Discover Mathematics P5: Term 2, Unit 11, Lessons 7–9
GoalFind the volume of a compound solid by splitting it into boxes and adding, or by taking a part away.
New words
compound solid · a solid made by joining two or more simple solidsdecompose · to split a solid into parts you know how to measurecross-section · the flat shape you see when you cut a solid straight through
Explanation
A compound solid is made of two or more boxes joined together. To find its volume, split it into boxes whose dimensions you know, find each volume with V = l × w × h, and add. Suppose a bottom block is 6 cm × 4 cm × 2 cm and a smaller block 3 cm × 4 cm × 3 cm sits on top. The bottom block has volume 6 × 4 × 2 = 48 cm³, the top block has 3 × 4 × 3 = 36 cm³, and the whole solid has 48 + 36 = 84 cm³. Another way is to take a bigger box and subtract the missing part: a block 10 cm × 6 cm × 4 cm with a corner 4 cm × 6 cm × 2 cm cut away has volume 240 − 48 = 192 cm³. A solid with an L-shaped side that has the same shape all the way through has volume = area of the L × depth. There is often more than one way to split a solid. Different ways must give the same volume, which is a good check.
Part
Length
Width
Height
Volume
Bottom block
6
4
2
48
Top block
3
4
3
36
Whole solid
84
Worked examples
Two blocks side by side: the first is 5 × 3 × 2 and the second is 4 × 3 × 2 (centimetres). Volumes: 5 × 3 × 2 = 30 and 4 × 3 × 2 = 24. Total: 30 + 24 = 54 cm³.
A concrete step has an L-shaped side (shown) and is 5 cm deep. First find the area of the L: 6 × 2 = 12 and 3 × 3 = 9, so 12 + 9 = 21 cm². Then V = 21 × 5 = 105 cm³.
A block is 10 cm × 6 cm × 4 cm. A corner 4 cm × 6 cm × 2 cm is cut away. Whole block: 10 × 6 × 4 = 240. Missing corner: 4 × 6 × 2 = 48. What is left: 240 − 48 = 192 cm³.
Class activity
Build and split. Use 20 or more sugar cubes or blocks. Build a compound solid from two boxes, for example 4 × 2 × 1 and 2 × 2 × 3 joined. Find the volume by adding the two parts, then count all the cubes to check. Draw your solid and write the two volumes and the total.
Practice
1
A step is made of a bottom block 8 × 5 × 2 and a top block 4 × 5 × 3. Find its volume.
Find the volume of this L-shaped step if it is 4 cm deep.
Area of the L: 7 × 2 = 14 and 3 × 3 = 9, so 23 cm². Volume: 23 × 4 = 92 cm³.
5
In the example with the 6 × 4 × 2 bottom block and 3 × 4 × 3 top block, split the solid a different way (a tall part and a low part) and check the answer.
The tall part is 3 × 4 × 5 = 60. The low part is 3 × 4 × 2 = 24. 60 + 24 = 84 cm³, the same.
