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.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
How much cardboard does it take to make a box, and how much will the box hold? Questions like these combine two different measures of a solid: surface area, the total area of all its faces, and volume, the space inside. Seventh graders solve real-world and mathematical problems using area for two-dimensional shapes and surface area and volume for cubes and right prisms.
The shapes can be composed of triangles, quadrilaterals and other polygons, so students break a complicated figure into familiar parts, add areas together, or subtract a missing piece. For a right prism of any cross-section, the volume is the area of the base times the height, which covers rectangular and triangular prisms alike. Surface area is found by adding the areas of every face, often with the help of a net. Choosing the right measure for the situation matters just as much: paint and wrapping paper need surface area, while sand, water and storage space need volume. Units should match the measure, square units for area and cubic units for volume.
Students sometimes multiply length × width × height when asked for the cardboard needed. Asking 'are we covering it or filling it?' points to the right measure.
A rectangular prism has six faces in three matching pairs. Listing the pairs, or drawing a net and labeling each face, prevents leaving some out.
For a triangle or trapezoid, the height must be perpendicular to the base. Using a slanted side gives an area that is too large.
Writing cm for an area or cm² for a volume hides confusion about what is being measured. Square units cover; cubic units fill.
A box is a right rectangular prism 8 cm long, 5 cm wide and 3 cm tall. How much space is inside it, and how much paper covers it exactly?
Answer: The box holds 120 cm³ and needs 158 cm² of paper to cover it exactly.
Unfolding real cereal boxes into nets connects surface area to something students can hold, and filling small boxes with unit cubes builds the meaning of volume before the formula. For composite areas, have students show two different ways of decomposing the same figure and confirm they get the same answer.
Assessment items often hide the needed measure inside a context, such as painting a shed or filling a planter, and some ask for a missing dimension given the volume. A labeled sketch with every dimension marked is the best first step on any of these problems.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 64 (also accepted: 64 cm³, 64 cm3)
V = 4 × 4 × 4 = 64 cm³.
Answer: 40 (also accepted: 40 m², 40 m2)
Area = (6 + 10) ÷ 2 × 5 = 8 × 5 = 40 m².
Answer: A) 52 ft²
The face pairs have areas 6, 8 and 12 ft². Doubling their sum gives 2 × 26 = 52 ft². The volume, 24 ft³, is a different measure.
Answer: D) 120 cm³
Base area = 1/2 × 6 × 4 = 12 cm². Volume = 12 × 10 = 120 cm³.
Answer: 68 (also accepted: 68 ft², 68 ft2, 68 sq ft)
Whole rectangle: 10 × 8 = 80 ft². Remove the corner: 80 - 12 = 68 ft².
Answer: B) 40
Each layer holds 5 × 4 = 20 cubes, and there are 2 layers, so 40 cubes fill the box.
Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
A full lesson with slides, activities and an exit ticket on area, volume and surface area problems, pitched to grade 7 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 7.G.B.6 with an answer key, ready in about a minute.
Make a worksheet →Turn area, volume and surface area problems into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Cubes and right prisms, plus two-dimensional figures made of triangles, quadrilaterals and other polygons. Cylinders, cones and spheres come in eighth grade (8.G.C.9).
Volume equals the area of the base times the height of the prism, V = Bh. It works whether the base is a rectangle, a triangle or any other polygon.