Multi-Step
Measurement Problems
Master combining area, surface area, and volume in multi-step problems: from flat shapes to 3D solids.
What Are Measurement Problems?
Measurement problems ask you to find quantities like how much space a shape covers (area), how much material wraps around a solid (surface area), or how much a container holds (volume). Multi-step problems combine two or more of these ideas in a single question.
Visualizing Area, Surface Area, and Volume
Why Multi-Step?
Real-world problems rarely ask for just one calculation. Painting a room requires surface area of the walls minus the area of windows. Filling an aquarium needs volume, but buying glass for it needs surface area. You must chain steps together and keep track of units.
Core Formulas You Need
| Shape | Area / SA | Volume |
|---|---|---|
| Rectangle | \( A = l \times w \) | — |
| Triangle | \( A = \tfrac{1}{2}bh \) | — |
| Circle | \( A = \pi r^2 \) | — |
| Rectangular Prism | \( SA = 2(lw + lh + wh) \) | \( V = lwh \) |
| Cylinder | \( SA = 2\pi r^2 + 2\pi rh \) | \( V = \pi r^2 h \) |
| Triangular Prism | \( \text{Area of 2 triangles} + \text{Area of 3 rectangles} \) | \( (\text{Area of base}) \times \text{length} \) |
The Rectangular Prism Up Close
Surface Area vs Volume
| Surface Area | Volume | |
|---|---|---|
| Measures | Outside covering | Inside capacity |
| Units | \( \text{units}^2 \) (square) | \( \text{units}^3 \) (cubic) |
| Real-world use | Wrapping paper, paint | Water in a pool, soil in a pot |
| Keyword clues | "cover", "wrap", "coat" | "fill", "hold", "capacity" |
Step-by-Step Strategy
Example 1: Painting a Box
A wooden box is 12 cm long, 5 cm wide, and 8 cm tall. You need to paint all faces except the bottom. How much area do you need to paint?
Example 2: Filling a Cylinder
A cylindrical tank has radius 3 ft and height 10 ft. It is three-quarters full of water. How many cubic feet of water are in the tank?
Example 3: Composite Area
A garden is shaped like a rectangle with a semicircle on one end. The rectangle is 20 m long and 10 m wide. Find the total area of the garden.
Example 4: SA and Volume Together
A gift box is 8 in \(\times\) 6 in \(\times\) 4 in. Find both the volume (to know how much it holds) and the surface area (to buy wrapping paper).
Handling Unit Conversions
Sometimes dimensions are given in different units. Always convert to the same unit before plugging into formulas.
Composite Solids
A composite solid is made of two or more basic shapes joined together. To find total volume, add the volumes of each part. For surface area, add all exposed faces: do not count faces where shapes touch.
Common Mistakes to Avoid
Summary Reference Table
| Question Type | Formula Needed | Units |
|---|---|---|
| Floor tiling | Area (\( l \times w \)) | \(\text{ft}^2\) |
| Gift wrapping | SA (\( 2(lw+lh+wh) \)) | \(\text{in}^2\) |
| Filling a pool | Volume (\( l \times w \times h \)) | \(\text{m}^3\) |
| Painting walls only | SA minus floor & ceiling | \(\text{ft}^2\) |
| Composite garden area | Add areas of each shape | \(\text{m}^2\) |
| Half-full tank | Volume \(\times\) fraction | \(\text{ft}^3\) |
Question 1 of 5
A rectangular prism is 6 cm long, 4 cm wide, and 3 cm tall. What is its surface area?
Question 2 of 5
A cylinder has radius 5 cm and height 12 cm. What is its volume? (Use \(\pi \approx 3.14\))
Question 3 of 5
A box is 10 in \(\times\) 8 in \(\times\) 6 in. You paint all faces except the bottom. How much area do you paint?
Question 4 of 5
A garden is a rectangle 14 m by 10 m with a semicircular flowerbed (diameter 10 m) added to one short side. What is the total area? (Use \(\pi \approx 3.14\))
Question 5 of 5
A fish tank is 2 ft long, 1 ft wide, and 1.5 ft tall. It is filled to \(\frac{2}{3}\) of its height. How many cubic feet of water are in the tank?