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Reveal Math · Unit 5 · Supplemental Resources

Geometry & Measurement

Standards: 6.G.A.1 (Area of polygons) · 6.G.A.2 (Volume) · 6.G.A.4 (Surface Area)

🖨️ Differentiated Practice Worksheets

Ready-to-print practice at three levels — pick the right fit for each student.

Level 0 · Extra Support Scaffolded steps, worked examples, extra space. Level 1 · Support Guided practice on the core skills. Level 2 · Enrichment Stretch problems and deeper reasoning.

Visual Vocabulary

h b
Parallelogram
Paralelogramo
A shape with 2 pairs of parallel sides. Area = base × height
h b
Triangle
Triángulo
A shape with 3 sides. Area = ½ × base × height
h b₁ b₂
Trapezoid
Trapecio
A shape with exactly 1 pair of parallel sides. Area = ½(b₁ + b₂) × h
h w l
Rectangular Prism
Prisma rectangular
A 3D box shape. Volume = length × width × height
Volume = how much space inside a 3D shape units³
Volume
Volumen
The space inside a 3D shape. Measured in cubic units.
face face
Surface Area
Área de superficie
The total area of all faces of a 3D shape. Measured in square units.

Sentence Frames

The area of this parallelogram is   because the base is   and the height is  .
To find the area of a triangle, I multiply   ×   and then divide by  .
A trapezoid has two bases. I add   +  , multiply by the height, then take   of that.
The volume of this rectangular prism is   cubic units because   ×   ×   =  .
Surface area is different from volume because surface area measures   and volume measures  .
This shape is a   because it has   sides and   pair(s) of parallel sides.

Step-by-Step: Find the Area of a Triangle

h = 8 cm b = 12 cm
1

Find the base (b). The base is the bottom side. Here, b = 12 cm.

2

Find the height (h). The height goes straight up from the base. Here, h = 8 cm.

3

Use the formula: Area = ½ × b × h

4

Plug in numbers: Area = ½ × 12 × 8 = ½ × 96 = 48 cm²

Step-by-Step: Find the Volume of a Rectangular Prism

5 in 10 in 3 in
1

Find length, width, height. l = 10 in, w = 5 in, h = 3 in.

2

Use the formula: V = l × w × h

3

Multiply: V = 10 × 5 × 3 = 150 in³

Practice Problems

Problem 1

A parallelogram has a base of 6 cm and a height of 4 cm. What is the area?

Show Hint Show Answer
Area = base × height. Multiply 6 × 4.
24 cm² — Area = 6 × 4 = 24
Problem 2

A triangle has a base of 10 in and a height of 6 in. What is the area?

Show Hint Show Answer
Area = ½ × base × height. First find 10 × 6, then divide by 2.
30 in² — ½ × 10 × 6 = ½ × 60 = 30
Problem 3

A trapezoid has bases of 8 m and 4 m, and a height of 5 m. What is the area?

Show Hint Show Answer
Area = ½(b₁ + b₂) × h. Add the two bases first: 8 + 4 = 12.
30 m² — ½(8 + 4) × 5 = ½ × 12 × 5 = 30
Problem 4

A box is 4 cm long, 3 cm wide, and 2 cm tall. What is its volume?

Show Hint Show Answer
V = length × width × height. Multiply 4 × 3 × 2.
24 cm³ — 4 × 3 × 2 = 24
Problem 5

A rectangular prism is 10 ft long, 5 ft wide, and 2 ft tall. What is its volume?

Show Hint Show Answer
V = l × w × h. Multiply all three numbers.
100 ft³ — 10 × 5 × 2 = 100
Problem 6

A cube has sides of 3 inches. What is its surface area?

Show Hint Show Answer
A cube has 6 equal faces. Each face is 3 × 3 = 9. Multiply 9 × 6.
54 in² — Each face = 3 × 3 = 9, and 9 × 6 = 54
Problem 7

A triangle has a base of 8 cm and a height of 3 cm. What is the area?

Show Hint Show Answer
Area = ½ × 8 × 3.
12 cm² — ½ × 8 × 3 = 12
Problem 8

A parallelogram has a base of 9 m and a height of 5 m. What is the area?

Show Hint Show Answer
Area = base × height.
45 m² — 9 × 5 = 45
Problem 9

A rectangular prism is 6 in long, 4 in wide, and 3 in tall. What is the surface area?

