7.G.B.6: Area, Volume and Surface Area of Polygons, Cubes and Prisms
In plain English: 7.G.B.6 is the Common Core grade 7 math standard that asks students to solve real-world and mathematical problems about area, volume and surface area. The shapes are polygons (flat shapes with straight sides), figures built from cubes, and right prisms, whose volume is the area of the base times the height. Students split a complicated shape into simple ones and add or subtract the parts.
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.
Common Core State Standards for Mathematics · Domain: Geometry (G) · Cluster: Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. Also written as 7.G.6 · Official standard
Students find three measurements of shapes. Area is the number of square units that cover a flat shape. Volume is the number of cubic units (small cubes) that fill a solid. Surface area is the total area of all the flat faces on the outside of a solid. The shapes in this standard are polygons (flat shapes with straight sides, such as triangles, quadrilaterals with 4 sides, pentagons with 5 sides and hexagons with 6 sides), solids built from cubes, and right prisms. A composite figure is a shape made of simpler shapes, and students find its area by splitting it into rectangles and triangles, or by subtracting a missing piece.
A prism is a solid with two matching, parallel faces called the bases, joined by faces called lateral faces. In a right prism, the lateral faces are rectangles that stand straight up from the base. The base can be any polygon, not only a rectangle, so grade 7 uses one volume rule for all right prisms: V = B × h, where B is the area of the base and h is the height of the prism (the distance between the two bases). Grade 6 found surface area with nets (6.G.A.4) and the volume of boxes (6.G.A.2). Circles belong to 7.G.B.4, and the volume of pyramids, cones and spheres comes later (8.G.C.9, HSG.GMD.A.3).
Learning Objectives
By the end of this lesson, students will be able to:
Find the area of a composite polygon by splitting it into triangles and quadrilaterals, or by subtracting a missing piece
Find the volume of a right prism with any polygon base using V = B × h, and work backward to find a missing length
Find the surface area of a right prism by adding the areas of both bases and every lateral face
Find the volume and surface area of a solid built from cubes by counting cubes and faces
Choose area, surface area or volume for a real problem and give the answer in the right units
Prior Knowledge Required
Students should already be comfortable with:
Finding the area of triangles, parallelograms (two pairs of parallel sides) and trapezoids (exactly one pair) by splitting them into rectangles and triangles 6.G.A.1
Finding the volume of a box with V = l × w × h 6.G.A.2
Using nets, flat patterns that fold into a solid, to find surface area 6.G.A.4
Adding volumes of two boxes put together 5.MD.C.5
Multiplying and dividing decimals and fractions 7.NS.A.3
Give each pair 4 snap cubes. Each cube is 1 unit on every edge. Ask them to build two shapes and compare them:
Warm-Up Prompt
"Build a row of 4 cubes. Then build a flat 2-by-2 square of 4 cubes. How many cubes does each shape use? If you painted every outside face, which shape would need more paint? Count the faces you would paint."
Both shapes use 4 cubes, so both have a volume of 4 cubic units. The row shows 18 square faces on the outside and the 2-by-2 square shows 16, so the row needs more paint: its surface area is 18 square units, and the square's is 16. Ask: "Why is the number of faces less than 4 × 6 = 24?" When two cubes touch, the two faces glued together are hidden. The row has 3 glued places (24 - 2 × 3 = 18), and the square has 4 (24 - 2 × 4 = 16). Same volume, different surface area: that is the main idea of today's lesson.
Direct Instruction20 minutes
Part 1: Area of polygons. Review the area rules from grade 6, and say each one in words:
Rectangle: length × width.
Triangle: 1/2 × base × height. The height is the straight-up distance from the base to the opposite corner, not a slanted side.
Parallelogram (a quadrilateral with two pairs of parallel sides): base × height.
Trapezoid (a quadrilateral with exactly one pair of parallel sides): 1/2 × (sum of the two parallel sides) × height.
