6.G.A.4: Nets and Surface Area of Prisms and Pyramids
In plain English: 6.G.A.4 is the Common Core grade 6 math standard that asks students to represent three-dimensional figures, such as prisms and pyramids, with nets made of rectangles and triangles, and to use those nets to find surface area. Students add the areas of all the faces and apply the idea to real problems like wrapping a box or sewing a tent.
Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.
Common Core State Standards for Mathematics · Domain: Geometry (G) · Cluster: Solve real-world and mathematical problems involving area, surface area, and volume. Also written as 6.G.4 · Official standard
Students learn to show a three-dimensional figure (a solid shape with length, width and height, such as a box) as a flat pattern. Each flat side of the figure is a face, two faces meet at an edge, and edges meet at a vertex. A net is a flat pattern of all the faces that folds up into the figure with no gaps and no overlaps. Students work with prisms (two matching bases joined by rectangles, such as a box or a tent) and pyramids (one base joined to a point by triangles), because their nets are made of rectangles and triangles.
Next, students use nets to find surface area: the total area of all the faces, measured in square units such as square centimeters (cm²). They find the area of each rectangle and triangle in the net and add them. The same method solves real problems: how much cardboard makes a box, how much wrapping paper covers a gift, or how much fabric a tent needs. Figures with curved surfaces, such as cylinders, are not part of this standard, because their nets need circles.
Learning Objectives
By the end of this lesson, students will be able to:
Identify and draw nets of rectangular prisms, cubes, triangular prisms and pyramids made of rectangles and triangles
Decide whether a flat pattern is a net that folds into a given figure
Use a net to find the surface area of a prism or pyramid by adding the areas of its faces
Solve real-world problems about covering, wrapping or painting objects with surface area
Prior Knowledge Required
Students should already be comfortable with:
Finding the area of rectangles 4.MD.A.3
Recognizing solid figures and measuring with unit cubes 5.MD.C.3
Multiplying decimals 5.NBT.B.7
Finding the area of right and other triangles 6.G.A.1
Hold up an empty tissue box. Carefully cut it along some edges so it lies flat in one piece, and show the class the flat shape.
Warm-Up Prompt
"How many flat sides did the box have? What shape is each one? Which pieces are exactly the same size? If you wanted to cover the whole box with paper, what would you need to measure?"
Collect answers. The box has 6 faces, all rectangles, and they come in 3 matching pairs: top and bottom, front and back, left and right. To cover it, you would need the area of every face. Introduce the words face, edge, net and surface area. Some students may say you need the space inside the box. Point out that the space inside is the volume, which is measured in cubic units, and that covering the outside is a different question.
Direct Instruction20 minutes
Part 1: Nets. A net shows every face exactly once, and the faces are joined along edges so the pattern folds up. Show the nets of five figures and count the faces:
Rectangular prism (a box): 6 rectangles in 3 matching pairs. A cube is a rectangular prism whose 6 faces are all squares.
Triangular prism (like a tent): 2 matching triangles (the bases) and 3 rectangles.
Square pyramid: 1 square base and 4 triangles that meet at the top point.
Triangular pyramid: 4 triangles.
Check a net: count the faces, check that matching edges have the same length, and picture the folding. Six squares in a row, for example, cannot fold into a cube: the ends overlap and two faces are left open.
Part 2: Surface area from a net. Label each face with its length and width (or base and height for a triangle), find each area, and add. For a triangle, area = 1/2 × base × height. On a pyramid, use the height of each triangular face, measured along the face from the middle of its base edge to the top point. Diagram 1 and Diagram 2 show the first, third and fourth examples.
Rectangular prism
A box is 8 cm long, 5 cm wide and 3 cm tall. Draw its net and find its surface area.
A tent is a triangular prism 7 ft long. Each end is a triangle with a base of 6 ft, a height of 4 ft and slanted sides of 5 ft. How much fabric covers the whole tent, including the floor?
A toy chest is 3 ft long, 2 ft wide and 2 ft tall. Leo paints the outside, but not the bottom. How many square feet does he paint?
