Surface Area of 3-D Objects

Calculate surface area of 3D shapes including prisms and cylinders. Math 10-C Alberta lesson.

Lesson 1.3 of Measurement in Math 10C — Alberta curriculum lessons.

What you'll learn in this lesson

• Rectangular Prisms. Apply the formula SA = 2lw + 2lh + 2wh to find the surface area of boxes, then work backwards from a given surface area to find the side length of a cube.
• Cylinders. Build the cylinder formula from its net — two circular ends plus a wrapped rectangle — and practise finding surface area when only the diameter is given.
• Cones. Discover the hidden right triangle inside every cone, use the Pythagorean theorem to find the slant height, and apply SA = πrs + πr² to solve forward and backward problems.
• Pyramids and Tetrahedrons. Find the surface area of square pyramids, rectangular pyramids (which need two slant heights), and regular tetrahedrons by adding the base area to the area of all triangular faces.

Inside the lesson — a free preview

What Is a Rectangular Prism?

Definition. A rectangular prism is a box with 6 rectangular faces arranged in 3 matching pairs: front and back, left and right, and top and bottom.

Surface area is the total area of all the outside faces — imagine unfolding the box flat and measuring every piece.

Why All the 2s?. Every face has an identical twin on the opposite side of the box.

Front matches back, left matches right, top matches bottom.

So you find the area of three faces, then double each one.

The Formula

Variables. l = length

w = width

h = height

Building the Formula from the Net

The Three Pieces of a Cylinder. Unroll a cylinder flat and you get three pieces:

Two circles (top and bottom), each with area π r^2. Together they give 2π r^2.

One rectangle (the curved side). Its height is h and its width equals the circumference of the circle, which is 2π r. Its area is 2π r h.

Worked example: Wrapping a Present

A box measures 15 cm long, 10 cm deep, and 8 cm high. How much wrapping paper is needed to cover it?

Solution. "How much paper to cover it" means surface area. Use l = 15, w = 10, h = 8.

Step 1: Write the formula.

SA = 2lw + 2lh + 2wh

Step 2: Substitute the values.

SA = 2(15)(10) + 2(15)(8) + 2(10)(8)

Step 3: Evaluate each term.

SA = 300 + 240 + 160

Step 4: Add.

Answer: SA = 700 cm^2

Create a free Studyio account to take the full lesson with voice-over teaching, practice questions and instant feedback.