Surface Area Word Problems and Error Analysis | Grade 7 Math | LevelKite

Some problems ask how much flat material covers a solid shape: fabric on a tent, paper on a gift box, paint on a ramp. To answer, you measure faces. A face is one flat surface on the outside of a solid, and the surface area of a solid is the total area of all its faces.

A word problem adds a twist. The story tells you which faces get covered. A tent's fabric might skip the floor, and paint might skip the face that rests on the ground. So the answer is the total area of the covered faces only.

Every length in these problems is a positive number, and each problem keeps one unit of length, such as feet. That is why every area comes out in the matching square unit, such as square feet.

In this lesson you will solve these problems from a written list of faces. Then you will use the same list to find and fix mistakes in finished work.

You do not need a drawing for these problems. You need a face list: every face, its shape, its dimensions (the lengths that give its size), and whether the material covers it.

Face names help you make the list complete. The solids in this lesson have $5$ faces each:

Each triangular prism here has $2$ matching triangles (the same shape and size), called its bases, and $3$ rectangles, called its lateral faces. That is $2 + 3 = 5$ faces.
Each square pyramid here has $1$ square, called its base, and $4$ matching triangles, called its lateral faces, which meet at one top point. That is $1 + 4 = 5$ faces.

If your list for one of these solids has more or fewer than $5$ faces, recheck it before you calculate.

Key concept

List every face, then add only the covered ones

In a surface area word problem, the area that the material covers is the sum of the areas of the covered faces. Faces do not overlap, so their areas add.

So write the face list before you add anything. Then find the area of each covered face and add those areas.

This works when every length is a positive number and all lengths use one unit. Lengths are in linear units such as feet (ft), areas are in square units such as square feet ( $\text{ft}^2$ ), and cubic units measure volume, the space inside a solid. Any height you use to find an area must be perpendicular to its base side: it meets that side at a right angle. On a pyramid, each triangular face uses its own height, called the slant height, which is not the height of the solid.

\text{area covered} = \text{sum of the areas of the covered faces}

Worked example

Example 1: Fabric for a tent

A tent is a triangular prism. Its $2$ triangular ends each have a base side of $6$ feet (ft) and a height of $4$ ft, perpendicular to that side. Its $2$ sloping walls are rectangles, each $8$ ft by $5$ ft. Its floor is a rectangle, $8$ ft by $6$ ft. Fabric covers the ends and the sloping walls, but not the floor. What total area does the fabric cover?

Write the face list and mark what the fabric covers, because only covered faces go in the sum. The 2 triangular ends are the prism's bases: covered. The 2 sloping walls are lateral faces: covered. The floor is the third lateral face: not covered. That is 2 + 2 + 1 = 5 faces, with 4 covered.
Find the area of one triangular end. Its 4 ft height is perpendicular to its 6 ft base side, so the area is half of base side times height. Feet times feet gives square feet.
$\frac{1}{2} \times 6 \times 4 = \frac{1}{2} \times 24 = 12 \text{ ft}^2$
Find the area of one sloping wall. It is a rectangle, so multiply its two side lengths, 8 ft and 5 ft.
$8 \times 5 = 40 \text{ ft}^2$
Add the areas of the 4 covered faces: 2 ends and 2 sloping walls. Leave out the floor, which is 8 times 6 = 48 square feet, because no fabric covers it.
$2 \times 12 + 2 \times 40 = 24 + 80 = 104 \text{ ft}^2$
Check the sum against the face list. Written out, it has 4 terms (the numbers being added), one for each covered face, and no floor term. Every length was in feet, so the total is in square feet.
$\underbrace{12 + 12}_{\text{2 ends}} + \underbrace{40 + 40}_{\text{2 sloping walls}} = 104 \text{ ft}^2$

Answer

The fabric covers a total area of $104$ square feet.

The face list for Example 1. Each covered row gives one term of the sum.

Face	Shape and dimensions	Covered by fabric?	Area
Triangular end 1 (base)	Triangle: base side $6$ ft, height $4$ ft	Yes	$12$ square feet
Triangular end 2 (base)	Triangle: base side $6$ ft, height $4$ ft	Yes	$12$ square feet
Sloping wall 1 (lateral face)	Rectangle: $8$ ft by $5$ ft	Yes	$40$ square feet
Sloping wall 2 (lateral face)	Rectangle: $8$ ft by $5$ ft	Yes	$40$ square feet
Floor (lateral face)	Rectangle: $8$ ft by $6$ ft	No	$48$ square feet, not added

A pyramid problem can name two heights, and they do different jobs.

The slant height is the height of one triangular lateral face. It is the distance along that face from the top point to the triangle's base side, perpendicular to that side.
The height of the pyramid is the distance from the top point to the base face, perpendicular to that face.

In the next example the slant height is $13$ centimeters (cm) and the height of the pyramid is $12$ cm. Of these two, only the $13$ cm is a dimension of a face. So only the $13$ cm can go into a face's area, which is measured in square centimeters ( $\text{cm}^2$ ).

Worked example

Example 2: Paper for a pyramid gift box

A gift box is a square pyramid. Its base is a square, $10$ cm by $10$ cm. Its $4$ lateral faces are matching triangles, each with a base side of $10$ cm and a slant height of $13$ cm. The height of the pyramid is $12$ cm. Paper covers all $5$ faces once, with no overlap. What total area does the paper cover?