Equation: Top 3 × 2 = 6; front and back 2 × (3 × 2) = 12; ends 2 × (2 × 2) = 8; 6 + 12 + 8 = 26 square feet
Part 3: Real-world problems. Before computing, ask which faces the problem needs. Wrapping a gift uses every face. Painting a chest that sits on the floor skips the bottom. A tent may or may not include a floor. Write the unit with every answer, and remember that surface area is in square units, never cubic units.
Guided Practice15 minutes
Pairs sketch a net for each figure on grid paper, label every face with its dimensions and area, and find the surface area. After each figure, one pair explains how it made sure no face was missing or counted twice.
Guided practice figures
Figure
Measurements
Faces in the net
Surface area
1. Rectangular prism
10 in by 4 in by 3 in
Two 10 × 4, two 10 × 3, two 4 × 3 rectangles
80 + 60 + 24 = 164 square inches
2. Triangular prism
Right-triangle ends (triangles with a square corner) with sides 5 cm, 12 cm and 13 cm; length 6 cm
For Figure 2, point out that the three rectangles all have the same length, 6 cm, and that their widths are the three sides of the triangle: 5 + 12 + 13 = 30, and 30 × 6 = 180 cm², the same as 30 + 72 + 78. Listen for students who forget the 1/2 in the triangle area, and for students who give only one rectangle from each matching pair.
Independent Practice10-15 minutes
Students sketch a net and find the surface area on their own. (1) A cube with edges of 9 cm. (6 × 81 = 486 cm².) (2) A box 7 in long, 5 in wide and 2 in tall. (2 × 35 + 2 × 14 + 2 × 10 = 118 square inches.) (3) A triangular pyramid made of 4 matching triangles, each with a base of 8 cm and a height of 7 cm. (4 × 28 = 112 cm².) (4) Draw two different nets for the same box from Problem 2 and explain why both give the same surface area. (The faces are the same six rectangles, only arranged differently.)
Closure5 minutes
Exit ticket: (1) Name the faces in a net of a triangular prism. (2 triangles and 3 rectangles.) (2) Find the surface area of a box 5 cm long, 2 cm wide and 1 cm tall. (2 × 10 + 2 × 5 + 2 × 2 = 34 cm².) (3) Why is surface area measured in square units and not cubic units? (It measures flat area on the outside, not the space inside.)
Differentiation Strategies
For Struggling Students
Give printed nets on grid paper to cut out and fold before drawing their own
Use a face table with one row per face (name, length, width, area) so no face is missed
Color matching faces the same color so students see the pairs in a box net
For Advanced Students
Find all 11 different nets of a cube on grid paper and explain why a 2-by-3 block of squares is not one of them
Use the formula A = 6s² from 6.EE.A.2 to find the surface area of a cube with edges of 1/2 foot, and check it with a net
Challenge: two boxes hold the same 24 unit cubes. Which box shape uses the least cardboard? (Activity 3 extends this)
Assessment Guidance
What to Look For
Check that every net has the right number and kind of faces and that matching edges have matching lengths. When students find surface area, look for a label on every face, the 1/2 in each triangle's area, both faces of each matching pair in a prism, and square units in the answer. In word problems, ask students to say which faces the situation needs before they compute. Watch for students who multiply length × width × height, which gives volume.
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Classroom Activities
3 Activities
1
Unfold the Box
20 minGroups of 3
Each group gets an empty cardboard box, such as a cereal, tissue or toothpaste box. Groups measure the box, predict its surface area, then cut it open into a net and check their prediction with the flat net.
Procedure
Measure the length, width and height of the closed box to the nearest half centimeter
Predict the surface area by finding the area of each pair of faces and adding
Cut along edges so the box lies flat in one piece, then cut off the glue flaps
Label every face of the net with its dimensions and area, add them, and compare with the prediction
Sample Measurements
A toothpaste box about 19 cm long, 4 cm wide and 3.5 cm tall: 2 × 76 + 2 × 66.5 + 2 × 14 = 313 cm²
A small cereal box about 19 cm wide, 5 cm deep and 27 cm tall: 2 × 95 + 2 × 513 + 2 × 135 = 1,486 cm²
Discussion Questions
Did your prediction match the net? If not, which face was missed or measured differently?