Composite figure: split it into these shapes and add the areas, or draw a rectangle around it and subtract the missing pieces.
Work Example 1 with Diagram 1. Ask students to find the trapezoid part a second way, with the trapezoid rule: 1/2 × (20 + 12) × 4 = 64 ft², the same as 48 + 8 + 8.
Part 2: Volume of right prisms. Hold up a stack of paper. Each sheet is a copy of the base, and the stack is as tall as the prism. So the volume is the area of one base times the height: V = B × h. A box is a prism whose base is a rectangle, so V = l × w × h is the same rule. Volume is in cubic units, such as cm³. For liquids, 1 cm³ holds 1 milliliter (mL), and 1,000 cm³ is 1 liter.
Part 3: Surface area of right prisms. Add the two bases and every lateral face. Each lateral face is a rectangle as long as the prism's height, and its width is one side of the base. So all the lateral faces together make one long rectangle: the perimeter of the base (the distance around it) times the height. Surface area = 2 × B + perimeter × h. For a solid built from cubes, count the outside faces, or use: faces = 6 × (number of cubes) - 2 × (number of places where two cubes touch).
Area of a composite polygon (Diagram 1)
A school stage is a rectangle 20 ft wide and 12 ft deep, with a trapezoid-shaped front: its back edge is 20 ft, its front edge is 12 ft, and it sticks out 4 ft. How much floor needs a new coat of paint?
Equation: Split the trapezoid into a 12 × 4 rectangle and two triangles with base 4 ft and height 4 ft. Area = 20 × 12 + 12 × 4 + 2 × (1/2 × 4 × 4) = 240 + 48 + 16 = 304 ft².
Volume of a triangular prism
A rubber doorstop is a right prism. Its base is a right triangle (a triangle with a square corner) whose legs, the two sides that meet at the square corner, are 15 cm and 4 cm, and the doorstop is 5 cm wide. How much rubber is in it?
Equation: B = 1/2 × 15 × 4 = 30 cm². V = B × h = 30 × 5 = 150 cm³.
Volume and surface area of a pentagonal prism (Diagram 2)
A birdhouse is a right prism 7 in deep. Its front is a pentagon: a 6 in by 6 in square with a triangle on top that is 4 in tall, and the two slanted roof edges are each 5 in. Ignore the entrance hole.
Equation: B = 36 + 1/2 × 6 × 4 = 48 in². V = 48 × 7 = 336 in³. The lateral faces are 5 rectangles, each 7 in long, with widths 6, 6, 6, 5 and 5 in: 28 × 7 = 196 in². Surface area = 2 × 48 + 196 = 292 in².
Volume and surface area of a figure made of cubes
A store builds a 3-step display from 6 wooden cubes, each 10 cm on an edge: 3 cubes on the bottom row, 2 on the middle row and 1 on top, all lined up against the left end.
Equation: V = 6 × (10 × 10 × 10) = 6,000 cm³. Faces: 6 cubes × 6 = 36, minus 2 for each of the 6 places where cubes touch: 36 - 12 = 24 faces. Surface area = 24 × 100 = 2,400 cm².
Working backward from the volume
A planter is a right prism whose base is a trapezoid with parallel sides 30 cm and 20 cm, 15 cm apart. It holds 15,000 cm³ of soil when full. How long is the planter?
Equation: B = 1/2 × (30 + 20) × 15 = 375 cm². 375 × h = 15,000, so h = 15,000 ÷ 375 = 40 cm long.
After each example, ask: "Is this answer an area, a surface area or a volume? What are its units?" Students should say square units for the area in Example 1 and the surface areas in Examples 3 and 4, and cubic units for the volumes in Examples 2 to 5. In Example 5, point out that the "height" of a prism lying on its side is its length: h is the distance between the two bases, whichever way the prism sits.
Guided Practice15 minutes
Pairs solve four problems. For each one, they first sketch the shape and write whether the question asks for area, volume or surface area. Circulate and ask each pair to name the base of every prism.
