الأسبوع 27 · الحجم نفسه بشكل مختلف؛ قانون الحجم · 3 دروس
2
Discovering the volume formula
Discover Mathematics P5: Term 2, Unit 11, Lessons 4–6
GoalDiscover and write the formula V = l × w × h (or V = B × h) for a rectangular prism.
New words
formula · a rule that uses letters to show how to calculatebase area · the area of the bottom face, B = l × wright rectangular prism · a box whose faces are all rectangles and whose corners are right angles
Explanation
Count the cubes in layers: the number of cubes in one layer equals the area of the base, which is length × width. The number of layers equals the height. So the volume is the base area times the height: V = B × h, and since B = l × w, V = l × w × h. Test it. A box 5 cm long, 3 cm wide and 2 cm high has a base area of 5 × 3 = 15, so V = 15 × 2 = 30 cm³. The formula works with any units, but all three measures must be in the same unit. It also does not matter in which order you multiply: 5 × 3 × 2 = 3 × 2 × 5 = 2 × 5 × 3. Write the answer with a cubic unit.
ƒFormula
V = l × w × h = B × h
B is the area of the base, B = l × w.
Worked examples
Count by layers: a box 5 × 3 × 2. Layer: 5 × 3 = 15 cubes. 2 layers: 15 × 2 = 30. Formula: V = 5 × 3 × 2 = 30 cm³. The count and the formula agree.
A box is 7 cm long, 4 cm wide and 5 cm high. B = 7 × 4 = 28 cm². V = 28 × 5 = 140 cm³. Turn the box so that 5 cm is the length: 5 × 4 × 7 = 140 cm³ again.
A cube has edges of 4 cm, so l = w = h = 4. V = 4 × 4 × 4 = 64 cm³.
Class activity
Test the formula. Build a box from 24 sugar cubes (for example 4 × 3 × 2). Count the cubes in one layer and the number of layers. Then multiply length × width × height and check that you get 24. Build another box from 18 cubes and check the formula again.
Practice
1
Find the volume of a box 8 cm long, 3 cm wide and 2 cm high.
8 × 3 × 2 = 48 cm³.
2
The base of a box has area 20 cm² and its height is 6 cm. What is its volume?
V = B × h = 20 × 6 = 120 cm³.
3
Find the volume of a cube with edges of 5 cm.
5 × 5 × 5 = 125 cm³.
4
Seif says V = l + w + h. Show with the box 4 × 3 × 2 that he is wrong.
4 + 3 + 2 = 9, but the box holds 4 × 3 × 2 = 24 cubes. We must multiply.
5
A box has volume 60 cm³, length 5 cm and width 4 cm. Find its height.
