Study guide · Illustrative Mathematics · Grade 6 · Unit 1
Area & Surface Area
Every part of the unit in one place: what to know, a worked example, the mistake to watch for, and practice. A 15-question practice test and a worked answer key are at the end.
To print: use your browser’s Print (Ctrl+P, or ⌘P on a Mac). It prints in dark ink on white letter paper, and the practice test and the answer key each start on a new page, so you can print them separately.
- Night 1Formula sheet, then Parts A–C: area, parallelograms, triangles.
- Night 2Parts D–F: polygons, nets and surface area, squares and cubes.
- Night 3Take the practice test without notes, check it, and redo anything missed.
Formula sheet
Base: the side you measure from (any side works).
Height: straight-up distance to the base, at a right angle.
Polygon: closed flat shape with straight sides.
Polyhedron: 3D shape with flat polygon faces.
Face · edge · vertex: flat side · where faces meet · corner.
Net: flat pattern that folds into a 3D shape.
Surface area: total area of all faces, in square units.
Volume: space inside, in cubic units.
Part A · Lessons 1–3
Reasoning to find area
Know it
- Area is how many unit squares cover a flat shape, with no gaps and no overlaps.
- Cut a shape into pieces and move them around: the area stays the same.
- Two ways to find an odd shape’s area: break it into pieces and add, or surround it with a rectangle and subtract the extra.
- Area is in square units: cm², in², ft², m².
Example
Break it apart: 6 × 2 = 12 and 2 × 3 = 6, so 12 + 6 = 18.
Or surround it with a 6 × 5 rectangle and subtract the empty corner: 30 − 4 × 3 = 30 − 12 = 18 square units.
Watch out: count squares, not the lines around the outside. And when you break a shape apart, don’t let the pieces overlap, or you count some squares twice.
Try it
Each square is 1 square unit. What is the area of the T shape?
A garden is 8 m by 6 m. A pond 3 m by 2 m sits in the middle. How much of the garden is not pond?
Noah cuts a paper shape with an area of 10 in² into 3 pieces and tapes them together into a new shape, with no gaps or overlaps. What is the area of the new shape? Why?
Answer:Part B · Lessons 4–6
Parallelograms
Know it
- A parallelogram has two pairs of parallel sides.
- Cut the triangle off one end and slide it to the other: you get a rectangle. So Area = base × height.
- The base can be any side. The height is the straight-up distance from the base to the opposite side. It meets the base at a right angle (the little square).
- The slanted side is not the height.
- The height can be drawn outside the shape. Extend the base with a dashed line.
Example
A = b × h = 6 × 4 = 24 cm².
The 5 cm side is slanted, so it isn’t used.
Watch out: 6 × 5 = 30 is wrong. It uses the slanted side instead of the height.
Try it
Find the area.
Find the area. (The height is drawn outside.)
A parallelogram has an area of 54 in² and a height of 6 in. How long is its base?
Answer:A parallelogram has a base of 7.5 m and a height of 4 m. What is its area?
Answer:Part C · Lessons 7–10
Triangles
Know it
- Two copies of any triangle fit together into a parallelogram. So a triangle is half of base × height: Area = ½ × b × h (or b × h ÷ 2).
- The height goes from the corner opposite the base straight down to the base, at a right angle.
- In a right triangle, the two sides that make the square corner are the base and the height.
- In a wide (obtuse) triangle, the height may fall outside. Extend the base to meet it.
- Triangles with the same base and height have the same area, whatever their shape.
Example
A = ½ × 10 × 6 = ½ × 60 = 30 m².
Watch out: don’t forget the ½. 10 × 6 = 60 is the area of the whole parallelogram, twice the triangle.
Try it
Find the area of the right triangle.
Find the area. (The height is outside.)
A triangle has a base of 7 cm and a height of 3 cm. What is its area?
Answer:A triangle has an area of 24 ft² and a base of 8 ft. What is its height?
Answer:Part D · Lesson 11
Polygons
Know it
- A polygon is a closed, flat shape with straight sides that meet only at their ends (vertices).
- To find its area, break it into rectangles and triangles and add their areas.
- Or surround it with a rectangle and subtract the corners you don’t need.
- For a shape with a hole, find the big area and subtract the hole.
Example
Rectangle: 6 × 4 = 24. Each triangle: ½ × 2 × 4 = 4.
Total: 24 + 4 + 4 = 32 square units.
