Composite 3D Figures

Solid Geometry — Combining and Carving Standard Shapes

In this lesson:

• Decompose composite figures into standard shapes
• Compute volume by adding or subtracting
• Find surface area while avoiding the contact-area trap

What You Will Learn Today

After this lesson, you will be able to:

1. Decompose a composite figure into standard shapes
2. Compute volume by adding component volumes
3. Compute volume by subtracting removed volumes
4. Find surface area, subtracting contact areas at joints
5. Solve ACT composite problems under time pressure

What Shapes Do You See Here

A composite figure is two or more standard shapes combined.

Two Types of Composite 3D Figures

• Additive: shapes joined together (hemisphere on cylinder)
• Subtractive: shape with part removed (block with hole)

Ask: "Is the material there, or is it empty space?"

Four Steps to Decompose Any Figure

1. Identify each standard shape (cylinder, cone, hemisphere...)
2. Note shared dimensions — the radius at the joint is the same
3. Classify — are you adding shapes or subtracting one?
4. Label every dimension needed for each formula

Always decompose before you compute anything.

Identify the Components in Each Figure

Figure A: Cylinder topped with a cone (same radius)
Figure B: Rectangular prism with a hemispherical scoop
Figure C: Large cylinder with a smaller cylinder drilled through

For each: name the components, classify as additive or subtractive, and identify shared dimensions.

Volume Is Simply Add or Subtract

For volume, the joint between shapes does not matter.

Additive composite:

Subtractive composite:

No contact-area complications — just combine the volumes.

Hemisphere on Cylinder: Additive Volume

A hemisphere sits on a cylinder. Both have cm, cylinder height cm.

Cylinder:

Hemisphere:

Prism With Cylindrical Hole Subtracted

Block: cm. Hole radius: 2 cm, drilled through the 12 cm length.

Prism:

Hole:

Cone on Cylinder: Silo Volume Example

A silo has a cylinder ( m, m) topped with a cone ( m, m).

Cylinder:

Cone:

ACT Practice: Tank With Hemispherical Caps

A cylindrical tank ( m, m) has hemispherical caps on both ends. Find the total volume.

Identify the components and compute before advancing...

Solution: Tank With Hemispherical Caps

Cylinder:

Two hemispheres = one full sphere:

Two hemispheres combine into one sphere — a useful shortcut.

Surface Area Has a Hidden Trap

The contact-area rule: When shapes join, the faces at the joint are hidden — they are no longer exterior.

You must subtract hidden faces from the total surface area.

Hemisphere on Cylinder Correct Surface Area

Same figure: , . Exterior surfaces only:

• Cylinder lateral:
• Cylinder bottom:
• Hemisphere curved:

Wrong vs Right the Overcounting Error

Wrong — adding full surface areas:

• Full cylinder SA:
• Full hemisphere SA:
• Wrong total: (overcounts by )

Why? The contact circle () was counted twice — once per shape.

Cone on Cylinder Surface Area Worked

Cylinder: , . Cone: , slant .

Exterior surfaces only:

• Cylinder lateral:
• Cylinder bottom:
• Cone lateral:

Listing Exterior Surfaces Prevents Errors

For each face, ask: exterior or hidden?

• Cylinder top at joint → Hidden
• Hemisphere flat face → Hidden
• Cylinder lateral → Exterior
• Cylinder bottom → Exterior
• Hemisphere curved → Exterior

List exterior surfaces first, then compute.

ACT Strategy for Composite Problems

1. Sketch and decompose (15 sec) — name shapes
2. Recall formulas (5 sec) — write them down
3. Compute (30-40 sec) — plug in, simplify
4. Check the question (5 sec) — volume or SA?

Traps: SA vs volume switch, diameter given instead of radius.

Try These Two ACT Practice Problems

Problem 1: Cone (, ) atop cylinder (, ). Find total volume.

Problem 2: Hemisphere () on a cube, edge 10. Find total surface area.

Decompose, classify, and compute before advancing...

Solutions to Both Practice Problems

Problem 1:

Problem 2: Exterior surfaces:

• Five cube faces:
• Top minus circle:
• Hemisphere curved:

Key Takeaways and Common Mistakes

• Volume: add or subtract — no contact-area issue
• Surface area: remove hidden faces at joints
• Hemisphere: and

Watch out:

• Contact area hides from both shapes
• Always decompose before computing

Coming Up Next in Solid Geometry

Up next: Review and mixed practice

• Volume and SA problems combined
• Multi-component composites with 3+ shapes
• Timed ACT practice across all topics