Volume is the amount of SPACE INSIDE a 3D shape

How much the shape can HOLD

Measured in CUBIC units (cm³, m³, in³)

Remember: Volume = cubic units (cm³, m³), Area = square units (cm², m²), Perimeter = linear units (cm, m)

V = l × w × h

l = length

w = width

h = height

V = s³

or V = s × s × s

s = side length

Problem: Find the volume of a rectangular prism with length 8 cm, width 5 cm, and height 6 cm.

Problem: Find the volume of a cube with side length 2.5 cm.

Surface area is the TOTAL AREA of ALL FACES

of a 3D shape

Measured in SQUARE units (cm², m²)

SA = 2(lw + lh + wh)

or

SA = 2lw + 2lh + 2wh

l = length, w = width, h = height

SA = 6s²

s = side length

(6 faces, each with area s²)

Problem: Find the surface area of a rectangular prism with length 8 cm, width 5 cm, and height 6 cm.

V = ½ × b × h × l

or

V = (Area of base triangle) × length

b = base of triangle

h = height of triangle

l = length (or height) of prism

Step 1: Find area of the triangular base (½ × base × height)

Step 2: Multiply by the length of the prism

Problem: A triangular prism has a base triangle with base 6 cm and height 4 cm. The prism length is 10 cm. Find the volume.

SA = (s₁ + s₂ + s₃) × l + bh

or

SA = (Perimeter of base) × length + (2 × Area of triangle)

s₁, s₂, s₃ = three sides of triangle

l = length of prism

b = base, h = height of triangle

A triangular prism has:

• 2 triangular faces (the bases)

• 3 rectangular faces (the sides)

Add up ALL these areas!

Problem: Find surface area of a triangular prism. Base triangle has base 6 cm, height 4 cm, and sides 5 cm, 5 cm, 6 cm. Prism length is 10 cm.

SA = B + ½Pl

or

SA = Base Area + (½ × Perimeter × Slant Height)

B = area of base

P = perimeter of base

l = slant height

SA = s² + 2sl

s = side of square base

l = slant height

Important: Use SLANT HEIGHT (l), not the perpendicular height!

Problem: A square pyramid has base side 6 cm and slant height 8 cm. Find surface area.

Volume and Surface Area are INDEPENDENT

• Different shapes can have SAME volume but DIFFERENT surface area

• Different shapes can have SAME surface area but DIFFERENT volume

• They measure different things: Volume = space inside, SA = area outside

SA:V = Surface Area ÷ Volume

This ratio compares the outside to the inside

✓ Volume = cubic units (cm³, m³, in³)

✓ Surface Area = square units (cm², m², in²)

✓ Rectangular prism volume: multiply all three dimensions

✓ Cube: all sides equal, use s³ for volume, 6s² for SA

✓ Fractional lengths work the same way - just multiply carefully

✓ Surface area: add up ALL faces

✓ Triangular prism: Find triangle area first, then multiply by length

✓ Pyramid: Use SLANT HEIGHT (not perpendicular height)

✓ Volume ≠ Surface Area - they're independent!

✓ Always include units! Check if answer makes sense

Volume vs Surface Area:

"Volume is what you FILL, Surface Area is what you COVER with paint!"

Rectangular Prism Volume:

"Length Width Height - multiply all three for volume right!"

Cube Formulas:

"Cube to the 3rd for volume you need, 6 faces squared for surface area indeed!"

Triangular Prism:

"Triangle area first, times the length - that gives volume strength!"

Surface Area:

"Add up all the faces you can see - that's surface area for you and me!"

Units:

"3D means cubic (cm³), 2D means square (cm²) - remember this cue!"