Watch out: 10 × 4 = 40 counts space outside the trapezoid, and 6 × 4 = 24 leaves out the two triangles.
Try it
Find the area of the house shape.
Each square is 1 square unit. Find the area of the diamond. (Hint: subtract the four corner triangles from the rectangle.)
Find the area of the trapezoid.
A lawn has a triangular flower bed in it. How many square meters of grass are there?
Part E · Lessons 12–15
Polyhedra, nets, and surface area
Know it
- A polyhedron is a closed 3D shape made of flat polygon faces. Faces meet at edges, and edges meet at vertices (corners). Cylinders, cones, and spheres are not polyhedra because they have curved surfaces.
- A prism has two matching, parallel bases joined by rectangles. A pyramid has one base and triangles that meet at a point. Both are named for their base: triangular prism, square pyramid, …
- A net is a flat pattern that folds up into the solid: one piece for each face.
- Surface area = the total area of all the faces = the area of the net, in square units.
| Shape | Faces | Edges | Vertices |
|---|---|---|---|
| Cube / rectangular prism | 6 | 12 | 8 |
| Triangular pyramid | 4 | 6 | 4 |
| Square pyramid | 5 | 8 | 5 |
| Triangular prism | 5 | 9 | 6 |
Example
Surface area of a box 5 cm by 3 cm by 2 cm. Unfold it into a net:
Two faces of each kind: 2 × 15 + 2 × 10 + 2 × 6
= 30 + 20 + 12 = 62 cm².
Watch out: a box has 6 faces, so add all of them, not just the 3 you can see. And 5 × 3 × 2 = 30 is the box’s volume (cubic units), not its surface area.
Try it
For each shape, how many faces, edges, and vertices? Name each shape.
Which of these are polyhedra? Circle them: cube · cylinder · cone · triangular pyramid · sphere · pentagonal prism
Answer:This net folds into a square pyramid. Each triangle has a base of 4 and a height of 5. What is the surface area?
Which nets fold into a cube? Circle every one that works.
Answer:This net folds into a triangular prism (the triangles have legs 3 and 4, and the long side is 5). Find the surface area.
Part F · Lessons 16–18
Surface area vs. volume, squares and cubes
Know it
- Surface area is how much it takes to cover the outside: square units (cm²). Volume is how much space is inside: cubic units (cm³).
- An exponent says how many times to multiply a number by itself: 5² = 5 × 5 = 25 (“5 squared”), 5³ = 5 × 5 × 5 = 125 (“5 cubed”).
- A square with side s: Area = s².
- A cube with edge s: Volume = s³ and Surface area = 6 × s² (six square faces).
Example
Each face: 4² = 16 cm². Six faces: 6 × 16 = 96 cm².
Volume: 4³ = 4 × 4 × 4 = 64 cm³.
Watch out: 4² means 4 × 4 = 16, not 4 × 2 = 8. And 4³ is 4 × 4 × 4 = 64, not 4 × 3 = 12.
Try it
Find the value: 3² 2³ 10² 1³ 6²
Answer:A cube has edges of 5 in. Find its surface area and its volume. Include the units.
Answer:Square units or cubic units? (a) wrapping paper to cover a gift box (b) sand to fill a sandbox (c) carpet for a floor (d) water in a fish tank
A cube has a surface area of 54 cm². How long is each edge?
Answer:Write 7 × 7 × 7 with an exponent. Then find the area of a square with sides of 11 m.
Answer:Practice test · about 40 minutes
Practice test
Show your work. Write units with every answer: cm² for area, cm³ for volume.
Find the area of the parallelogram.
A triangle has a base of 9 in and a height of 4 in. What is its area?
Answer:Find the area of the triangle.
Which segment is a height for the base shown: a, b, or c? Why aren’t the others?
Find the area of the trapezoid.
Each square is 1 square unit. Find the area of the triangle. (Surround it with a rectangle and subtract.)
A parallelogram has an area of 42 ft² and a height of 6 ft. What is its base?
Answer:A triangle has an area of 18 cm² and a base of 4 cm. What is its height?
Answer:A pentagonal prism has a pentagon for each base. How many faces, edges, and vertices does it have?
Answer:What polyhedron does this net fold into?
A box is 6 cm long, 4 cm wide, and 3 cm tall. What is its surface area?
Answer:A cube has edges of 7 m. Find its surface area and its volume.
