Volume of 3D Figures

Solid Geometry — Prisms, Cylinders, Pyramids, Cones, Spheres

In this lesson:

• Apply volume formulas for all five solid types
• Select the right formula from shape clues
• Find a missing radius or height from a volume

What You Will Learn Today

After this lesson, you will:

1. Prisms and cylinders:
2. Pyramids and cones:
3. Spheres:
4. Identify the formula from name, description, or diagram
5. Find a missing radius or height from a volume

One Formula Unifies Prisms and Cylinders

Recall , , . Then is really

Box 8 × 5 × 3 cm: , so .

V = Bh: The Cross-Section Idea

The base is the shape you see when you slice perpendicular to the height.

Triangular Prism and Unusual Bases

Triangle base 6 in, triangle height 4 in, length 10 in.

, so .

Hexagonal prism, , cm:

Worked Example With a Right Cylinder

A cylinder has radius 5 cm and height 8 cm.

Step 1: Compute the base area

Step 2: Multiply by height

What Happens When Solids Taper to a Point

• A pyramid or cone has the volume of the matching prism or cylinder
• Same base, same height, but tapered

Worked Example With a Square Pyramid

A square pyramid has base edge 10 m and height 15 m.

Step 1: Compute the base area

Step 2: Apply the 1/3 rule

Worked Example With a Right Cone

A cone has radius 3 cm and height 8 cm.

Step 1: Compute the base area

Step 2: Apply the 1/3 rule

Cone Versus Its Matching Cylinder

Same and :

• Cylinder:
• Cone:

The cone is exactly one-third of the cylinder.

Spheres: The Odd One Out

The sphere formula doesn't follow the pattern. It depends only on the radius.

Example: A sphere with cm:

Watch Out for the Diameter Trap

A sphere has diameter 10 cm. Find the volume.

Step 1: cm

Step 2: Apply the formula

Make your first written line.

A Sphere Inside a Cylinder

• Cylinder:
• Sphere:
• Ratio:

Choose the Formula From the Solid

Memory aid: Tapers to a point? Multiply by .

Solve the Volume Formula Backward

Cone: , . , so .

Cylinder: , .

Key Takeaways and Common Mistakes

• Full solids (prisms, cylinders):
• Pointed solids (pyramids, cones):
• Sphere:

Watch out:

• Pointed solid? Don't forget the
• Given diameter? Convert to radius first
• Compute first; the base is the repeated cross-section

Coming Up Next in Solid Geometry

Up next: Surface area of 3D figures

• Same five shapes, but now we're wrapping, not filling
• Lateral surface area vs. total surface area
• The slant height trap (different from regular height!)