Why do we cut off the glue flaps before we compare?
Of the two sample boxes, which faces of the cereal box have the largest area? (The front and back, 19 × 27 = 513 cm² each)
Modification for Distance Learning
Students unfold a box at home, photograph the net with a ruler next to it, and post the photo with each face labeled. Classmates check one another's totals.
2
Net or Not? Cube Card Sort
15 minPairs
Pairs get 8 cards. Each card shows 6 squares on a grid. Pairs predict whether each pattern folds into a cube, then cut out and fold the card to check. Rows are listed from top to bottom, and columns are numbered from left to right.
Sort the cards into "net" and "not a net" before folding, and write one reason for each card
Cut out and fold each card to check, taping it closed if it works
For each card that works, color opposite faces the same color
Discussion Questions
Which cards fold into a cube? (Cards 1, 2, 3, 4 and 8)
On Card 5, the two top squares sit on the same side of a row of 4. What goes wrong when you fold it? (Both land on the same face, and the opposite face is left open)
Every card has 6 squares. Why is having 6 squares not enough to make a net?
3
Design a Package
15 minPairs
A company packs 24 small cube-shaped erasers, each 1 cm on every edge, into one box with no empty space. Pairs list every box shape with whole-number sides that holds exactly 24 erasers, draw a net for each on centimeter grid paper, and find which box uses the least cardboard.
For each box, draw a net and label every face with its area
Add the areas to find the surface area, and record it in a table
Circle the box with the least surface area
Discussion Questions
Which box uses the least cardboard? (The 2 × 3 × 4 box, with a surface area of 52 cm²)
Which box uses the most? (The 1 × 1 × 24 box, with 98 cm²)
All six boxes hold the same 24 erasers. Why do they need different amounts of cardboard?
Extension Variation
Pairs repeat the task for 36 erasers and describe which box shapes use the least cardboard.
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Diagrams & Visual Aids
2 diagrams
Diagram 1: The Net of a Rectangular Prism
The net of a box 8 cm long, 5 cm wide and 3 cm tall, drawn to scale. The six rectangles form three matching pairs. The surface area is the total area of the net: 80 + 48 + 30 = 158 cm².
Diagram 2: Nets of a Square Pyramid and a Triangular Prism
Left: the net of a square pyramid, one square and four matching triangles, drawn to scale. Right: the net of a tent shaped like a triangular prism, two triangles and three rectangles, drawn to scale. Each slanted side of a triangle is 5 ft, the same as the short edge of the side rectangle it folds against.
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Homework Assignment
~30 min
6.G.A.4 Homework: Nets and Surface Area
Directions: Sketch a net for every figure and label each face with its dimensions and area. Show the sum you use for surface area, and write square units with every answer.
Part 1: Nets (Problems 1-2)
Name the figure each net folds into: (a) 4 matching triangles; (b) 2 matching triangles and 3 rectangles; (c) 1 square and 4 matching triangles. Then sketch a net for a box 4 in long, 3 in wide and 1 in tall, and find its surface area.
Rosa draws 7 squares in a cross shape and says it is a net of a cube. Explain what is wrong. Then draw a correct net of a cube with edges of 7 cm and find its surface area.
Part 2: Surface Area from Nets (Problems 3-4)
A square pyramid has a base 8 in on each side. Each triangular face has a base of 8 in and a height of 5 in. Draw the net and find the surface area.
A triangular prism is 12 cm long. Its bases are right triangles with sides of 6 cm, 8 cm and 10 cm. Draw the net and find the surface area.
Part 3: Real-World Problems (Problems 5-6)
A gift box is 12 in long, 10 in wide and 4 in tall. How many square inches of wrapping paper cover the box exactly, with no overlap?
A wooden planter box with no top is 5 ft long, 2 ft wide and 1.5 ft tall. How many square feet of wood are needed for the bottom and the four sides?
