Volume of 3D Solids

SAT Geometry and Trigonometry

In this lesson:

• Calculate volume of prisms, cylinders, cones, pyramids
• Calculate volume of spheres
• Solve applied and reverse volume problems

Your Learning Goals for This Lesson

You will be able to:

1. Find the volume of a prism, cylinder, cone, pyramid, or sphere by selecting its formula
2. Find a missing dimension (a radius or a height) given the volume
3. Solve real-world problems: compare capacities, fill at a given rate

What You Already Know About Area

Ten-second check: the area of a circle with radius 5?

• Rectangle:
• Circle:

Volume extends area into the third dimension — it is area times height.

Volume Equals Base Area Times Height

A box: a layer holds unit cubes; 2 layers give .

For any solid with a uniform cross-section:

• Rectangular prism:
• Cylinder:

Cylinder Example:

A cylinder has radius 5 cm and height 12 cm.

Cylinder Example: When Diameter Is Given

A cylinder has diameter 10 cm and height 12 cm.

Step 1: cm

Step 2:

Substituting 10 as gives — four times too big.

Pointed Solids: The 1/3 Factor

Cones and pyramids taper to a point — they hold less than the corresponding prism or cylinder.

If it points, multiply by .

Cone and Pyramid: Apply One-Third

Cone, in, in:

Square pyramid, edge 8 m, m:
, so

A square base has no .

Sphere Formula:

A sphere has no flat base or height — its volume depends only on radius.

• Sphere fits inside a cylinder of radius and height
• Sphere volume = of that cylinder's volume
• Formula: — note the cubed radius

Sphere Example:

A sphere has diameter 18 cm.

Step 1: cm ← Always find r first

Step 2:

Strategy: Solve Any Volume Problem

For every volume question:

1. Identify the solid type
2. Select the formula from the reference sheet
3. Check radius or diameter? Write if needed
4. Substitute and simplify
5. Keep if answer choices use

Reverse Problem: Find the Radius

A cylindrical tank holds cubic feet. The height is 20 feet. Find the radius.

Set up:

Solve:

Compare the Capacity of Two Containers

Box: . Cylinder: .

The box holds more.

Fill Rate: How Long to Fill?

A cone-shaped cup: cm, cm. Water flows in at per second.

Key Takeaways: Volume of 3D Solids

• ✓ for prisms and cylinders
• ✓ for pointed solids: cones, pyramids
• ✓ for spheres — exponent is 3

Diameter given? Write first

Sphere volume uses , not

All Five Formulas on Reference Sheet

What You Will Learn Next

Next lesson: Effects of Scaling on Area and Volume

You will learn:

• Area scales by when dimensions multiply by
• Volume scales by when dimensions multiply by
• Reverse: given a ratio, find the scale factor

These rules give the change without recomputing either figure.