Show Hint Show Answer
Find area of each pair of faces: top/bottom = 6×4, front/back = 6×3, left/right = 4×3. Multiply each by 2 and add.
108 in² — 2(6×4) + 2(6×3) + 2(4×3) = 48 + 36 + 24 = 108
Problem 10

A trapezoid has bases of 6 ft and 10 ft and a height of 4 ft. What is the area?

Show Hint Show Answer
Area = ½(b₁ + b₂) × h = ½(6 + 10) × 4.
32 ft² — ½(16) × 4 = 8 × 4 = 32

Real-World Connections

Wrapping a Gift Box

When you wrap a present, you need to cover all the faces. That is surface area! A box that is 12 in × 8 in × 6 in needs 2(12×8) + 2(12×6) + 2(8×6) = 432 in² of wrapping paper.

Painting a Wall

A wall is shaped like a rectangle. If it is 10 ft wide and 8 ft tall, you need paint for 80 ft². If the wall has a triangle-shaped window (base 3 ft, height 2 ft), subtract ½ × 3 × 2 = 3 ft². You need 77 ft² of paint.

Filling a Fish Tank

A fish tank is a rectangular prism. If it is 20 in long, 10 in wide, and 12 in tall, the volume = 20 × 10 × 12 = 2,400 in³ of water.

Garden Bed

A garden bed shaped like a trapezoid with bases 6 ft and 10 ft and height 4 ft has an area of ½(6+10) × 4 = 32 ft² for planting.

Challenge Problems

Challenge 1

A composite figure is made of a rectangle (12 cm × 8 cm) with a triangle on top (base 12 cm, height 5 cm). Find the total area.

Show Hint Show Answer
Find each area separately and add them together.
126 cm² — Rectangle: 12 × 8 = 96. Triangle: ½ × 12 × 5 = 30. Total: 96 + 30 = 126.
Challenge 2

A rectangular prism has a volume of 360 cm³. Its length is 10 cm and width is 6 cm. What is its height?

Show Hint Show Answer
V = l × w × h. Divide both sides by l × w to find h.
6 cm — 360 ÷ (10 × 6) = 360 ÷ 60 = 6
Challenge 3

A triangular prism has a triangular face with base 6 in and height 4 in, and the prism is 10 in long. Find the volume.

Show Hint Show Answer
Volume of a triangular prism = area of triangle base × length.
120 in³ — Triangle area = ½ × 6 × 4 = 12. Volume = 12 × 10 = 120.
Challenge 4

A room is 15 ft long, 12 ft wide, and 9 ft tall. How much paint is needed to cover all four walls (ignore windows and doors)?

Show Hint Show Answer
Find the area of each wall: two walls are 15 × 9 and two walls are 12 × 9.
486 ft² — 2(15 × 9) + 2(12 × 9) = 270 + 216 = 486
Challenge 5

A box has a surface area of 214 cm². Its length is 7 cm and width is 5 cm. Find the height.

Show Hint Show Answer
SA = 2(lw + lh + wh). Plug in known values: 214 = 2(35 + 7h + 5h). Solve for h.
6 cm — 214 = 2(35 + 7h + 5h) = 70 + 24h. 144 = 24h. h = 6 cm.
Challenge 6

Two rectangular prisms are glued together. Prism A is 4 × 3 × 2 and Prism B is 4 × 3 × 5. They share a 4 × 3 face. What is the total surface area of the combined shape?

Show Hint Show Answer
Find each SA separately, then subtract 2 × the shared face area (one from each prism).
122 cm² — SA(A) = 2(4×3 + 4×2 + 3×2) = 2(12+8+6) = 52. SA(B) = 2(4×3 + 4×5 + 3×5) = 2(12+20+15) = 94. Subtract 2 shared faces: 52 + 94 − 2(12) = 122.
Challenge 7

A trapezoid has an area of 84 m². One base is 10 m and the height is 7 m. What is the other base?

Show Hint Show Answer
84 = ½(10 + b₂) × 7. Solve for b₂.
14 m — 84 = ½(10 + b₂)(7) → 84 = 3.5(10 + b₂) → 24 = 10 + b₂ → b₂ = 14
Challenge 8

A rectangular prism is made of unit cubes. It uses 120 cubes total. If the base is 6 cubes × 5 cubes, how many layers tall is it?

