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
• Solve multi-step ACT problems

What You Will Learn Today

After this lesson, you will:

1. Find volume of prisms and cylinders:
2. Find volume of pyramids and cones:
3. Find volume of spheres:
4. Select the right formula from clues
5. Solve multi-step ACT problems

One Formula Unifies All Prism Shapes

You know for a box — that's really:

where = base area, = height

• Rectangular prism:
• Triangular prism:
• Cylinder:

V = Bh: The Cross-Section Idea

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

Worked Example With a Rectangular Prism

A box measures 8 cm × 5 cm × 3 cm. Find the volume.

Step 1: Identify the base area

Step 2: Multiply by height

Worked Example With a Triangular Prism

Base triangle: in, in. Prism length: 10 in.

Step 1: Compute the base area

Step 2: Multiply by the prism height

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

Quick Check on Prism Volume Formula

A hexagonal prism has a base area of 54 cm² and a height of 7 cm.

What is the volume?

Think about it before advancing...

Answer to the Hexagonal Prism Check

The hexagonal base doesn't change the formula — still .

The ACT may give unusual base shapes — just find first.

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

Quick Check on the One-Third Rule

A cylinder has and . A cone has the same dimensions.

What fraction of the cylinder's volume is the cone?

Think about it...

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

ACT twist: The problem gives diameter, not radius!

A sphere has diameter 10 cm. Find the volume.

Step 1: Convert to radius: cm

Step 2: Apply the formula

Quick Check on Sphere With Diameter

A sphere has diameter 8 cm. What is its volume?

Convert to radius first...

Complete Volume Formula Summary by Category

Memory aid: Does it taper to a point? Multiply by .

ACT Practice Problems to Identify and Solve

Problem 1: A triangular prism has a right-triangle base with legs 3 and 4, and length 12. Find the volume.

Problem 2: A solid has a circular base with radius 5 and tapers to a point at height 9. Find the volume.

Identify the shape and formula before computing.

ACT Practice: Find the Missing Dimension

Problem 3: A cone has volume and radius 4. Find the height.

Solutions to the ACT Practice Problems

Problem 1: Right triangle base → ,

Problem 2: Circular base + point = cone →

Problem 3: (shown on previous slide)

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
• Always compute as a separate step

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!)