Surface Area of Three-Dimensional Figures

Solid Geometry — Wrapping, Not Filling

In this lesson:

• Choose lateral or total surface area, and the right formula
• Find slant height when only the height is given
• Handle open solids and volume-to-surface-area items

What You Will Learn About Today

1. Find surface area: lateral or total, the right formula, the right bases
2. Compute SA of right prisms (rectangular, triangular) and cylinders
3. Compute SA of regular pyramids and cones, finding slant height from the height; TSA adds the base
4. Apply (no LSA/TSA)

Surface Versus Volume: Wrapping Paper

Recall: , , , , .

• Volume: how much a solid holds (cubic units)
• Surface area: paper covering it (square units)
• LSA: sides only. TSA: plus bases.
• Plain "surface area" means TSA.

Rectangular Prism Surface Area Step by Step

A box is 8 × 5 × 3 cm. Find TSA.

Step 1:

Step 2:

Step 3:

Check by faces: .

Triangular Prism: Base From the Legs

• , so
• , from the legs, not the 13

Unwrapping a Cylinder Reveals the Formula

The lateral surface is a rectangle: width , height . A prism unfolds the same way, with width .

Cylinder Surface Area and Label Area

Can: , cm. ; .

Label around a can (, cm) covers the lateral surface only:

Slant Height Changes Everything for Pyramids

• Height : straight down from apex to base center
• Slant height : along the surface to the base circle (cone) or a base-edge midpoint (pyramid)
• . Check:

Slant Height Given: Pyramid and Cone

Pyramid: base edge 6 m, m. .
; .

Cone: cm, cm.
; .

Pyramid When Only Height Is Given

• Base edge 8 cm, cm; apothem
• ;

Cone With Radius and Height

A cone: , cm. Find TSA.

Step 1:

Step 2:

Step 3:

Wrong: (height used for ).

A Cone Where the Slant Is Not Whole

A cone: , . Not a triple.

One base: , not .

Sphere Surface Area and the Diameter Trap

• :
• Basketball, diameter 9.4 in:

No flat faces, so no LSA/TSA. Convert to first.

Surface Area Versus Volume: The Exponent

is area (cm²); is volume (cm³). A cylinder around a sphere has .

Pick the Formula by Counting Bases

• Prisms, cylinders: , add 2 bases
• Pyramids, cones: use , add 1 base
• Sphere: , no bases
• Open at one end: that base is lost

Open-top can, , : .

Cube From Volume to Surface Area

A cube has volume . Find its surface area.

Step 1: cm

Step 2:

Check: .

Multi-Step Cone Problem With Slant Height

A cone: , . Find the total surface area.

Step 1:

Step 2:

Step 3:

Key Takeaways and Common Mistake Warnings

• Prisms, cylinders: , add 2 bases
• Pyramids, cones: use , add 1 base
• Sphere: , no bases

Watch out:

• Slant height is not height: find
• is area, is volume
• Halve a diameter; count the bases

Coming Up Next in Solid Geometry

Up next: Composite 3D figures

• Combining and subtracting standard shapes
• Volume of composite figures
• The contact-area trap in surface area