Surface Area and Volume of Composite Objects

Calculate volume and surface area of composite 3D objects. Math 10-C Alberta curriculum.

Lesson 1.6 of Measurement in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• Volume of Composite Objects. Learn the three-step process for finding the volume of composite objects — identify the shapes, find each volume using formulas from previous lessons, and add them together. Worked examples include a pyramid on a prism and a hemisphere on a cylinder.
• Surface Area of Composite Objects. Master the golden rule of composite surface area: find each piece's surface area then subtract hidden faces where shapes meet. Worked examples cover a pyramid-roof house and a hemisphere-topped cylinder, plus two full practice problems.

Inside the lesson — a free preview

What Is a Composite Object?

Definition — Composite Object. A composite object is an object made of two or more distinct 3-D shapes combined into one.

Common examples: a grain silo (a cylinder with a cone roof), an ice cream cone (a cone with a hemisphere on top), or a house (a rectangular prism with a pyramid roof).

The key insight: you already know every formula you need from earlier lessons. What is new here is deciding which faces to count (for surface area) and which pieces to add (for volume).

The Three-Step Volume Strategy

Three Steps for Composite Volume. Step 1 — Identify the separate shapes.

Step 2 — Find each shape's volume.

Step 3 — Add the volumes together.

Why Volume Is Always Just Addition. Volume measures the space inside an object. When you join two shapes, the total interior space is simply the sum of the two interiors — nothing gets hidden or removed.

Always add.

Worked example: Pyramid on a Prism

Example 1 — Pyramid on Top of a Rectangular Prism. A rectangular prism has a pyramid sitting on top (like a house with a peaked roof). The prism base is 6.7 m × 2.9 m and the prism is 2.9 m tall. The pyramid on top has the same 6.7 m × 2.9 m base and a height of 2.1 m. Find the total volume.

Step 1 — Volume of the prism.

Use V = (base area) × h:

V = (6.7 × 2.9) × 2.9

V = 19.43 × 2.9

V = 56.3 m^3

Step 2 — Volume of the pyramid.

Use V = (1)/(3) × (base area) × h:

V = (1)/(3) × (6.7 × 2.9) × 2.1

V = (1)/(3) × 19.43 × 2.1

V = (1)/(3) × 40.8

V = 13.6 m^3

Step 3 — Add the volumes.

V = 56.3 + 13.6

Answer: V = 69.9 m^3

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