Surface Area of Three-Dimensional Figures

Solid Geometry — Wrapping, Not Filling

In this lesson:

• Compute LSA and TSA for all solid types
• Master the slant height computation
• Solve multi-step ACT surface area problems

What You Will Learn About Today

After this lesson, you will:

1. Distinguish lateral from total surface area
2. Compute SA of prisms and cylinders
3. Compute SA of pyramids and cones using slant height
4. Apply the sphere formula
5. Solve multi-step ACT surface area problems

Wrapping Paper Not Filling — Surface Versus Volume

• Volume = how much fits inside (cubic units)
• Surface area = how much covers the outside (square units)
• LSA = lateral faces only (sides, no top or bottom)
• TSA = all faces including base(s)

On the ACT, "surface area" means TSA unless stated otherwise.

LSA Versus TSA and When Each Applies

• Prisms and cylinders: 2 bases →
• Pyramids and cones: 1 base →
• Spheres: 0 bases → (just one formula)

Count the flat faces to know how many bases to add.

Unwrapping a Cylinder Reveals the Formula

The lateral surface is a rectangle: width , height .

Rectangular Prism Surface Area Step by Step

A box is 8 × 5 × 3 cm. Find TSA.

Step 1:

Step 2:

Step 3:

Cylinder Surface Area With Radius and Height

A cylinder has cm and cm.

Step 1:

Step 2: Base area:

Step 3:

Quick Check — Find the Label Area

A cylindrical can has cm and cm. A label wraps the lateral surface. What is the label's area?

Think: do you need LSA or TSA?

Slant Height Changes Everything for Pyramids

• Height : straight down from apex to base center
• Slant height : along the surface from apex to base edge
• Key relationship:

Square Pyramid With Given Slant Height

Base edge m, slant height m. Find TSA.

Step 1: m

Step 2: m²

Step 3: m²

Square Pyramid Requiring Slant Height Computation

Base edge cm, height cm. Find TSA.

Step 1: Find : apothem ,

Step 2: cm²

Step 3: cm²

Cone Surface Area With Radius and Height

A cone: cm, cm. Find TSA.

Step 1: cm

Step 2:

Step 3: cm²

Quick Check — Find Total Surface Area

A cone has and . Find the total surface area.

Remember: cones have one base, not two.

Sphere Surface Area Is Four Pi R Squared

Example: Sphere with :

No LSA/TSA distinction — spheres have no flat faces.

Sphere Example With the Diameter Trap

A basketball has diameter 9.4 in. Find the surface area.

Step 1: in

Step 2: in²

Always convert diameter to radius first!

Surface Area Versus Volume — The Exponent Tells

→ area (square units) | → volume (cubic units)

ACT Practice — Identify Shape and Formula

Problem 1: A cylinder has and . Find the TSA.

Problem 2: A square pyramid has base edge 10 and . Find the TSA.

Identify what you need — and check for slant height!

Multi-Step Cone Problem With Slant Height

A cone: , . Find the total surface area.

Step 1:

Step 2:

Step 3:

Solutions to the ACT Practice Problems

Problem 1:

Problem 2: Apothem , .

Key Takeaways and Common Mistake Warnings

• Prisms/cylinders: , add 2 bases
• Pyramids/cones: use slant height , add 1 base
• Sphere: (no bases)

Watch out:

• Slant height height — find first
• = area, = volume
• Count bases: 2, 1, or 0

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