Guided practice problems with answers
Problem
Answer
The front wall of a house is a rectangle 24 ft wide and 10 ft tall with a triangle on top that is 5 ft tall. A window 3 ft by 4 ft is not painted. How many square feet are painted?
A storage bin is a cube 40 cm on each edge. What is its volume, and how many 10-cm cubes fit inside?
40 × 40 × 40 = 64,000 cm³; 4 × 4 × 4 = 64 cubes
A skateboard ramp is a right prism 20 in wide. Its base is a right triangle with legs 7 in and 24 in and a slanted side of 25 in. Find its volume and surface area.
A 2 × 2 × 2 cube is built from 8 small cubes, each 1 cm on an edge. One corner cube is taken away. Find the new volume and surface area.
V = 7 cm³. SA = 24 cm², the same as before: the corner cube took away 3 outside faces and uncovered 3 new ones.
Listen for these errors: forgetting the 1/2 for a triangle, using a slanted side as the height, adding only one base to the surface area, and giving a volume in square units. For the ramp, ask: "Which faces are the bases?" (The two triangles, not the bottom rectangle.)
Independent Practice15 minutes
Students solve on their own, with a sketch and units for every answer:
Independent practice problems with answers
Problem
Answer
A garden bed is a trapezoid with parallel sides 18 m and 12 m, 7 m apart. Find its area.
1/2 × (18 + 12) × 7 = 105 m²
A block of cheese is a right prism 5 cm thick. Its base is a triangle with a base of 8 cm and a height of 6 cm. Find its volume.
B = 24 cm²; V = 24 × 5 = 120 cm³
A gift box is a cube 12 cm on each edge. Find its volume and surface area.
V = 1,728 cm³; SA = 6 × 144 = 864 cm²
A pencil is close to a right prism with a hexagon base. The base area is about 0.4 cm², and the pencil is 18 cm long. About how much wood and graphite does it hold?
V = 0.4 × 18 = 7.2 cm³
Closure5 minutes
Exit ticket: (1) A right prism has a triangular base with an area of 14 in² and a height of 9 in. Find its volume. (126 in³.) (2) An L-shaped patio is made of a 6 m by 2 m rectangle and a 3 m by 4 m rectangle that do not overlap. Find its area. (12 + 12 = 24 m².) (3) In one sentence: why is surface area in square units and volume in cubic units?
Differentiation Strategies
For Struggling Students
Give a formula card with a small picture next to each area rule, and color the base of every prism
Let students build prisms from snap cubes and count, before they use V = B × h
Start with whole-number measurements, then change one measurement to a decimal
For Advanced Students
Ask for two different right prisms that have the same volume but different surface areas, and explain which uses less material
Ask what happens to the volume and the surface area of a prism when every length doubles
Give a floor plan with a missing length and ask students to find it from the total area
Assessment Guidance
What to Look For
Check that students name the base of each prism before they compute, and that the base can be a triangle, trapezoid or pentagon, not only a rectangle. In composite figures, look for a sketch that shows how the shape was split, with no piece counted twice. Every answer should have the right units: square units for area and surface area, cubic units for volume. For cube figures, students should explain why the glued faces are not part of the surface area.
02
Classroom Activities
3 Activities
1
Cube Builders
15 minPairs
Pairs build four figures from 1-cm snap cubes, then find the volume and the surface area of each. They count the outside faces one by one, then check with the rule: faces = 6 × cubes - 2 × glued places.
