LESSON 18

VOLUME & SURFACE AREA

Measuring Three-Dimensional Shapes

From Flat to Three-Dimensional

Volume measures how much three-dimensional space an object occupies — how much it can hold. Surface area measures the total area of all the outer faces of a 3D shape. While area and perimeter describe flat (2D) figures, volume and surface area describe solid (3D) objects.

These measurements appear constantly in real life. Volume tells you how much water fills a tank, how much concrete a foundation needs, or how much air fills a room. Surface area tells you how much paint covers a box, how much wrapping paper covers a gift, or how much material is needed to build a container.

Cubes and Rectangular Prisms

A rectangular prism (box) has length l, width w, and height h. Its volume is V = l × w × h — multiply all three dimensions. Its surface area is SA = 2(lw + lh + wh) — find the area of each pair of opposite faces and add them. A cube is a special prism where all sides are equal (s), so V = s³ and SA = 6s².

Example:Rectangular prism: l = 5, w = 3, h = 4 Volume: 5 × 3 × 4 = 60 cubic units Surface Area: 2(15 + 20 + 12) = 2(47) = 94 square units

Cylinders

A cylinder has two circular bases and a curved side. The volume is V = πr²h — the area of the circular base times the height. The surface area is SA = 2πr² + 2πrh — two circles (top and bottom) plus the rectangle that wraps around the side (its width is the circumference 2πr and its height is h).

Example:Cylinder: r = 3, h = 10 Volume: π(3²)(10) = 90π ≈ 282.74 cubic units Surface Area: 2π(9) + 2π(3)(10) = 18π + 60π = 78π ≈ 245.04 square units

Pyramids and Cones

A pyramid has a polygonal base and triangular faces that meet at a point (apex). Its volume is V = ⅓ × Base Area × height — one-third of what a prism with the same base and height would hold. A cone is like a circular pyramid: V = ⅓πr²h. The surface area of a cone is SA = πr² + πrl, where l is the slant height (the distance from the apex to the edge of the base).

Example:Square pyramid: base 6×6, height = 8 Volume: ⅓ × 36 × 8 = 96 cubic units Cone: r = 4, h = 3, slant height l = 5 Volume: ⅓π(16)(3) = 16π ≈ 50.27 cubic units Surface Area: π(16) + π(4)(5) = 16π + 20π = 36π ≈ 113.1 sq units

Spheres

A sphere is a perfectly round 3D shape where every point on the surface is the same distance (radius r) from the center. Its volume is V = (4/3)πr³ and its surface area is SA = 4πr². Notice that the surface area of a sphere equals exactly four times the area of a circle with the same radius — a beautiful relationship discovered by Archimedes.

Example:Sphere: r = 6 Volume: (4/3)π(216) = 288π ≈ 904.78 cubic units Surface Area: 4π(36) = 144π ≈ 452.39 square units

Key Facts

Volume is always measured in cubic units (cm³, m³, ft³) because it fills three-dimensional space. Surface area is in square units (cm², m²) — it is still a 2D measurement wrapped around a 3D shape.

Archimedes discovered that the volume of a sphere is exactly two-thirds the volume of the smallest cylinder that contains it. He considered this his greatest achievement and had it engraved on his tombstone.

A sphere has the smallest surface area for a given volume of any 3D shape. This is why soap bubbles are spherical — nature minimizes surface tension by minimizing surface area.

The Great Pyramid of Giza has a volume of about 2.6 million cubic meters. Ancient Egyptians calculated pyramid volumes using a formula equivalent to V = ⅓Bh — the same one we use today.

← LESSON 17 - AREA & PERIMETER LESSON 19 - THE COORDINATE PLANE →