7th Grade · 7.G.B.6

Volume of prisms in 7th grade

Finding the volume of prisms means measuring the space inside a solid with two matching ends, in cubic units. In 7th grade, your child finds the volume of prisms, the surface area of prisms and the area of flat shapes built from triangles and rectangles. These ideas come up in real problems about boxes, tents and rooms.

What the standard says

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

7.G.B.6 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

What it means, in plain words

A prism is a solid with two matching, parallel ends, called bases, joined by flat sides. In a right prism the sides stand straight up. A box is a rectangular prism, and a tent shape is a triangular prism. The base can be any polygon, a flat shape with straight sides.

Every problem on the volume of prisms uses one rule: volume is the area of the base times the height, $V = B \times h$ . Here $B$ is the area of one base, and the height is the distance between the two bases. For a box this is length times width times height. For other prisms, your child first finds the area of the base shape, such as a triangle, and then multiplies. Volume is in cubic units, such as cubic inches.

Surface area is the total area of all the faces, the amount of paper that would wrap the solid. A net helps here: it is the solid unfolded flat, so every face is visible. In 7th grade, surface area and volume problems sit next to area problems about flat shapes made of rectangles and triangles. Your child splits such a shape into simple parts, finds each area and adds them, or subtracts a part that is cut out.

Your child should be able to:

Find the area of a shape made of rectangles and triangles by splitting it into parts
Find the volume of a box or a triangular prism as base area times height
Find the surface area of a prism from a picture or a net
Decide whether a real problem asks for area, surface area or volume

Example

A tent is shaped like a triangular prism. Each end is a triangle 6 feet wide at the bottom and 4 feet tall, and the tent is 10 feet long. How much space is inside?

The bases are the two triangle ends. Area of one end: $\frac{1}{2} \times 6 \times 4 = 12$ square feet.
The distance between the two ends is the length of the tent, 10 feet.
$V = 12 \times 10 = 120$
The tent holds 120 cubic feet of space.

Example

A closed gift box is 5 inches long, 4 inches wide and 3 inches tall. How much paper covers it exactly, and how much space is inside?

Surface area: the box has three pairs of matching faces.
$2 \times (5 \times 4) + 2 \times (5 \times 3) + 2 \times (4 \times 3) = 40 + 30 + 24 = 94$
Volume: $5 \times 4 \times 3 = 60$
The paper covers 94 square inches, and the box holds 60 cubic inches.

Where children usually get stuck

Asked how much paper wraps a box, your child multiplies the length, width and height.
Surface area and volume are about the same box, and the volume rule is the shorter one to remember. Paper covers the faces, so the question is about area.
What helps. Ask whether the question is about covering the box or filling it. Covering is in square units. Filling is in cubic units.
For the tent, your child multiplies $6 \times 4 \times 10 = 240$ cubic feet.
The habit from boxes is to multiply the three numbers on the picture. The base here is a triangle, and a triangle's area is half of its base times its height.
What helps. Before any multiplying, ask your child to point to the two matching ends. Find the area of one end first, then multiply by the distance between the ends.
For the surface area of the gift box, your child adds only the three faces in the picture: $20 + 15 + 12 = 47$ square inches.
A drawing shows three faces. The bottom, the back and the far side are hidden, so they get left out.
What helps. Take a cereal box and put a sticky note on each face as you count together. There are six faces, in three matching pairs.

Try four practice questions

Real 7th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Find the area that remains after the cutout is removed from the whole shape.
Whole shape
Cutout
Remaining area:cm²
Show the answer and how to solve it
Answer. 56
The remaining area is $56 \text{ cm}^{2}$ because the whole parallelogram has area $72 \text{ cm}^{2}$ and the triangle cutout has area $16 \text{ cm}^{2}$ .
1. 1The whole shape is a parallelogram with base 12 cm and height 6 cm. Find its area using base × height.
2. 2The cutout is a triangle with base 8 cm and height 4 cm. Find its area using $\frac{1}{2}$ × base × height.
3. 3Subtract the cutout area from the whole area.
Common mistake. A common mistake is to add the two areas or to forget the $\frac{1}{2}$ in the triangle formula. Because the triangle is removed, you subtract $16$ from $72$ , and triangle area is always $\frac{1}{2} \times \text{base} \times \text{height}$ .
The idea underneath. A cutout-area problem becomes simpler when you treat it as whole area minus removed area.
2On level
Use the net to find the total surface area.
Surface area:square units
Show the answer and how to solve it
Answer. 142
The surface area is $142$ square units because the net has two $7 \times 5$ faces, two $7 \times 3$ faces, and two $5 \times 3$ faces.
1. 1Look at the net. A rectangular prism has 3 pairs of congruent faces, so there are 3 different face areas to find.
2. 2Find the area of one face in each pair.
3. 3Each of those face areas appears twice.
4. 4Add the areas of all 6 faces.
Common mistake. A common mistake is to add $35 + 21 + 15 = 71$ and stop. That counts only one face of each size, but a rectangular prism has 2 faces of each size.
The idea underneath. Opposite faces in a rectangular prism are congruent, so you can find the 3 distinct face areas and count each one twice.

3On level

A glass fish tank is 12 in. long, 8 in. wide, and 6 in. tall. It has no lid, so paint goes on the outside bottom and the 4 outside side faces. Priya wrote the calculation below. Which statement best explains her mistake?

Priya's setup	$\displaystyle SA = 2(12 \cdot 8) + 2(12 \cdot 6) + 2(8 \cdot 6)$
Priya's arithmetic	$\displaystyle = 192 + 144 + 96$
Priya's total	$\displaystyle = 432$

Priya counted both the top and bottom 12-by-8 faces, but the tank has no lid so only the outside bottom is painted.

Priya used volume instead of adding areas of faces.

Priya forgot one pair of side faces.

Priya should leave out the bottom face, not the top.

Show the answer and how to solve it

The best explanation is that Priya counted the top face even though the tank has no lid. The correct painted area is 336 square inches, not 432.

1Look at Priya's first line. The term 2(12 · 8) means she counted two 12-by-8 faces: both the top and the bottom.
2That formula works for a closed rectangular prism with six faces. But this fish tank has no lid, so there is no top face to paint.
3Only one 12-by-8 face should be counted: the outside bottom.
4So Priya's mistake was using 2(12 · 8) and counting the top face, even though the tank has no lid.

Common mistake. A common mistake is applying 2lw + 2lh + 2wh automatically to every box-shaped object. Before calculating, identify which faces are actually exposed. Here the top is open, so only one 12-by-8 face should be included.

The idea underneath. Surface area depends on the faces that are exposed, not automatically all 6 faces of a rectangular prism.

4Stretch
Find the total area of the partitioned figure. All measurements are in units.
Total area:square units
Show the answer and how to solve it
Answer. 63
The total area is 63 square units because the left triangle is 12, the middle trapezoid is 42, and the right triangle is 9.
1. 1Break the figure into its 3 parts: the left triangle, the middle trapezoid, and the right triangle. Find each area, then add them.
2. 2Find the area of the left triangle.
3. 3Find the area of the middle trapezoid.
4. 4Find the area of the right triangle.
5. 5Add the 3 part areas.
Common mistake. A common mistake is computing $(5 + 9) \times 6 = 84$ for the trapezoid and forgetting to divide by 2. That gives 84 instead of 42 and makes the total 21 too large.
The idea underneath. A composite figure's area is the sum of the areas of its non-overlapping parts, so you can use the correct formula for each part and then add.