Figure Cards
Card 1: a tower of 5 cubes stacked straight up
Card 2: an L of 5 cubes: 3 cubes in a row, and 2 more stacked on the left end
Card 3: a flat 2-by-3 rectangle of 6 cubes, with 1 more cube on top of a corner
Card 4: a flat row of 4 cubes, with 2 more cubes beside the first two (6 cubes)
Answer Key
Card 1: 5 cm³; 4 glued places, 30 - 8 = 22 cm²
Card 2: 5 cm³; 4 glued places, 22 cm²
Card 3: 7 cm³; 8 glued places, 42 - 16 = 26 cm²
Card 4: 6 cm³; 6 glued places, 36 - 12 = 24 cm²
Discussion Questions
Cards 1 and 2 look very different. Why do they have the same volume and the same surface area?
Card 4 and a flat 2-by-3 rectangle both use 6 cubes. Which has less surface area, and why? (The rectangle, 22 cm², because it has 7 glued places instead of 6.)
Build your own 6-cube figure with the least surface area you can. Can anyone get below 22 cm²? (No: 22 cm² is the least for 6 cubes, and a flat 2-by-3 rectangle is one way to get it.)
Modification for Distance Learning
Students build the figures with sugar cubes or dice at home, or in a free online cube-building app, and post a photo with their face counts.
2
Floor Plan Split-Up
15 minGroups of 3
Each group gets three floor plans drawn on grid paper, where each square stands for 1 m by 1 m. Each plan is given by its vertices (corner points) on the grid. The group finds each area two ways: by splitting the shape into pieces and adding, and by drawing a rectangle around it and subtracting the empty corners.
Floor Plans
Plan A, a classroom with a pointed end (pentagon): (0, 0), (10, 0), (10, 6), (5, 9), (0, 6)
Plan B, a library with one cut corner (pentagon): (0, 0), (12, 0), (12, 4), (8, 8), (0, 8)
Plan C, a hallway shaped like a stretched hexagon: (0, 3), (4, 0), (10, 0), (14, 3), (10, 6), (4, 6)
Answer Key
Plan A: rectangle 10 × 6 = 60, plus triangle 1/2 × 10 × 3 = 15: 75 m²
Plan B: rectangle 12 × 8 = 96, minus corner triangle 1/2 × 4 × 4 = 8: 88 m²
Plan C: rectangle 6 × 6 = 36, plus two triangles 1/2 × 6 × 4 = 12 each: 60 m². Around it: 14 × 6 = 84, minus four corner triangles of 1/2 × 4 × 3 = 6 each: 60 m²
Discussion Questions
Which plan has the largest area? (Plan B.) Is it also the plan with the largest surrounding rectangle?
For which plan was subtracting easier than adding? Why?
New flooring costs $30 per square meter. Which room costs the least to floor, and how much? (Plan C, $1,800.)
Challenge Variation
Groups draw their own pentagon floor plan with an area of exactly 50 m², trade with another group, and check each other's area.
3
Build a Prism and Fill It
20 minGroups of 3-4
Each group gets one base card, builds a right prism 10 cm tall from card stock, predicts its volume and surface area, then fills it with rice and pours the rice into a measuring jug to test the volume prediction (1 cm³ = 1 mL).
Base Cards
Card T: a right triangle with legs 9 cm and 12 cm and a slanted side of 15 cm
Card Q: a trapezoid with parallel sides 10 cm and 4 cm, 4 cm apart, and two slanted sides of 5 cm
Card P: a pentagon made of an 8 cm by 5 cm rectangle with a triangle on top, 3 cm tall, whose slanted sides are 5 cm
Procedure
Trace the base twice on card stock. Cut a strip 10 cm wide and as long as the base's perimeter, fold it at each side length, and tape the bases to it
Predict: B, V = B × 10, and surface area = 2 × B + perimeter × 10
Leave one base open, fill the prism with rice, and pour it into the jug. Compare the milliliters with your predicted volume
Answer Key
Card T: B = 54 cm², V = 540 cm³, SA = 108 + 36 × 10 = 468 cm²
Card Q: B = 28 cm², V = 280 cm³, SA = 56 + 24 × 10 = 296 cm²
Card P: B = 40 + 12 = 52 cm², V = 520 cm³, SA = 104 + 28 × 10 = 384 cm²
Discussion Questions
Which prism holds the most rice? Does it also use the most card stock? (Card T in both cases.)