Answer:Find 4³ and 9². Which is greater, 3² or 2³?
Answer:Lin says a cube with edges of 3 cm has a surface area of 27 cm². What did Lin find instead? What is the real surface area?
Two triangles both have a base of 6 cm and a height of 5 cm, but one is tall and skinny and the other is wide. Compare their areas.
Answer:For checking
Answer key
Worked answers for every practice problem and the practice test.
Part A
- A1Top bar 7 × 2 = 14, stem 3 × 3 = 9. 23 square units.
- A28 × 6 = 48, pond 3 × 2 = 6. 48 − 6 = 42 m².
- A310 in². Moving pieces doesn’t change how much space they cover, as long as there are no gaps or overlaps.
Part B
- B19 × 5 = 45 cm² (the 6 cm slanted side isn’t used).
- B24 × 7 = 28 in².
- B354 ÷ 6 = 9 in, because 9 × 6 = 54.
- B47.5 × 4 = 30 m².
Part C
- C1The legs are the base and height: ½ × 8 × 5 = 20 ft². (9.4 ft isn’t used.)
- C2½ × 6 × 4 = 12 cm².
- C3½ × 7 × 3 = ½ × 21 = 10.5 cm².
- C4½ × 8 × h = 24, so 4 × h = 24 and h = 6 ft. (3 ft forgets the ½.)
Part D
- D1Rectangle 6 × 4 = 24, roof ½ × 6 × 3 = 9. 33 cm².
- D2Rectangle 6 × 4 = 24. Each corner triangle is ½ × 3 × 2 = 3, and there are 4: 24 − 12 = 12 square units.
- D3Rectangle 5 × 4 = 20, triangle ½ × 4 × 4 = 8. 28 in².
- D4Lawn 10 × 8 = 80, bed ½ × 4 × 3 = 6. 80 − 6 = 74 m².
Part E
- E1Triangular prism: 5 faces, 9 edges, 6 vertices. Square pyramid: 5 faces, 8 edges, 5 vertices.
- E2Cube, triangular pyramid, pentagonal prism. The cylinder, cone, and sphere have curved surfaces.
- E3Square 4 × 4 = 16. Each triangle ½ × 4 × 5 = 10, four of them: 40. 16 + 40 = 56 square units.
- E4A and C fold into a cube. B (a 2 × 3 block) and D (5 in a row) do not.
- E5Triangles: 2 × (½ × 3 × 4) = 12. Rectangles: 30 + 40 + 50 = 120. 132 square units.
Part F
- F19, 8, 100, 1, 36.
- F2Surface area 6 × 5² = 6 × 25 = 150 in². Volume 5³ = 125 in³.
- F3(a) square (b) cubic (c) square (d) cubic.
- F454 ÷ 6 = 9 cm² per face, and 3² = 9, so 3 cm.
- F57³. Area 11² = 11 × 11 = 121 m².
Practice test
- 112 × 5 = 60 cm². (Not 12 × 7: 7 cm is the slanted side.)
- 2½ × 9 × 4 = 18 in².
- 3½ × 5 × 8 = 20 in².
- 4b. It meets the base at a right angle. a is a slanted side, and c is slanted too.
- 5Rectangle 8 × 5 = 40, two triangles ½ × 3 × 5 = 7.5 each. 40 + 15 = 55 ft².
- 6Rectangle 6 × 5 = 30. Corner triangles: ½ × 6 × 2 = 6, ½ × 4 × 3 = 6, ½ × 2 × 5 = 5. 30 − 17 = 13 square units.
- 742 ÷ 6 = 7 ft.
- 8½ × 4 × h = 18, so 2 × h = 18 and h = 9 cm.
- 97 faces (2 pentagons + 5 rectangles), 15 edges, 10 vertices.
- 10A triangular pyramid: 4 triangle faces.
- 112 × (6 × 4) + 2 × (6 × 3) + 2 × (4 × 3) = 48 + 36 + 24 = 108 cm².
- 12Surface area 6 × 7² = 6 × 49 = 294 m². Volume 7³ = 343 m³.
- 134³ = 64, 9² = 81. 3² = 9 is greater than 2³ = 8.
- 14Lin found the volume, 3³ = 27 cm³. The surface area is 6 × 3² = 54 cm².
- 15The same: ½ × 6 × 5 = 15 cm² each. Same base and height means same area.