Solid Geometry

The geometry of three-dimensional figures — prisms, pyramids, spheres, cylinders, and cones — and their properties.

Common 3-D solids

Definition

Solid geometry is the geometry of three-dimensional shapes. While flat (2D) geometry deals with areas and lengths, solid geometry adds a third dimension and introduces volume (the space inside) and surface area (the total area of the outer faces).

Common 3D shapes:

Prism: two identical parallel bases connected by rectangular faces (e.g., a box, a triangular prism)
Pyramid: a polygonal base tapering to a point (apex)
Cylinder: circular base, curved side, circular top
Cone: circular base tapering to a point
Sphere: all points equidistant from the centre in 3D

Each shape has its own volume formula, but they follow patterns: prisms and cylinders multiply base area by height; pyramids and cones are one-third of that.

Volume of a rectangular prism

A rectangular box is $4$ cm long, $3$ cm wide, and $5$ cm tall.

$V = \text{length} \times \text{width} \times \text{height} = 4 \times 3 \times 5 = 60 \text{ cm}^3$

The surface area (the total area of all six faces):

$SA = 2(4 \times 3) + 2(4 \times 5) + 2(3 \times 5) = 24 + 40 + 30 = 94 \text{ cm}^2$

Try it

A cylindrical can has radius $3$ cm and height $10$ cm. Find its volume and surface area. Use $\pi \approx 3.14$ .

Solution

$V = \pi r^2 h = \pi (3)^2 (10) = 90\pi \approx 282.6 \text{ cm}^3$

$SA = 2\pi r^2 + 2\pi r h = 2\pi(9) + 2\pi(3)(10) = 18\pi + 60\pi = 78\pi \approx 245.0 \text{ cm}^2$

Related concepts

Geometry· Measurement

Surface Area & VolumeExtending measurement to three dimensions — the total exposed surface and the space enclosed by a solid.

Geometry· Triangles