Cards T and P have volumes only 20 cm³ apart but surface areas 84 cm² apart. Why can two prisms of the same height hold about the same amount but use different amounts of card stock?
Your measured rice was probably a little less than the prediction. What could cause the difference?
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Splitting the Stage Floor into Simple Shapes
The stage from Example 1 is a hexagon (a polygon with 6 sides). Dashed lines split it into a 20 ft by 12 ft rectangle, a 12 ft by 4 ft rectangle and two triangles with base 4 ft and height 4 ft. Adding the parts gives 304 ft². Drawn to scale, 20 pixels per foot.
Diagram 2: The Birdhouse as a Right Prism
The birdhouse from Example 3 is a right prism whose base is the shaded pentagon, a 6 in square with a triangle on top. The front face is drawn to scale at 24 pixels per inch. The 7 in depth is drawn going back at a slant, at half scale, as in many textbook drawings, and gray dashed edges are hidden behind the solid. The blue dashed lines on the front split the pentagon into the square and the triangle and show the 4 in height.
04
Homework Assignment
~30 min
7.G.B.6 Homework: Area, Volume and Surface Area
Directions: Show all work. Sketch every shape and label its measurements. Say whether each answer is an area, a surface area or a volume, and give the units.
Part 1: Area of Polygons (Problems 1-2)
A school banner is a rectangle 60 cm long and 40 cm tall. A triangle is cut out of one short end to make a swallowtail: the triangle's base is the whole 40 cm edge, and it reaches 15 cm into the banner. What is the area of the banner that is left?
A table top is a regular hexagon (6 equal sides and 6 equal angles). It splits into 6 matching triangles, each with a base of 50 cm and a height of about 43.3 cm. Find the area of the table top. The top is 3 cm thick. How much wood is in it?
Part 2: Right Prisms (Problems 3-4)
A water trough for horses is a right prism 200 cm long. Each end is a trapezoid 60 cm across the top, 40 cm across the bottom and 30 cm deep. How many liters of water does the trough hold when full? (1,000 cm³ = 1 liter.)
A poster tube is a right prism 60 cm long. Each end is a triangle with three 8 cm sides and a height of about 6.9 cm. Find the volume of the tube and the amount of cardboard needed to make it, including both ends.
Part 3: Cubes and Real-World Problems (Problems 5-6)
Maya glues 27 one-inch cubes into a large 3 × 3 × 3 cube. Find its volume and surface area. Then she pulls out the center cube of each of the 4 side faces (not the top or the bottom). Find the new volume and surface area, and explain why the surface area went up.
A fish tank with no lid is 60 cm long, 30 cm wide and 36 cm tall. How much glass is needed for the bottom and the four sides? How many liters of water does it hold when the water is 4 cm below the top?
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Sketch and Setup
Shape sketched and split correctly; base of each prism named
Sketch present but a piece or base is missing
No sketch or wrong shapes
Computation
All areas, volumes and surface areas correct
One computing error
Most answers incorrect
Units
Square units for area, cubic units or liters for volume, every time
Units missing or wrong once
Units missing or wrong throughout
Explanation
Problem 5 explains the change in surface area with face counts
Explanation is partly correct
No explanation
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. 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 bedroom is shaped like an 8 m by 5 m rectangle with a 3 m by 3 m square cut out of one corner. What is the floor area?
Answer: B
The full rectangle is 8 × 5 = 40 m², and the missing corner is 3 × 3 = 9 m², so the area is 40 - 9 = 31 m². Choice A ignores the missing corner. Choice C adds the corner instead of subtracting it. Choice D is the perimeter, 2 × (8 + 5) = 26 m, which measures the distance around, not the floor.