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Nets
Every net has the right faces, joined so it folds, with labels
One face missing, extra or unlabeled
Net does not match the figure
Face Areas
All rectangle and triangle areas correct, including the 1/2 for triangles
One area incorrect
Several areas incorrect
Surface Area
Correct totals with square units
Correct method with one addition or unit error
Volume or perimeter given instead
Real-World Use
Chooses exactly the faces the situation needs
Includes or leaves out one face
Method does not fit the situation
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Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order, and sketch a net on scrap paper when it helps. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A triangular prism is 10 cm long. Its triangular bases have sides of 4 cm, 5 cm and 6 cm. Which rectangles belong in its net?
Answer: A
Each side of the triangle is joined to one rectangle, and every rectangle runs the length of the prism, 10 cm. So the rectangles are 4 × 10, 5 × 10 and 6 × 10. Choice B pairs up the sides of the triangle, but no face of the prism measures 4 cm by 5 cm. Choice C uses the length for both dimensions. Choice D repeats the 4 cm side and leaves out the 6 cm side, so the net would not close.
Question 2 of 20 · Multiple Choice
A net of a box so far has two 6 × 5 rectangles and two 6 × 2 rectangles. What else does the net need?
Answer: C
A box has 6 faces in 3 matching pairs. The 6 × 5 and 6 × 2 pairs are there, so the missing pair is the two ends, each 5 × 2. Choice A adds only one end, so one side of the box stays open. Choices B and D use squares whose edges do not match the 5-unit and 2-unit edges of the other faces.
Question 3 of 20 · Multiple Choice
A net has 1 square and 4 matching triangles, one attached to each side of the square. What figure does it fold into?
Answer: B
The square is the base, and the 4 triangles fold up to meet at one point, so the figure is a square pyramid. Choice A needs 2 triangles and 3 rectangles. Choice C needs 6 squares. Choice D has a triangle as its base and 4 triangles in all, with no square.
Question 4 of 20 · Multiple Choice
What is the surface area of a box 6 cm long, 4 cm wide and 2 cm tall?
Answer: D
The faces come in 3 pairs: 2 × (6 × 4) + 2 × (6 × 2) + 2 × (4 × 2) = 48 + 24 + 16 = 88 cm². Choice A adds only one face from each pair: 24 + 12 + 8 = 44. Choice B multiplies 6 × 4 × 2, which is the volume in cubic centimeters. Choice C adds the three edge lengths.
Question 5 of 20 · Multiple Choice
What is the surface area of a cube with edges 4 inches long?
Answer: A
Each face is a 4 × 4 square with an area of 16 square inches, and a cube has 6 faces: 6 × 16 = 96 square inches. Choice B is 4 × 4 × 4 = 64, the volume. Choice C is the area of just one face. Choice D multiplies the edge by the number of faces, 4 × 6, and never finds a face's area.
Question 6 of 20 · Multiple Choice
A paperweight is a square pyramid. Its base is 6 cm on each side, and each triangular face has a height of 4 cm. What is its surface area?
Answer: C
Base: 6 × 6 = 36 cm². Each triangle: 1/2 × 6 × 4 = 12 cm², and there are 4 of them: 48 cm². Total: 36 + 48 = 84 cm². Choice A forgets the 1/2 and uses 6 × 4 = 24 for each triangle: 36 + 96 = 132. Choice B leaves out the square base. Choice D counts only 2 of the 4 triangles: 36 + 24 = 60.
Question 7 of 20 · Multiple Choice
A candy box is a triangular prism 8 in long. Each end is a right triangle with sides of 3 in, 4 in and 5 in. What is the surface area of the box?
Answer: B
Each triangle: 1/2 × 3 × 4 = 6 square inches, so the two ends are 12. The three rectangles are 3 × 8, 4 × 8 and 5 × 8, which add to 24 + 32 + 40 = 96. Total: 12 + 96 = 108 square inches. Choice A forgets the 1/2 and uses 12 for each triangle. Choice C leaves out both triangles. Choice D counts only one triangle.
Question 8 of 20 · Multiple Choice
In the net of a square pyramid, the square has an area of 25 cm² and each triangle has an area of 15 cm². What is the surface area?
