Surface Area and Volume | Grade 10
📐 Lateral Area of Prisms and Cylinders
What is Lateral Area?
Lateral area (LA) is the area of all the sides of a 3D figure, excluding the bases (top and bottom).
🔹 Prism Lateral Area
LA = Ph
P = Perimeter of the base
h = Height of the prism
🔹 Cylinder Lateral Area
LA = 2πrh
r = Radius of the base
h = Height of the cylinder
📐 Lateral Area of Pyramids and Cones
🔹 Pyramid Lateral Area
LA = ½Pℓ
P = Perimeter of the base
ℓ = Slant height (distance from apex to base edge)
🔹 Cone Lateral Area
LA = πrℓ
r = Radius of the base
ℓ = Slant height
Note: Slant height ℓ = √(r² + h²)
📦 Surface Area of Prisms and Cylinders
What is Surface Area?
Surface area (SA) is the total area of all surfaces of a 3D figure, including the bases.
Surface Area = Lateral Area + Area of Bases
🔹 General Prism Surface Area
SA = Ph + 2B
P = Perimeter of the base
h = Height
B = Area of one base
🔹 Rectangular Prism Surface Area
SA = 2(lw + lh + wh)
l = Length, w = Width, h = Height
🔹 Cylinder Surface Area
SA = 2πr² + 2πrh = 2πr(r + h)
r = Radius of the base
h = Height
🔺 Surface Area of Pyramids and Cones
🔹 Pyramid Surface Area
SA = ½Pℓ + B
P = Perimeter of the base
ℓ = Slant height
B = Area of the base
🔹 Cone Surface Area
SA = πrℓ + πr² = πr(r + ℓ)
r = Radius of the base
ℓ = Slant height
🌐 Surface Area of Spheres
🔹 Sphere Surface Area
SA = 4πr²
r = Radius of the sphere
🔹 Hemisphere Surface Area
SA = 3πr²
Includes the curved surface (2πr²) and the circular base (πr²)
📊 Volume of Prisms and Cylinders
What is Volume?
Volume (V) is the amount of three-dimensional space enclosed by a solid figure, measured in cubic units.
🔹 Prism Volume
V = Bh
B = Area of the base
h = Height of the prism
🔹 Rectangular Prism Volume
V = lwh
l = Length, w = Width, h = Height
🔹 Cylinder Volume
V = πr²h
r = Radius of the base
h = Height
🔺 Volume of Pyramids and Cones
⚠️ Key Concept: Pyramid and cone volumes are ⅓ of prism and cylinder volumes
🔹 Pyramid Volume
V = ⅓Bh
B = Area of the base
h = Height (perpendicular distance from base to apex)
🔹 Cone Volume
V = ⅓πr²h
r = Radius of the base
h = Height (perpendicular distance from base to apex)
🌐 Volume of Spheres
🔹 Sphere Volume
V = (4/3)πr³
r = Radius of the sphere
🔹 Hemisphere Volume
V = (2/3)πr³
Half the volume of a sphere
🔄 Volume of Compound Figures
What are Compound Figures?
Compound figures (composite solids) are 3D shapes made by combining two or more basic solids.
📝 Steps to Find Volume of Compound Figures
Step 1: Break down the compound figure into simple solids (prisms, cylinders, pyramids, cones, spheres)
Step 2: Find the volume of each simple solid separately
Step 3: Add the volumes together (or subtract if there are hollow sections)
Vtotal = V1 + V2 + V3 + ...
📏 Similar Solids
What are Similar Solids?
Two solids are similar if they have the same shape but different sizes. All corresponding linear dimensions are proportional.
🔹 Scale Factor (k)
The scale factor is the ratio of corresponding linear dimensions.
k = length₁ / length₂ = width₁ / width₂ = height₁ / height₂
🔹 Surface Area Ratio
If the scale factor is k, then the ratio of surface areas is k²
SA₁ / SA₂ = k²
🔹 Volume Ratio
If the scale factor is k, then the ratio of volumes is k³
V₁ / V₂ = k³
📈 Surface Area and Volume: Changes in Scale
🔹 Effects of Scaling
When all linear dimensions are multiplied by a factor k:
| Measurement | Scale Factor | New Value |
|---|---|---|
| Linear (length, width, height) | k | k × original |
| Perimeter | k | k × original |
| Area | k² | k² × original |
| Surface Area | k² | k² × original |
| Volume | k³ | k³ × original |
💡 Important Note
✓ Doubling all dimensions (k = 2): Surface area becomes 4× larger, Volume becomes 8× larger
✓ Tripling all dimensions (k = 3): Surface area becomes 9× larger, Volume becomes 27× larger
✓ Halving all dimensions (k = ½): Surface area becomes ¼ as large, Volume becomes ⅛ as large
📐 Complete Formula Summary
| 3D Shape | Lateral Area | Surface Area | Volume |
|---|---|---|---|
| Prism | Ph | Ph + 2B | Bh |
| Rectangular Prism | 2h(l + w) | 2(lw + lh + wh) | lwh |
| Cylinder | 2πrh | 2πr(r + h) | πr²h |
| Pyramid | ½Pℓ | ½Pℓ + B | ⅓Bh |
| Cone | πrℓ | πr(r + ℓ) | ⅓πr²h |
| Sphere | — | 4πr² | (4/3)πr³ |
| Hemisphere | 2πr² | 3πr² | (2/3)πr³ |
📝 Variable Key
B = Area of the base | P = Perimeter of the base | h = Height
r = Radius | ℓ = Slant height | l = Length | w = Width
💡 Quick Reference Tips
✅ Lateral Area: Only the sides (no bases)
✅ Surface Area: All surfaces including bases
✅ Volume: Prisms and cylinders use full base area × height
✅ Volume: Pyramids and cones use ⅓ × base area × height
✅ Slant height (ℓ): For cones: ℓ = √(r² + h²)
✅ Similar solids: Linear ratio = k, Area ratio = k², Volume ratio = k³
✅ Compound figures: Break into simple shapes and add/subtract volumes
📚 Master these formulas for success in Tenth Grade Geometry! 📚