Question 2 of 20 · Multiple Choice
A trapezoid has parallel sides of 9 in and 15 in, and they are 6 in apart. What is its area?
Answer: C
Area = 1/2 × (9 + 15) × 6 = 1/2 × 24 × 6 = 72 in². Choice A forgets the 1/2: 24 × 6 = 144. Choice B multiplies all three numbers: 9 × 15 × 6 = 810. Choice D uses only the longer side: 15 × 6 = 90.
Question 3 of 20 · Multiple Choice
A small sailboat's sail is a triangle with a base of 3 m and a height of 5 m. How much sail cloth does it use?
Answer: A
Area = 1/2 × 3 × 5 = 7.5 m². Choice B forgets the 1/2. Choice C adds the base and the height: 3 + 5 = 8. Choice D takes half of that sum.
Question 4 of 20 · Multiple Choice
A right prism has a triangular base with a base of 10 cm and a height of 6 cm. The prism is 25 cm long. What is its volume?
Answer: D
B = 1/2 × 10 × 6 = 30 cm², and V = B × h = 30 × 25 = 750 cm³. Choice A forgets the 1/2 in the triangle's area: 10 × 6 × 25 = 1,500. Choice B adds the three lengths: 10 + 6 + 25 = 41. Choice C is the area of the base only, not multiplied by the length.
Question 5 of 20 · Multiple Choice
In the volume rule V = B × h for a right prism, what does B stand for?
Answer: B
B is the area of one base, the number of square units in it. Multiplying by the height stacks that many layers. Choice A, the perimeter, is used for the lateral faces in surface area, not for volume. Choice C is a length, not an area. Choice D is always 2 for a prism, and the rule does not use it.
Question 6 of 20 · Multiple Choice
A gift box is a cube with edges 15 cm long. How much wrapping paper covers it exactly, with no overlap?
Answer: B
Each face is 15 × 15 = 225 cm², and a cube has 6 faces: 6 × 225 = 1,350 cm². Choice A is the volume, 15 × 15 × 15 = 3,375, which is in cubic units. Choice C is one face only. Choice D covers only 4 faces, leaving out the top and bottom.
Question 7 of 20 · Multiple Choice
Four 1-cm cubes are glued into a flat 2-by-2 square, and a fifth cube is glued on top of one of them. What is the surface area of the figure?
Answer: C
There are 5 × 6 = 30 faces. The square has 4 places where cubes touch, and the top cube adds 1 more, so 5 glued places hide 2 faces each: 30 - 10 = 20 cm². Choice A counts every face of every cube, including the hidden ones. Choice B subtracts only 1 face for each glued place, but each glued place hides a face on both cubes. Choice D is the volume, 5 cm³.
Question 8 of 20 · Multiple Choice
A right prism is 15 in long. Its bases are triangles with sides 10 in, 10 in and 12 in, and the height of each triangle from the 12 in side is 8 in. What is the surface area of the prism?
Answer: D
Each base is 1/2 × 12 × 8 = 48 in². The lateral faces are the perimeter times the length: (10 + 10 + 12) × 15 = 480 in². Surface area = 2 × 48 + 480 = 576 in². Choice A adds only one base. Choice B is the volume, 48 × 15 = 720 in³. Choice C leaves out both bases.
Question 9 of 20 · Multiple Choice
Jada is sewing a cover for a cube-shaped footstool, to go over every face. Which measurement tells her how much fabric she needs?
Answer: A
Fabric covers the outside, so she needs the total area of all the faces: the surface area, in square units. Choice B measures how much the stool holds inside. Choice C covers only one of the 6 faces. Choice D measures the distance around one face, a length.
Question 10 of 20 · Multiple Choice
A hexagonal paving stone is a right prism 6 cm thick. Its top is a regular hexagon (6 equal sides and 6 equal angles) made of 6 matching triangles, each with a base of 12 cm and a height of about 10.4 cm. About how much concrete is in the stone?
