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6th Grade · 6.G.A.4

Surface area in 6th grade

Surface area is the total area of all the outside faces of a 3D shape, like the amount of paper needed to wrap a box. In 6th grade math, your child learns to unfold a shape into a flat pattern called a net. Adding the areas of the rectangles and triangles in the net gives the surface area.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

6.G.A.4 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.

What it means, in plain words

A three-dimensional (3D) shape takes up space, like a box or a tent. Its flat sides are called faces. Surface area is the total area of all the faces, the paper it would take to cover the shape with no gaps and no overlap. It is measured in square units, such as square inches.

In 6th grade, children find surface area using nets. A net is the flat pattern you get when you cut a box along some of its edges and lay it flat. A box (a rectangular prism) opens out into six rectangles. A shape like a tent (a triangular prism) opens into three rectangles and two triangles. A square pyramid opens into one square and four triangles.

Nets and surface area go together because the net shows every face at once. Your child finds the area of each rectangle and triangle and adds them, so no face is missed or counted twice. Children also notice shortcuts, such as the faces of a box coming in matching pairs. Real problems include wrapping a gift, painting a shed and making a box from cardboard.

Your child should be able to:

  • Match a 3D shape to its net and tell whether a net will fold into that shape
  • Find the area of each rectangle and triangle in a net
  • Add the areas of the faces to find the surface area of a prism or a pyramid
  • Solve real problems with surface area, such as how much paper wraps a box

Example

How much wrapping paper covers a box that is 6 inches long, 4 inches wide and 2 inches tall, with no overlap?

  1. The net of the box has six rectangles in three matching pairs.
  2. Top and bottom: 6×4=246 \times 4 = 24 square inches each, so 48 square inches.
  3. Front and back: 6×2=126 \times 2 = 12 square inches each, so 24 square inches.
  4. The two ends: 4×2=84 \times 2 = 8 square inches each, so 16 square inches.
  5. 48+24+16=8848 + 24 + 16 = 88, so it takes 88 square inches of paper to cover the box.

Example

A square pyramid has a square base with 8 cm sides. Each of its four triangle faces has a base of 8 cm and a height of 7 cm. What is its surface area?

  1. The net has one square and four triangles.
  2. Square: 8×8=648 \times 8 = 64 square centimeters.
  3. Each triangle: 12×8×7=28\frac{1}{2} \times 8 \times 7 = 28 square centimeters. Four triangles: 4×28=1124 \times 28 = 112 square centimeters.
  4. 64+112=17664 + 112 = 176, so the surface area of the pyramid is 176 square centimeters.

Where children usually get stuck

  • For a box 6 inches long, 4 inches wide and 2 inches tall, your child writes 6×4×2=486 \times 4 \times 2 = 48 as the surface area.

    Multiplying the three edges is the volume rule from 5th grade, and it is still fresh. Volume is the space that fills the box. Surface area is the cardboard that covers it.

    What helps. Cut open an empty cereal box and lay it flat. Ask which number describes the cardboard: the one you get by adding the flat pieces.

  • For the same box, your child adds only three faces (24+12+8=4424 + 12 + 8 = 44) and stops.

    A drawing of a box shows only three faces. The bottom, back and far end are hidden, so they get left out.

    What helps. Have your child label each face of the net with a letter and tick it off once its area is written down. Six faces need six ticks.

  • For a triangle face with a base of 8 cm and a height of 7 cm, your child writes 8×7=568 \times 7 = 56 square centimeters.

    Most faces in a net are rectangles, so the rectangle rule carries over to the triangles too, and the half is lost.

    What helps. Before any multiplying, have your child point to each face and say "rectangle" or "triangle". Every triangle gets the 12\frac{1}{2}.

Try four practice questions

Real 6th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

  1. 1Warm-up
    Select all nets that will fold into a triangular prism.
    A
    B
    C
    D
    Show the answer and how to solve it

    The valid net has 3 rectangles and 2 triangles. Its rectangles wrap around the sides, and its triangles close opposite ends.

    1. 1A triangular prism has 3 rectangular side faces and 2 triangular end faces. A valid net must include exactly those 5 faces — no more and no fewer.
    2. 2In the complete net, the 3 rectangles form a strip. Their widths match the 3 sides of each triangle. Fold the outer rectangles toward each other along the shared edges to form the prism's sides. The matching triangles above and below the middle rectangle close opposite ends.
    3. 3The net with only 1 triangle is missing one end face. The net with 3 triangles has one extra end face. The net built from only rectangles has no triangular faces at all and folds into a different kind of prism.