Show Hint Show Answer
Each layer has 6 × 5 = 30 cubes. How many layers to reach 120?
4 layers — 120 ÷ 30 = 4
Challenge 9

A square pyramid has a base edge of 8 cm and 4 triangular faces, each with a slant height of 10 cm. What is the total surface area?

Show Hint Show Answer
SA = base area + 4 × triangle face area. Base = 8². Each triangle = ½ × 8 × 10.
224 cm² — Base = 64. Each face = ½ × 8 × 10 = 40. Total = 64 + 4(40) = 64 + 160 = 224.
Challenge 10

An L-shaped room can be split into two rectangles: 10 ft × 6 ft and 8 ft × 4 ft. What is the total floor area?

Show Hint Show Answer
Find each area, then add.
92 ft² — 10 × 6 = 60, 8 × 4 = 32, Total = 60 + 32 = 92

Real-World Investigations

Investigation 1: Design Your Dream Room

Draw a floor plan of your dream bedroom on grid paper. Include at least one non-rectangular feature (triangular reading nook, trapezoid closet). Calculate the total floor area of your room. Then calculate how much carpet you would need, and how much paint for all walls (assume 8 ft ceilings). Present your calculations showing all formulas used.

Investigation 2: Package Design Challenge

A cereal company wants to redesign their box. The current box is 12 in × 8 in × 2.5 in. Design a new box with the same volume but a different shape. Calculate the surface area of both boxes. Which design uses less cardboard? Which would be easier to stack on a shelf? Write a proposal to the company explaining your recommendation.

Investigation 3: School Mural

Your school wants a mural on a wall that is 20 ft wide and 10 ft tall. The mural design includes 3 triangles, 2 parallelograms, and 1 trapezoid. Sketch a design, label all dimensions, and calculate the area of each shape. How much total area does the mural cover? If paint costs $0.50 per ft², what is the total cost?

Brain Teasers

Teaser 1: The Doubling Puzzle

If you double all three dimensions of a rectangular prism, how many times larger is the new volume? How many times larger is the new surface area? Explain why they scale differently.

Show Answer
Volume increases by a factor of 8 (2³), while surface area increases by a factor of 4 (2²). Volume scales with the cube of the factor because it measures 3D space; surface area scales with the square because it measures 2D faces.

Teaser 2: Same Area, Different Shapes

Find three different shapes (a rectangle, a triangle, and a parallelogram) that all have an area of exactly 36 cm². Give the dimensions for each.

Show Answer
Many answers work. Example: Rectangle 9 × 4, Triangle with base 12 and height 6 (½ × 12 × 6 = 36), Parallelogram with base 6 and height 6.

Teaser 3: The Impossible Net

A student drew a net with 6 squares arranged in a cross shape (+). Can this net fold into a cube? What if you move one square? How many different nets can make a cube? (There are exactly 11!)

Show Answer
Yes, the cross-shaped net (a plus sign) is one valid net for a cube. There are exactly 11 unique nets that fold into a cube. Try finding all 11!

Teaser 4: Maximum Volume

You have a 24 cm × 18 cm sheet of cardboard. You cut equal squares from each corner and fold up the sides to make an open box. If you cut 3 cm squares, what is the volume? Can you find a cut size that gives a larger volume?

Show Answer
With 3 cm cuts: l = 18, w = 12, h = 3, V = 648 cm³. With 4 cm cuts: l = 16, w = 10, h = 4, V = 640 cm³. With 2 cm cuts: l = 20, w = 14, h = 2, V = 560 cm³. The 3 cm cut gives the best integer solution at 648 cm³.

Where This Math Leads Next

7th Grade: Cross Sections

In 7th grade, you will slice 3D shapes and study the 2D cross-sections. Imagine cutting a rectangular prism at an angle — what shape do you get?

7th Grade: Circles & Cylinders

You will learn the area of circles (A = πr²) and use it to find the volume of cylinders (V = πr²h) and surface area of curved shapes.

8th Grade: Cones, Spheres, & Pythagorean Theorem

Area and volume extend to cones (V = ⅓πr²h) and spheres (V = &frac43;πr³). The Pythagorean theorem connects to finding slant heights and diagonals of prisms.

High School: Trigonometry & Calculus

The area formulas you learn now are the foundation for trigonometric area calculations and integral calculus, where you find areas of any curved shape.

Self-Assessment Checklist