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

🔹 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)

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

🔹 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

🔹 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)

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

⚠️ 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)

🔹 Sphere Volume

V = (4/3)πr³

r = Radius of the sphere

🔹 Hemisphere Volume

V = (2/3)πr³

Half the volume of a sphere

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)

V_total = V₁ + V₂ + V₃ + ...

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

🔹 Effects of Scaling

When all linear dimensions are multiplied by a factor k:

💡 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

📝 Variable Key

B = Area of the base | P = Perimeter of the base | h = Height

r = Radius | ℓ = Slant height | l = Length | w = Width