    Common mistake. A common mistake is choosing any net that has triangles without counting them. A triangular prism needs exactly 2 triangles — a net with only 1 triangle is incomplete, and a net with 3 triangles has one too many. After counting, check that the rectangles can wrap around the triangle edges and the triangles can close opposite ends.

    The idea underneath. A triangular prism net needs 3 rectangles and 2 matching triangles. The faces must also fit together: the rectangles wrap around the sides, and the triangles close opposite ends without gaps or overlap.

  2. 2On level
    Mia claims this net will fold into a triangular prism. Look at the net below.
    Select the statement that best evaluates the claim.
    A
    The claim is incorrect because the net has only 1 triangle.
    B
    The claim is incorrect because the net has 4 rectangles.
    C
    The claim is correct because the net has 3 rectangles and 2 triangles.
    D
    The claim is incorrect because the net has 3 triangles.
    Show the answer and how to solve it

    The claim is correct. The 3 rectangles wrap around the sides, and the 2 triangles close the ends.

    1. 1Look at the net. A triangular prism net must have exactly 3 rectangles and 2 triangles.
    2. 2Count the rectangles in the strip. This net has 3 rectangles.
    3. 3Count the triangular ends. This net has 2 triangles.
    4. 4Imagine folding along the lines between the rectangles. The 3 rectangles wrap around to make the sides. The 2 triangles fold over to close the ends. This forms a triangular prism, so the claim is correct.

    Common mistake. Counting 3 rectangles and 2 triangles is only the first check. Also check how they join: here the rectangles wrap around the sides, and the triangles close the two ends.

    The idea underneath. A triangular prism has 3 rectangular sides and 2 triangular ends. A net must let these faces fold together to make the prism.

  3. 3On level
    The rectangular prism below has dimensions 5, 3, and 3, as labeled. Complete the table with the area of each opposite-face pair and the total surface area.
    533
    Enter each area in square units. The top and bottom are the horizontal faces at the base and top of the prism.
    Top and bottom
    square units
    Front and back
    square units
    Left and right
    square units
    Total surface area
    square units
    Show the answer and how to solve it

    The completed table has 3030 square units for top and bottom, 7878 square units for the total surface area, and the two vertical face pairs have areas 3030 and 1818 square units (in either order): one pair is 5×3×2=305 \times 3 \times 2 = 30 and the other is 3×3×2=183 \times 3 \times 2 = 18.

    1. 1Look at the 3D prism and its three labeled dimensions. A rectangular prism has 3 opposite-face pairs: top and bottom (the horizontal faces), plus two pairs of vertical faces. The vertical dimension (3) is the height of both vertical face pairs. The two horizontal dimensions (5 and 3) determine the top-and-bottom pair, while each vertical face pair uses the height together with one horizontal dimension.
    2. 2Top and bottom each measure 5×3=155 \times 3 = 15 square units (the two horizontal dimensions), so the pair area is 3030.
    3. 3One vertical face pair uses dimension 55 and the height 33, so each face is 5×3=155 \times 3 = 15 and the pair area is 3030.
    4. 4The other vertical face pair uses dimension 33 and the height 33, so each face is 3×3=93 \times 3 = 9 and the pair area is 1818.
    5. 5Add the three pair areas to get the total surface area: 30+30+18=7830 + 30 + 18 = 78 square units.

    Common mistake. A common mistake is entering the area of one face instead of the whole opposite-face pair. For example, 5×3=155 \times 3 = 15 is only one face, so the Top and bottom row must be 3030, not 1515.

    The idea underneath. Opposite faces on a rectangular prism are congruent, so you can find one face area, double it for that pair, and then add the three pair areas. Top and bottom are the two horizontal faces at the base and top of the prism, so they are unambiguous. The two vertical face pairs (front-back and left-right) are interchangeable depending on how you orient the prism, so either mapping of 3030 and 1818 to those two rows is accepted.

  4. 4Stretch
    Look at the net below. Some side lengths are hidden.
    Enter the total surface area in square units.
    Surface area:square units
    Show the answer and how to solve it

    Answer. 336

    The total surface area is 336 square units because the 4 side triangles total 192 square units and the square base is 144 square units.

    1. 1The net has 1 square base and 4 side triangles. Only 1 triangle still shows its measurements, so use that triangle to find the area of every side triangle.
    2. 2The labeled triangle has base 12 and height 8.
    3. 3All 4 side triangles have the same size and shape.
    4. 4The base is a square with side length 12.
    5. 5Add the area of the 4 triangles and the square base.

    Common mistake. A common mistake is using the slanted side length 10 instead of the triangle height 8 when finding the triangle area. Another mistake is finding the area of just 1 side triangle and forgetting to multiply by 4.

    The idea underneath. Surface area comes from adding the areas of all faces in the net, and repeated faces let you find 1 face area and multiply.