
cm²
cm²
Understanding Volume
Volume is the amount of 3D space inside a shape. It's measured in cubic units (cm³, m³).
Volume is the amount of 3D space inside a shape. It's measured in cubic units (cm³, m³).
⚡ Volume Formulas:
Cuboid
V = length × width × height
V = lwh
Prism
V = cross-section area × length
Any prism: base area × height
Cylinder
V = πr²h
Circle area × height
Key Principle: For any prism or cylinder:
$$\text{Volume} = \text{Area of cross-section} \times \text{Length/Height}$$
$$\text{Volume} = \text{Area of cross-section} \times \text{Length/Height}$$
Example 1: Cuboid Volume
A box has length 10 cm, width 6 cm, and height 4 cm. Find its volume.
Formula:
$$V = l \times w \times h$$
Solution:
$V = 10 \times 6 \times 4$
$V = 240$ cm³
Answer: 240 cm³
Formula:
$$V = l \times w \times h$$
Solution:
$V = 10 \times 6 \times 4$
$V = 240$ cm³
Answer: 240 cm³
Example 2: Triangular Prism Volume
A triangular prism has a triangular face with base 8 cm and height 5 cm. The prism is 12 cm long. Find its volume.
Step 1: Find the cross-section area (triangle)
$\text{Triangle area} = \frac{1}{2} \times 8 \times 5 = 20$ cm²
Step 2: Multiply by length
$V = 20 \times 12 = 240$ cm³
Answer: 240 cm³
Step 1: Find the cross-section area (triangle)
$\text{Triangle area} = \frac{1}{2} \times 8 \times 5 = 20$ cm²
Step 2: Multiply by length
$V = 20 \times 12 = 240$ cm³
Answer: 240 cm³
Example 3: Cylinder Volume
A cylinder has radius 4 cm and height 10 cm. Find its volume. Use $\pi = 3.14$
Formula:
$$V = \pi r^2 h$$
Solution:
$V = 3.14 \times 4^2 \times 10$
$V = 3.14 \times 16 \times 10$
$V = 502.4$ cm³
Answer: 502.4 cm³
$$V = \pi r^2 h$$
Solution:
$V = 3.14 \times 4^2 \times 10$
$V = 3.14 \times 16 \times 10$
$V = 502.4$ cm³
Answer: 502.4 cm³
💡 Volume Tips:
• Prism rule: Always cross-section area × length
• Units: Always cubed (cm³, m³) because it's 3D
• Cylinder is a circular prism: Circle area × height
• Check dimensions: Make sure all measurements use the same units
• Units: Always cubed (cm³, m³) because it's 3D
• Cylinder is a circular prism: Circle area × height
• Check dimensions: Make sure all measurements use the same units
🎯 Volume Calculations Practice
cm³
cm³
Advanced 3D Shapes
These shapes have curved surfaces and use more complex formulas involving π.
These shapes have curved surfaces and use more complex formulas involving π.
⚡ Advanced Volume Formulas:
Cone
V = ⅓πr²h
One-third of a cylinder
Sphere
V = ⁴⁄₃πr³
Perfect ball shape
Frustum
V = ⅓πh(R² + Rr + r²)
Cone with top cut off
Example 1: Cone Volume
A cone has radius 6 cm and perpendicular height 8 cm. Find its volume. Use $\pi = 3.14$
Formula:
$$V = \frac{1}{3}\pi r^2 h$$
Solution:
$V = \frac{1}{3} \times 3.14 \times 6^2 \times 8$
$V = \frac{1}{3} \times 3.14 \times 36 \times 8$
$V = \frac{1}{3} \times 904.32$
$V = 301.44$ cm³
Answer: 301.44 cm³
$$V = \frac{1}{3}\pi r^2 h$$
Solution:
$V = \frac{1}{3} \times 3.14 \times 6^2 \times 8$
$V = \frac{1}{3} \times 3.14 \times 36 \times 8$
$V = \frac{1}{3} \times 904.32$
$V = 301.44$ cm³
Answer: 301.44 cm³
Example 2: Sphere Volume
A sphere has radius 5 cm. Calculate its volume. Use $\pi = 3.14$
Formula:
$$V = \frac{4}{3}\pi r^3$$
Solution:
$V = \frac{4}{3} \times 3.14 \times 5^3$
$V = \frac{4}{3} \times 3.14 \times 125$
$V = \frac{4}{3} \times 392.5$
$V = 523.33$ cm³
Answer: 523.33 cm³
Formula:
$$V = \frac{4}{3}\pi r^3$$
Solution:
$V = \frac{4}{3} \times 3.14 \times 5^3$
$V = \frac{4}{3} \times 3.14 \times 125$
$V = \frac{4}{3} \times 392.5$
$V = 523.33$ cm³
Answer: 523.33 cm³
Example 3: Frustum Volume
A frustum (truncated cone) has top radius 3 cm, bottom radius 6 cm, and height 4 cm. Find its volume. Use $\pi = 3.14$
Formula:
$$V = \frac{1}{3}\pi h(R^2 + Rr + r^2)$$
where $R$ = large radius, $r$ = small radius
Solution:
$V = \frac{1}{3} \times 3.14 \times 4 \times (6^2 + 6 \times 3 + 3^2)$
$V = \frac{1}{3} \times 3.14 \times 4 \times (36 + 18 + 9)$
$V = \frac{1}{3} \times 3.14 \times 4 \times 63$
$V = \frac{1}{3} \times 791.28$
$V = 263.76$ cm³
Answer: 263.76 cm³
Formula:
$$V = \frac{1}{3}\pi h(R^2 + Rr + r^2)$$
where $R$ = large radius, $r$ = small radius
Solution:
$V = \frac{1}{3} \times 3.14 \times 4 \times (6^2 + 6 \times 3 + 3^2)$
$V = \frac{1}{3} \times 3.14 \times 4 \times (36 + 18 + 9)$
$V = \frac{1}{3} \times 3.14 \times 4 \times 63$
$V = \frac{1}{3} \times 791.28$
$V = 263.76$ cm³
Answer: 263.76 cm³
Cylinder vs Cone
Cone volume = ⅓ of cylinder volume
(same base & height)
Hemisphere
Half a sphere
Volume = $\frac{2}{3}\pi r^3$
🎯 Volume Practice:
cm³
cm³
Real Life Uses:
• Engineering: Calculating material amounts for manufacturing
• Architecture: Designing domes, towers, water tanks
• Engineering: Calculating material amounts for manufacturing
• Architecture: Designing domes, towers, water tanks