Answer: C
The base is 6 × (1/2 × 12 × 10.4) = 374.4 cm², so V = 374.4 × 6 = 2,246.4 cm³. Choice A is the area of the base, not multiplied by the thickness. Choice B forgets the 1/2 in each triangle. Choice D multiplies the base area by 2, as if counting the two bases for surface area, instead of by the 6 cm thickness.
Question 11 of 20 · Multiple Choice
Two concrete steps are poured as one piece. The bottom step is 4 ft wide, 2 ft deep and 0.5 ft tall. The top step sits on the back half of it and is 4 ft wide, 1 ft deep and 0.5 ft tall. How much concrete is used?
Answer: A
Bottom step: 4 × 2 × 0.5 = 4 ft³. Top step: 4 × 1 × 0.5 = 2 ft³. Total: 4 + 2 = 6 ft³. Choice B treats the steps as one full 4 × 2 × 1 block, which includes the empty space in front of the top step. Choice C is the bottom step only. Choice D is the top step only.
Question 12 of 20 · Multiple Choice
A right prism has a volume of 360 cm³ and a base with an area of 24 cm². How tall is the prism?
Answer: D
V = B × h, so 24 × h = 360 and h = 360 ÷ 24 = 15 cm. Choice A multiplies instead of dividing. Choice B subtracts 24 from 360. Choice C divides by half of 24, as if the prism rule had a 1/2 like the triangle rule.
Question 13 of 20 · Multiple Choice
Ben paints the four walls and the ceiling of a bedroom that is 12 ft long, 10 ft wide and 8 ft tall. He does not paint the door (3 ft by 7 ft) or the window (4 ft by 3 ft). How many square feet does he paint?
Answer: B
Walls: 2 × (12 × 8) + 2 × (10 × 8) = 192 + 160 = 352 ft². Ceiling: 12 × 10 = 120 ft². Door and window: 21 + 12 = 33 ft². Painted: 352 + 120 - 33 = 439 ft². Choice A does not subtract the door and window. Choice C is the four walls only, with nothing subtracted. Choice D also paints the floor and subtracts nothing.
Question 14 of 20 · Multiple Choice
A garden bed is a parallelogram with a base of 14 ft, slanted sides of 9 ft and a height of 8 ft. What is its area?
Answer: C
Area = base × height = 14 × 8 = 112 ft². Choice A uses the slanted side instead of the height: 14 × 9 = 126. Choice B takes half, as for a triangle. Choice D is the perimeter, 2 × (14 + 9) = 46 ft.
Question 15 of 20 · Short Answer
A store sign is shaped like an arrow: a rectangle 30 in long and 12 in tall, with a triangle on one end whose base is 24 in and whose height is 16 in. Find the area of the sign.
Rectangle: 30 × 12 = 360 in². Triangle: 1/2 × 24 × 16 = 192 in². Area = 360 + 192 = 552 in². The sign is a composite polygon with 7 sides.
Question 16 of 20 · Short Answer
A garden shed is a right prism 10 ft deep. Its front is a pentagon: a rectangle 12 ft wide and 8 ft tall, with a triangle on top that is 2.5 ft tall and has slanted sides of 6.5 ft. Find the volume of the shed and the area of its roof (the two slanted rectangles).
A 3 × 2 × 2 block is built from 12 one-inch cubes. Find its volume and surface area. Then one cube is taken from the middle of a long top edge. Find the new volume and surface area.
Block: V = 12 in³ and SA = 2 × (3 × 2 + 3 × 2 + 2 × 2) = 32 in². The removed cube showed 2 outside faces (top and side) and uncovers 4 faces of its neighbors (left, right, below and behind). New block: V = 11 in³ and SA = 32 - 2 + 4 = 34 in².
Question 18 of 20 · Short Answer
A juice box is a right prism with a 5 cm by 4 cm rectangular base. It holds 250 mL of juice when full. How tall is the box? (1 mL = 1 cm³.)