Answer: D
Add every face in the net: 25 + 4 × 15 = 25 + 60 = 85 cm². Choice A adds only one triangle. Choice B multiplies the square's area by 4 instead of the triangle's. Choice C multiplies the two areas, 25 × 15, which does not measure anything on the figure.
Question 9 of 20 · Multiple Choice
Which set of faces makes a net of a box 5 in long, 3 in wide and 2 in tall?
Answer: D
A box has 3 pairs of matching faces: top and bottom (5 × 3), front and back (5 × 2), and the two ends (3 × 2). Choice A uses the largest face six times. Choice B has the right number of faces but no 5 × 2 faces, so the front and back are missing. Choice C also leaves out the 5 × 2 faces.
Question 10 of 20 · Multiple Choice
A cereal box is 20 cm wide, 7 cm deep and 30 cm tall. How much cardboard covers the box, ignoring the flaps?
Answer: C
Front and back: 2 × (20 × 30) = 1,200. Sides: 2 × (7 × 30) = 420. Top and bottom: 2 × (20 × 7) = 280. Total: 1,200 + 420 + 280 = 1,900 cm². Choice A counts only one face from each pair. Choice B is the volume, 20 × 7 × 30 = 4,200 cubic centimeters. Choice D adds the three measurements.
Question 11 of 20 · Multiple Choice
A storage box with no lid is 3 ft long, 2 ft wide and 1 ft tall. How much cardboard is needed for the bottom and the four sides?
Answer: B
Bottom: 3 × 2 = 6. Front and back: 2 × (3 × 1) = 6. Ends: 2 × (2 × 1) = 4. Total: 6 + 6 + 4 = 16 square feet. Choice A includes a lid, which the box does not have. Choice C leaves out the bottom as well as the lid. Choice D multiplies 3 × 2 × 1, which is the volume.
Question 12 of 20 · Multiple Choice
A doghouse roof is shaped like a square pyramid with no base. The base edges are 4 ft, and each triangular face is 3 ft tall. How many square feet of shingles cover the roof?
Answer: A
The roof is only the 4 triangles: 4 × (1/2 × 4 × 3) = 4 × 6 = 24 square feet. Choice B adds the 4 × 4 square base, but the base is open, under the roof. Choice C forgets the 1/2 in each triangle's area. Choice D counts only 2 triangles.
Question 13 of 20 · Multiple Choice
What does the surface area of a three-dimensional figure measure?
Answer: D
Surface area is the total area of every face, which is the same as the area of the figure's net. Choice B describes volume. Choice C adds lengths, not areas. Choice A uses only one face.
Question 14 of 20 · Multiple Choice
A cube has a surface area of 54 square inches. How long is each edge?
Answer: A
A cube has 6 matching faces, so each face is 54 ÷ 6 = 9 square inches. A square with an area of 9 square inches has sides of 3 inches, because 3 × 3 = 9. Choice B is the area of one face, not the edge length. Choice C divides 54 by 2. Choice D uses the number of faces as the edge length.
Question 15 of 20 · Short Answer
Sketch a net for a box 9 in long, 4 in wide and 2 in tall. Label each face with its dimensions, and find the surface area.
The net has two 9 × 4 faces (36 each), two 9 × 2 faces (18 each) and two 4 × 2 faces (8 each). Surface area: 2 × 36 + 2 × 18 + 2 × 8 = 72 + 36 + 16 = 124 square inches.
Question 16 of 20 · Short Answer
A square pyramid has a base 12 cm on each side. Each triangular face has a height of 10 cm. Find the surface area and show your net.
The net is 1 square and 4 triangles. Square: 12 × 12 = 144 cm². Each triangle: 1/2 × 12 × 10 = 60 cm², so 4 × 60 = 240 cm². Surface area: 144 + 240 = 384 cm².
Question 17 of 20 · Short Answer
A triangular prism is 20 cm long. Each base is a triangle with a base of 10 cm, a height of 12 cm and two slanted sides of 13 cm. Find the surface area.
Two triangles: 2 × (1/2 × 10 × 12) = 120 cm². Three rectangles, each 20 cm long: 10 × 20 = 200, 13 × 20 = 260 and 13 × 20 = 260, for 720 cm². Surface area: 120 + 720 = 840 cm².
