Introduction
Three-dimensional (3D) shapes are solid objects that have length, width, and height. In this lesson, we will focus on two important 3D shapes: cuboids and triangular prisms.
Understanding these shapes involves being able to identify them, visualize their nets (2D layouts that can be folded to form the 3D shape), and calculate their surface areas by adding up the areas of all their faces.
Identify Shapes
Cuboids and Triangular Prisms
Let's learn to identify these two important 3D shapes:
Cuboid
A box-shaped object with six rectangular faces. All angles are right angles.
Examples: Books, bricks, cereal boxes
Key Features:
- 6 rectangular faces
- 12 edges
- 8 vertices (corners)
- Opposite faces are identical
Triangular Prism
A prism with two triangular bases and three rectangular faces.
Examples: Toblerone package, roof structures
Key Features:
- 2 triangular bases
- 3 rectangular lateral faces
- 9 edges
- 6 vertices
Practice Identifying Shapes
Look at these objects and identify which are cuboids and which are triangular prisms:
Question 1
Identify the following shapes (cuboid or triangular prism):
- A shoebox
- A pyramid (square base)
- A wedge of cheese
- A rectangular aquarium
- A tent with triangular ends
Answers:
- Cuboid - It has six rectangular faces
- Not a triangular prism - This is a pyramid
- Triangular prism - Often wedge-shaped with triangular faces
- Cuboid - Rectangular faces all around
- Triangular prism - Has triangular ends and rectangular sides
Drawing Nets
Understanding Nets
A net is a 2D pattern that can be folded to form a 3D shape. It shows all the faces of the shape laid out flat.
Cuboid Net
A cuboid net consists of six rectangles arranged in a cross or other configuration where:
- Opposite faces are identical
- Each rectangle represents one face of the cuboid
- Adjacent rectangles share edges
Triangular Prism Net
A triangular prism net consists of:
- Two identical triangles (the bases)
- Three rectangles (the lateral faces)
- The rectangles connect to the sides of the triangles
Drawing Practice
Follow these steps to draw nets for cuboids and triangular prisms:
Drawing a Cuboid Net
- Draw a central rectangle (this will be the front face)
- Attach identical rectangles to the top, bottom, left and right
- Add one more rectangle to either the right or left of this arrangement
- Ensure all rectangles are properly aligned and sized
Drawing a Triangular Prism Net
- Draw a triangle (this will be one base)
- From each side of the triangle, draw a rectangle outward
- The three rectangles should be of equal length
- At the end of the rectangles, draw another identical triangle
Question 1
Draw the net of a cuboid with dimensions 4cm × 3cm × 2cm.
Steps:
- Draw a central rectangle 4cm × 3cm (front face)
- Above it, draw a 4cm × 2cm rectangle (top face)
- Below it, draw another 4cm × 2cm rectangle (bottom face)
- To the left, draw a 3cm × 2cm rectangle (left side face)
- To the right of the central rectangle, draw another 3cm × 2cm rectangle (right side face)
- To the right of that, draw a final 4cm × 3cm rectangle (back face)
Question 2
Draw the net of a triangular prism where the triangular bases are equilateral triangles with 3cm sides, and the length of the prism is 5cm.
Steps:
- Draw an equilateral triangle with 3cm sides (one base)
- From each side of the triangle, draw a rectangle 5cm long (lateral faces)
- At the ends of these rectangles, draw another identical equilateral triangle
- This creates a net with 2 triangles and 3 rectangles
Calculate Surface Area
Surface Area Concept
The surface area of a 3D shape is the total area of all its faces. Using nets makes it easier to calculate surface area because we can see all the faces at once.
General steps:
- Draw or visualize the net of the shape
- Identify all the individual faces
- Calculate the area of each face
- Add all the areas together
Calculation Methods
Cuboid Surface Area
For a cuboid with length (l), width (w), and height (h):
There are three pairs of identical faces:
- Front and back: l × h
- Top and bottom: l × w
- Left and right sides: w × h
Total Surface Area = 2(lw + lh + wh)
Triangular Prism Surface Area
For a triangular prism with:
- Base of triangle (b)
- Height of triangle (h)
- Length of prism (l)
Surface Area = 2 × (Area of triangular base) + (Perimeter of base × length)
Or more specifically:
= 2 × (½bh) + (a + b + c) × l
Where a, b, c are the sides of the triangular base
Practice Problems
Question 1
Calculate the surface area of a cuboid with dimensions 5cm × 4cm × 3cm.
Using the formula: 2(lw + lh + wh)
= 2[(5×4) + (5×3) + (4×3)]
= 2[20 + 15 + 12]
= 2[47]
= 94 cm²
Question 2
A triangular prism has a triangular base with sides 3cm, 4cm, 5cm and height 2.4cm (for the 4cm base). The length of the prism is 6cm. Calculate its surface area.
First, calculate the area of the triangular base:
Area = ½ × base × height = ½ × 4 × 2.4 = 4.8 cm²
There are two bases: 2 × 4.8 = 9.6 cm²
Now calculate the lateral (rectangular) faces:
Three rectangles with lengths 3cm, 4cm, 5cm and height 6cm:
Areas = 3×6 = 18 cm², 4×6 = 24 cm², 5×6 = 30 cm²
Total lateral area = 18 + 24 + 30 = 72 cm²
Total surface area = 9.6 + 72 = 81.6 cm²
Question 3
From the net shown below (imagine a cuboid net), calculate the surface area if the rectangles have dimensions: two 6cm×4cm, two 6cm×2cm, and two 4cm×2cm.
Surface Area = sum of all rectangular areas
= 2×(6×4) + 2×(6×2) + 2×(4×2)
= 2×24 + 2×12 + 2×8
= 48 + 24 + 16
= 88 cm²
Comprehensive Practice
Problem 1
A cereal box has dimensions 20cm × 10cm × 30cm.
- Identify the shape of the cereal box
- Draw its net
- Calculate its surface area
a) The cereal box is a cuboid.
b) Net would consist of six rectangles: two 20×10 (top/bottom), two 20×30 (front/back), and two 10×30 (sides).
c) Surface Area = 2[(20×10) + (20×30) + (10×30)] = 2[200 + 600 + 300] = 2[1100] = 2200 cm²
Problem 2
A triangular prism has an equilateral triangular base with 6cm sides and a length of 10cm.
- Draw its net
- Calculate the height of the triangular base
- Find the total surface area
a) Net consists of two equilateral triangles and three 6cm×10cm rectangles.
b) Height of equilateral triangle = (√3/2) × side = (1.732/2) × 6 ≈ 5.196 cm
c) Surface Area:
Area of two triangles = 2 × (½ × 6 × 5.196) ≈ 31.176 cm²
Area of three rectangles = 3 × (6 × 10) = 180 cm²
Total ≈ 31.176 + 180 = 211.176 cm²
Ready for more challenges?
Timed TestMath Challenge
Timed Test in 7 Levels
Test your math skills through progressively challenging levels