B = 5 × 4 = 20 cm². 20 × h = 250, so h = 12.5 cm.
Question 19 of 20 · Short Answer
Sam says, "If I double every edge of a cube, its volume doubles." Test his claim with a cube whose edges are 4 cm, and describe what happens to the surface area too.
4 cm cube: V = 64 cm³, SA = 6 × 16 = 96 cm². 8 cm cube: V = 512 cm³, SA = 6 × 64 = 384 cm². Sam is wrong: the volume is 8 times as large (512 ÷ 64 = 8), and the surface area is 4 times as large (384 ÷ 96 = 4).
Question 20 of 20 · Short Answer
Box A is a cube 10 cm on each edge. Box B is 20 cm long, 10 cm wide and 5 cm tall. Show that the boxes hold the same amount, and find which needs less cardboard and by how much.
7.G.B.6 means students can find area, volume and surface area in real problems. The shapes are polygons made of triangles and quadrilaterals, solids built from cubes, and right prisms with any polygon base. A typical task is to find the paint for a wall with a triangular top, or the water in a trough with trapezoid ends.
What grade is 7.G.B.6, and what comes after it?
It is a grade 7 standard in the Geometry domain. In grade 8, students use formulas for the volume of cylinders, cones and spheres (8.G.C.9). In high school geometry, they add pyramids and explain why the volume formulas work (HSG.GMD.A.3).
What is the difference between area, surface area and volume?
Area covers a flat shape, surface area covers the outside of a solid, and volume fills the inside of a solid. Area and surface area are in square units, such as ft². Volume is in cubic units, such as cm³, or in liters for liquids. A quick test: paint and wrapping paper are surface area, while soil, water and rice are volume.
Why is the volume of a prism the area of the base times the height?
A prism is a stack of identical layers, each one a copy of the base. A layer 1 unit thick holds B cubic units, where B is the area of the base, and h layers hold B × h. The same idea gives l × w × h for a box, because its base is an l by w rectangle.
How do you find the area of a composite figure?
Split it into rectangles, triangles and trapezoids, find each area, and add them. Another way is to draw a rectangle around the figure and subtract the pieces that are not part of it. Doing it both ways is a good check. Make sure no piece is counted twice.
Does 7.G.B.6 include circles or cylinders?
No. Circles have their own grade 7 standard, 7.G.B.4, which covers circumference and area. The volume of cylinders, cones and spheres comes in grade 8 (8.G.C.9). 7.G.B.6 uses only shapes with straight sides and flat faces.
What is a right prism?
A right prism is a solid with two matching polygon bases whose side faces are rectangles standing straight up from the base. A box, a cube, a triangular doorstop and a hexagon-shaped pencil are all close to right prisms. The bases are the two matching faces, which are not always the top and the bottom.
What mistakes should teachers watch for?
A common mistake is using a slanted side as the height of a triangle or parallelogram. Others are forgetting the 1/2 in a triangle's area, adding only one base to the surface area, counting hidden faces in cube figures, and writing a volume in square units. Asking students to sketch and label the base first prevents many of these.
How can parents help with area and volume at home?
Parents can use real objects. Measure a cereal box and ask how much cardboard it uses and how much it holds, or measure a room and ask how much paint the walls need. Ask your child to say whether each question is about area, surface area or volume, and which units the answer uses.
How is 7.G.B.6 different from 6.G.A.4?
6.G.A.4 finds surface area using nets, and 7.G.B.6 adds volume and more complex shapes. Grade 6 volume work (6.G.A.2) stays with boxes. In grade 7, students find the volume of prisms with triangle, trapezoid and pentagon bases, work with solids built from cubes, and work backward from a volume to a missing length.
07
Related Standards
6 standards
These standards connect to 7.G.B.6: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
6.G.A.1Prerequisite
Find areas of triangles, quadrilaterals and polygons by composing or decomposing
