Applies area and perimeter to composite figures by decomposing them into rectangles, and introduces volume as the space inside a three-dimensional solid. Students calculate area of L-shapes and composite rectangles, solve real-world perimeter problems, and find volume of rectangular prisms by counting unit cubes and using the length x width x height formula.
Upper Elementary • 3-5
You know how to find the area and perimeter of simple rectangles. Now we tackle composite figures (shapes made of multiple rectangles), tricky perimeter problems, and the exciting new concept of volume for 3D shapes.
| Measurement | What It Measures | Formula | Units |
|---|---|---|---|
| Perimeter | Distance around the outside | P = 2L + 2W | cm, m, ft, in |
| Area | Space inside (covering) | A = L x W | cm², m², ft², in² |
| Volume | Space inside a 3D shape (filling) | V = L x W x H | cm³, m³, ft³, in³ |
A composite figure is a shape that can be split into two or more simpler shapes. To find its area, break it apart, find each area, and add them up.
An L-shaped room has these measurements: the top part is 8 m long and 3 m wide. The bottom part is 5 m long and 4 m wide.
There is often more than one way to split a composite figure. Choose the split that gives you the easiest rectangles to work with. Either way, you will get the same total area.
A rectangle has a perimeter of 36 cm and a length of 11 cm. What is the width?
Check: 2(11) + 2(7) = 22 + 14 = 36 cm. Correct!
Volume measures how much space a 3D shape takes up. Think of it as how many unit cubes fit inside.
A rectangular prism (box shape) has length, width, and height:
A fish tank is 60 cm long, 30 cm wide, and 40 cm tall. Find its volume.
Volume is measured in cubic units. One cubic centimeter (1 cm³) is a cube that is 1 cm on every edge. You can think of filling a box with these tiny cubes -- the number of cubes that fit is the volume.
A box that is 4 x 3 x 2:
Bottom layer: 4 x 3 = 12 cubes
Number of layers: 2
Total: 12 x 2 = 24 cubes
V = L x W x H
V = 4 x 3 x 2
V = 24 cubic units
Same answer!
Do not confuse area units with volume units! Area is measured in square units (cm²), while volume is measured in cubic units (cm³). Area covers a flat surface; volume fills a 3D space.
When you buy soil for a garden bed, you need to know the volume. A raised bed that is 6 ft long, 4 ft wide, and 1 ft deep needs 6 x 4 x 1 = 24 cubic feet of soil.
1. A composite shape is made of two rectangles: one is 10 in by 4 in, and the other is 6 in by 4 in. What is the total area?
10 x 4 = 40 in² and 6 x 4 = 24 in². Total: 40 + 24 = 64 in².
2. A square has a perimeter of 48 cm. What is its side length? What is its area?
A square has 4 equal sides: 48 / 4 = 12 cm per side. Area: 12 x 12 = 144 cm².
3. Find the volume of a box that is 5 m long, 3 m wide, and 2 m high.
V = 5 x 3 x 2 = 30 m³.
4. A rectangular prism is built from unit cubes. The bottom layer has 6 rows of 4 cubes. There are 3 layers. What is the volume?
Bottom layer: 6 x 4 = 24 cubes. Three layers: 24 x 3 = 72 cubic units. Using the formula: V = 6 x 4 x 3 = 72.
5. A swimming pool is 25 m long, 10 m wide, and 2 m deep. How many cubic meters of water does it hold?
V = 25 x 10 x 2 = 500 m³.