Composite 3D Figures

Solid Geometry — Combining and Carving Standard Shapes

In this lesson:

• Halve the sphere's formulas to get a hemisphere's
• Decompose a figure, then add or subtract volumes
• Find surface area from exposed faces only

What You Will Learn Today

1. Find the volume or surface area asked for
2. Decompose into standard shapes (hemispheres by halving the sphere); label shared dimensions; classify additive or subtractive
3. Compute volume by adding or subtracting
4. Compute surface area from exposed faces: remove contact areas; for a removed solid, add interior surface

Recall the Formulas From Memory

• Prism and cylinder:
• Pyramid and cone:
• Sphere: and

Say each one aloud before checking.

A Hemisphere Is Half a Sphere

• Volume: ; curved surface:
• Flat face: , counted only if exposed
• : , curved , total

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 radius and height; halve any diameter

Always decompose before you compute anything.

Identify the Components in Each Figure

A: Cone on top of a cylinder
B: Rectangular prism with a hemispherical scoop removed
C: Cylinder with a smaller cylinder drilled through its center
D: Triangular prism with a pyramid on top

Name components, classify, and find shared dimensions.

Additive Volume: Hemisphere on Cylinder

Volume adds or subtracts; the joint never matters. Both cm, cylinder cm.

Cylinder:
Hemisphere:

Prism With Cylindrical Hole Subtracted

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

Prism:

Hole: height 12, so

Cone on Cylinder: Silo Volume Example

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

Cylinder:

Cone:

Three Components: Tank With Caps

A cylindrical tank ( m, m) has hemispherical caps on both ends.

Cylinder:
Two caps = one sphere:

Surface Area Has a Hidden Trap

• Contact-area rule: a face where two shapes meet is exterior to neither
• Volume of this figure (, ): the joint plays no part
• Surface area: the joint must be reasoned about

Wrong First: The Naive Sum

• Full cylinder SA:
• Full hemisphere SA:
• Naive total: , which overcounts by

Why? The contact circle () was counted once for each shape.

Correct Surface Area: Exterior Faces Only

Same figure: , . Draw an arrow on each exposed face; the joint gets none.

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

Cone on Cylinder Surface Area Worked

Cylinder: , . Cone: , , slant .

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

Subtractive Surface Area: Block With Hole

Block cm, hole cm through the 12 cm length.

• Block surface:
• Remove two openings:
• Add tunnel wall:

A Four-Step Routine for Composites

1. Sketch and decompose (15 sec): name shapes, label radii
2. Recall formulas (5 sec): hemisphere is half a sphere
3. Compute (30-40 sec): plug in, simplify
4. Check the question (5 sec): volume or SA, exact or approximate?

No formula sheet: rebuild hemisphere formulas by halving.

Timed Demo: Volume and Surface Area

Cylinder (diameter 6, ) topped by a cone (). Decompose once: , slant .

Volume:
Surface area:

Three Traps in the Answer Choices

The question asks for volume. Choices: , , , .

• : the other measure (surface area)
• : cone's left off
• : diameter 6 used as the radius

Surface area asked? leaves both contact circles in.

Key Takeaways and Common Mistakes

• Volume: add or subtract; the joint never matters
• Surface area: exposed faces only; add a removed solid's new surface
• Hemisphere: and , flat face

Watch out:

• Contact area hides from both shapes
• Decompose first; halve any diameter

Coming Up Next in Solid Geometry

Up next: Cross-sections and solids of revolution

• The shape a flat cut makes in a solid
• How the shape changes with the cut
• The solid a spinning flat shape makes