Write the face list and mark what the paper covers. Base: 1 square, 10 cm by 10 cm, covered. Lateral faces: 4 triangles, each with base side 10 cm and height 13 cm (the slant height), covered. That is 1 + 4 = 5 faces, all covered. The 12 cm is the height of the solid, perpendicular to the base face. It is not a dimension of any face, so it stays out.
Find the area of the square base by multiplying its two side lengths. Centimeters times centimeters gives square centimeters.
$10 \times 10 = 100 \text{ cm}^2$
Find the area of one triangle. The 13 cm slant height is perpendicular to the 10 cm base side, so the area is half of base side times height.
$\frac{1}{2} \times 10 \times 13 = \frac{1}{2} \times 130 = 65 \text{ cm}^2$
Add the areas of all 5 covered faces: 1 square and 4 triangles.
$100 + 4 \times 65 = 100 + 260 = 360 \text{ cm}^2$
Check the sum against the face list. Written out, it has 1 square term and 4 triangle terms, one for each of the 5 faces. Every length was in centimeters, so the total is in square centimeters.
$100 + 65 + 65 + 65 + 65 = 360 \text{ cm}^2$

Answer

The paper covers a total area of $360$ square centimeters.

Error analysis means reading finished work, finding each mistake, and explaining why it is a mistake. For surface area, you can do this from the face list alone. Match each term in the sum to a face, or to a group of matching faces. Then ask three questions:

Faces. Is every covered face counted the right number of times? Is any uncovered face in the sum?
Dimensions. Does each area use lengths from its own face? Is each height perpendicular to its base side? For a pyramid's triangles, is it the slant height and not the height of the solid?
Units and arithmetic. Is every length positive and in the same linear unit? Is the answer in square units? Linear units measure length and cubic units measure volume, so neither can label an area. Then redo each calculation.

For each error you find, name the term, say why it is wrong, and write the corrected term.

One mistake for each audit question, using Examples 1 and 2

Audit question	Mistaken work	What the audit finds	Corrected work
Faces	Tent: $12 + 40 + 40 = 92$ square feet	The face list has $2$ covered triangular ends, but the sum has only one $12$ .	$12 + 12 + 40 + 40 = 104$ square feet
Dimensions	Box: $100 + 4 \times 10 \times 13 = 620$ square centimeters	Each triangle is half of $10 \times 13$ , so its area is $65$ , not $130$ .	$100 + 4 \times 65 = 360$ square centimeters
Units and arithmetic	Box: $100 + 260 = 360$ centimeters	Centimeters measure a length. Areas add to an area, so the unit is square centimeters.	$100 + 260 = 360$ square centimeters

Worked example

Example 3: Find and fix the errors

A skate ramp is a triangular prism. Its $2$ triangular faces each have a base side of $4$ ft, a height of $3$ ft perpendicular to it, and a third side of $5$ ft that slopes. Its $3$ rectangles are the bottom, $6$ ft by $4$ ft; the back, $6$ ft by $3$ ft; and the riding surface, $6$ ft by $5$ ft. Paint covers every face except the bottom. Dev writes: $2 \times \frac{1}{2} \times 4 \times 5 + 24 + 18 + 30 = 92$ square feet. Find and fix each error.

Write the face list first. Bases: 2 triangles, each with base side 4 ft and height 3 ft, covered. Lateral faces: the back, 6 ft by 3 ft, covered; the riding surface, 6 ft by 5 ft, covered; the bottom, 6 ft by 4 ft, not covered. That is 2 + 3 = 5 faces, with 4 covered.
Ask the Faces question. Dev's sum has one term for the 2 triangles, then 24, 18 and 30. The 18 and the 30 match the back and the riding surface, which are covered. The 24 is 6 times 4, the area of the bottom. Paint does not cover the bottom, so the 24 does not belong. That is the first error.
$6 \times 4 = 24 \text{ ft}^2 \text{ (bottom, not painted)}$
Ask the Dimensions question. In the triangle term, Dev multiplied the 4 ft base side by 5 ft. The problem says the height perpendicular to that side is 3 ft, and the 5 ft side slopes. Using 5 ft is the second error. Redo the 2 triangles with 3 ft.
$2 \times \frac{1}{2} \times 4 \times 3 = 1 \times 12 = 12 \text{ ft}^2$
Ask the Units and arithmetic question. Every length is positive and in feet, and the answer is in square feet, so the units pass. The products 6 times 3 = 18 and 6 times 5 = 30 are right. Dev's addition, 20 + 24 + 18 + 30 = 92, is right too. The errors are in what was added, not in how.
Rebuild the sum with only the 4 covered faces: the 2 corrected triangles, the back and the riding surface.
$12 + 18 + 30 = 60 \text{ ft}^2$

Answer

Dev made two errors: adding the unpainted bottom, and using the $5$ ft sloping side as a triangle's height. The paint covers a total area of $60$ square feet, not $92$ .

Takeaway. In a surface area word problem, the story decides which faces count. Write the face list first: every face, its dimensions, and whether it is covered. Then add the areas of the covered faces only. Keep every length positive and in one unit so the total is in square units, and use only heights that are perpendicular to their base sides. On a pyramid, the triangular faces use the slant height, not the height of the solid.

The same list lets you check finished work. Match each term to a face, then test the faces, the dimensions, and the units and arithmetic.