Surface Area of 3D Solids

SAT Geometry and Trigonometry

In this lesson:

• Calculate surface area of prisms and cylinders
• Calculate surface area of pyramids, cones, and spheres
• Solve applied problems on the SAT

Your Learning Goals for This Lesson

By the end of this lesson, you will be able to:

1. Calculate surface area of prisms and cylinders
2. Calculate surface area of pyramids and cones
3. Calculate the surface area of spheres
4. Solve applied surface area problems in context

You Already Know What You Need

You already know how to find areas of 2D shapes.

• Rectangle:
• Circle:
• Triangle:

Surface area of a 3D solid is just adding up 2D areas — one face at a time.

A Solid Unfolds Into a Net

Surface area = total area of all faces when the solid is unfolded flat

Rectangular Prism: Three Pairs of Faces

Three pairs of congruent opposite faces: top/bottom, front/back, left/right

Breaking Down the Prism SA Formula

• = top and bottom faces
• = front and back faces
• = left and right faces

Find each pair, double it, and add.

Example 1: Box with Whole Numbers

A box: , , cm. Find SA.

Top/bottom:

Front/back:

Left/right:

Example 2: Box with Decimals

A box: 12.5 × 4.2 × 6.0 cm. Find SA.

Top/bottom:

Front/back:

Left/right:

Quick Check: A Storage Cube Problem

A storage cube has side length 7 cm.

What is its surface area?

All six faces are identical — use what you know.

(Think before the next slide…)

Cube Check: Walking Through the Solution

Side = 7 cm → all faces are 7 × 7 = 49 cm²

Or use the formula:

Cylinder: Net is Two Circles + Rectangle

• Two circular bases:
• Lateral surface (rectangle): width = , height =

The Cylinder Formula from the Net

• = two circular bases
• = lateral (curved) surface

SAT note: Problems may ask for lateral SA only — read carefully.

Cylinder Example: Total and Lateral SA

A cylinder has cm and cm.

Total SA:

Lateral SA only:

Check-In: Watch the Diameter Trap

A cylindrical can has diameter 6 cm and height 8 cm.

Find the total surface area. Leave your answer in terms of .

Step 1: What is the radius?

(Pause and solve — next slide shows the answer)

Diameter Trap: Here Is the Answer

Step 1 always: If given diameter, write immediately.

Pyramids and Cones: Slant Height

• = vertical height (straight up from center)
• = slant height (along the face — what the formula needs)
• by the Pythagorean theorem

Pyramid Formula: Base Plus Lateral Faces

• = area of the base
• = perimeter of the base
• = slant height (not vertical height)

For a square pyramid: and

Square Pyramid: Base Edge Six Meters

A square pyramid has base edge m and slant height m.

Base:

Lateral:

Cone: Find Slant Height from Vertical Height

A cone: cm, cm (vertical). Find total SA.

Step 1 — slant height:

Step 2 — apply formula:

The Sphere: One Elegant Curved Surface

The sphere's surface equals exactly four great circles.

Sphere Example: Diameter Given Again

A sphere has diameter 10 in. Find the surface area.

Applied Problem 1: Painting a Tank

A cylindrical water tank has m and m. Paint covers only the curved side (no top or bottom). How many square meters need painting?

Only the lateral surface — bases are excluded.

Applied Problem 2: Wrapping a Gift Box

A box is 30 cm × 20 cm × 10 cm. How much wrapping paper is needed (no overlap)?

Total surface area — all six faces.

Applied Problem 3: Material Cost

A metal cone (no base) has cm and cm. Metal costs $0.50 per cm².

Lateral SA (no base):

Cost:

Key Takeaways and SAT Watch-Outs

Slant height — use , not , in formulas
Diameter — halve it first:
Lateral vs. total SA — read the question twice
Sphere vs. volume — for SA, for volume
Units — SA is always in cm², m², in²

What Comes Next: Volume of Solids

Next lesson: Volume of 3D Solids

• Same five solid types, formulas on the SAT reference sheet
• SA vs. volume — often paired in the same SAT problem

Surface area = the outside. Volume = what's inside.