Question 18 of 20 · Short Answer
Jay says the surface area of a cube with edges of 5 units is 125 square units. What did he find, and what is the correct surface area?
Jay found 5 × 5 × 5 = 125, which is the volume in cubic units. The surface area is 6 faces × (5 × 5) = 150 square units.
Question 19 of 20 · Short Answer
A cardboard shipping box is 2 ft long, 1.5 ft wide and 1 ft tall. How many square feet of cardboard make the box, ignoring the flaps?
Top and bottom: 2 × (2 × 1.5) = 6. Front and back: 2 × (2 × 1) = 4. Ends: 2 × (1.5 × 1) = 3. Surface area: 6 + 4 + 3 = 13 square feet.
Question 20 of 20 · Short Answer
A net is made of 4 matching triangles. Each triangle has a base of 6 in and a height of 5.2 in. Name the figure and find its surface area.
Four triangles fold into a triangular pyramid. Each triangle: 1/2 × 6 × 5.2 = 15.6 square inches. Surface area: 4 × 15.6 = 62.4 square inches.
0 of 20 answered · 0 correct
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Frequently Asked Questions
10 Questions
What does 6.G.A.4 mean?
6.G.A.4 means students can show a three-dimensional figure as a net, a flat pattern of rectangles and triangles that folds into the figure, and use the net to find the figure's surface area. They then use surface area in real problems, such as how much paper wraps a gift.
Is 6.G.A.4 taught in grade 6 or grade 7?
It is a grade 6 standard in the Geometry domain. In grade 7, students extend it to surface area of objects made of several prisms and pyramids (7.G.B.6). In grade 6, the figures are single prisms and pyramids whose nets use only rectangles and triangles.
What is a net in math?
A net is a flat pattern that folds into a three-dimensional figure, with every face shown once and no overlaps. For example, a cereal box cut along some edges and laid flat is a net of a rectangular prism: 6 rectangles joined along their edges.
What is the difference between surface area and volume?
Surface area measures the outside of a figure, the total area of its faces, in square units. Volume measures the space inside, in cubic units. Wrapping paper for a gift box is a surface area question; how much sand fills the box is a volume question (6.G.A.2).
How many different nets does a cube have?
A cube has 11 different nets, not counting ones that are just turned or flipped. Every one has 6 squares, but not every arrangement of 6 squares works: 6 squares in a row, or a 2-by-3 block, cannot fold into a cube. Folding paper cutouts is the quickest way to check.
What mistakes do students often make with 6.G.A.4?
Common mistakes are leaving out a face or counting one twice, forgetting the 1/2 when finding a triangle's area, multiplying length × width × height (which is volume), and writing cubic units instead of square units. A face table with one row per face helps students catch these.
Do students need a formula for surface area in 6.G.A.4?
No. The standard asks students to use nets, so they add the areas of the faces they see. Some students notice shortcuts, such as doubling each pair of faces in a box, and in 6.EE.A.2 they meet the formula A = 6s² for a cube. Those shortcuts should always match the net.
Why does 6.G.A.4 only use rectangles and triangles?
Because the standard covers prisms and pyramids with flat faces, and those faces are rectangles and triangles that students already know how to measure (6.G.A.1). Figures with curved surfaces, such as cylinders and cones, need circles in their nets and come later.
Where is surface area used in real life?
Any time you cover the outside of an object: wrapping gifts, designing boxes and packages, painting furniture, sewing tents and covers, or putting shingles on a roof. Real problems also ask which faces to include, for example a box with no lid or a roof with no floor.
How can parents help with nets and surface area at home?
Cut open an empty cereal or tissue box together and lay it flat. Ask your child to name each face, measure it with a ruler, and find the total area. Then ask which faces you would skip if the box had no lid.
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Related Standards
6 standards
These standards connect to 6.G.A.4: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
4.MD.A.3Prerequisite
Apply the area and perimeter formulas for rectangles in real-world problems
Lesson coming soon
6.G.A.1Prerequisite
Find areas of triangles, quadrilaterals and polygons by composing or decomposing
