1. VOLUME OF SOLIDS

2. THINK OF THIS…
Imagine transferring a 12-ounce soft drink from the bottle to an ice plastic container.
Will the volume change?
If the ice plastic is twisted a little, will the volume change?

3. DEFINITION: -is the amount of space enclosed in a solid figure.
VOLUME OF SOLIDS
DEFINITION:
-is the amount of space enclosed in a solid figure.
The volume of a solid is the number of cubic units contained in the solid.

4. VOLUME OF SOLIDS
In finding volume of solids, you have to consider the area of a base and height of the solid.
If the base is triangular, you have to make use of the area of a triangle, if rectangular, make use of the area of a rectangle and so on.

5. VOLUME OF A CUBE
The volume V of a cube with edge s is the cube of s. That is,
V = s3
s=h s

6. Find the volume of a cube whose sides 8 cm.
Example 1.
Find the volume of a cube whose sides 8 cm.
Solution:
V = s³
= (8cm) ³
V = 512 cm³
8 cm

7. VOLUME OF A RECTANGULAR PRISM

8. V = lwh Rectangular prism
The volume V of a rectangular prism is the product of its altitude h, the length l and the width w of the base. That is,
V = lwh h w l

9. Example 2. Find the volume of a rectangular prism. V = lwh V = 160 cm³
Solution:
V = lwh
= (8cm)(4 cm) (5 cm)
V = 160 cm³
5cm
4 cm
8 cm

10. VOLUME OF A SQUARE PRISM

11. Square prism
It is a prism whose bases are squares and the other faces are rectangles.
Square
base
Height (H)

12. Square prism
The volume V of a square prism is the product of its altitude H and the area of the base, s². That is,
V = s²H h s s

13. Example 3. Find the volume of a square prism. Solution: V = 80 cm³
V = s²H
= (4 cm) ² (5 cm)
= (16 cm²) 5 cm
V = 80 cm³
4 cm
4 cm

14. VOLUME OF A TRIANGULAR PRISM

15. V = (½bh) H. Triangular prism
The volume V of a triangular prism is the product of its altitude H and the area of the base(B), ½bh. That is,
V = (½bh) H. h b
Base H

16. Example 4. Find the volume of a Triangular prism
Solution:
V = (½bh) H
= ½(3.9 cm)(4.5 cm)(2.8 cm)
= ½ (49.14 cm³)
V = cm³
3.9 cm
4.5cm
2.8 cm

17. ANOTHER KIND OF POLYHEDRON
PYRAMIDS

18. Slant Height (height of the Triangular face) Altitude Or Height
pyramid

19. Pyramids are classified according to their base.
SquarePyramid- the base is square.
Rectangular Pyramid- the base is rectangle.
Triangular Pyramid- the base is triangle.

20. VOLUME of PYRAMIDS
w = 4 cm
l = 9 cm
h = 6 cm
base
Consider a pyramid and a prism having equal altitudes and bases with equal areas.
If the pyramid is filled with water or sand and its contents poured into a prism, only one- third of the prism will be filled. Thus the volume of a pyramid is ⅓ the volume of the prism.

21. Volume of pyramids V = ⅓Bh. SQUARE PYRAMID V = ⅓(s²)H
The volume V of a pyramid is one third the product of its altitude h and the area B of its base. That is,
V = ⅓Bh.
SQUARE PYRAMID
V = ⅓(s²)H
RECTANGULAR PYRAMID
V = ⅓(lw)H
TRIANGULAR PYRAMID
V = ⅓(½bh)H

22. EXAMPLE 5: FIND THE VOLUME OF A RECTANGULAR PYRAMID
Height
( 10 cm )
Solution:
V = ⅓(lwH)
= ⅓(6 cm)(4 cm)(10 cm)
= ⅓ (240 cm³)
V = 80 cm³
l= 6 cm
W= 4 cm

23. EXAMPLE 6: FIND THE VOLUME OF A SQUARE PYRAMID
Height
( 8 cm )
Solution:
V = ⅓(s²H)
= ⅓(6 cm)²(8 cm)
= ⅓ (36 cm²)(8 cm)
= ⅓ (288 cm³)
V = 96 cm³
6 cm
6 cm

24. EXAMPLE 7: Find the volume of a regular triangular pyramid.
Solution:
V =⅓(½bh)H
8 cm
h=3cm
6 cm

25. CYLINDERS

26. CYLINDER
-is a space figure with two circular bases that are parallel and congruent. .
Circular base .
Height
Radius
Circular base

27. CYLINDERS
Guide questions: What is the geometric figure represented by the bases of the cylinder? How do you compute its area?

28. Volume of a cylinder ANSWERS Circles (circular bases) A = r² Height
radius

29. V= πr²h Volume of a cylinder
How can the volume of a cylinder be computed?
V = Bh, where B is the area of the base and h is the height of the cylinder.
by substitution,
V= πr²h

30. EXAMPLE 8: Find the volume of a cylinder. Use π = 3.14
Solution:
V = πr²h
=(3.14)(5 cm)² 10 cm
= 3.14( 25cm²) (10 cm)
= 3.14( 250 cm³)
V = 785 cm³
10 cm
5 cm

31. REFLECTIVE TRAFFIC CONE
VOLUME OF A CONE
REFLECTIVE TRAFFIC CONE

32. . CONE -is a space figure with one circular base and a vertex Vertex
Height
Of the cone
Slant Height
Of the cone .
Radius
Circular base

33. VOLUME of a CONE
Consider a CONE and a CYLINDER having equal altitudes and bases with equal areas.
If the CONE is filled with water or sand and its contents poured into a CYLIDER, only one- third of the CYLINDER will be filled. Thus the volume of a CONE is ⅓ the volume of the CYLINDER. h r

34. V= ⅓ πr²h Volume of a cone How can the volume of a cone be computed?
V = ⅓Bh, where B is the area of the base and h is the height of the cone.
by substitution,
V= ⅓ πr²h

35. Find the volume of a cone. Use π = 3.14
Solution:
V = ⅓ πr²h
= ⅓(3.14)(5 cm) ²(10 cm)
= ⅓ (3.14)(25 cm²) (10 cm)
= ⅓ (785 cm³)
V= cm³
10 cm
5 cm

36. Find the volume of a cone. Use π = 3.14
Solution:
V = ⅓ πr²h
= ⅓(3.14)(3 cm) ²(4 cm)
= ⅓ (3.14)(9 cm²) (4 cm)
= ⅓ ( cm³)
V= cm³
Solution:
Step 1. find h.
Using Pythagorean theorem,
h²= 5² - 3²
=25-9
h² = 16
h = 4 cm
5 cm
4 cm
3 cm

37. VOLUME OF A SPHERE
THE EARTH
BALLS

38. This distance is the length of the radius of a sphere.
A sphere is a solid where every point is equally distant from its center.
This distance is the length of the radius of a sphere.
radius
radius

39. How can the volume of a sphere be computed?
BALL
The formula to find the VOLUMEof a sphere is V = πr³, where r is the length of its radius.
How can the volume of a sphere be computed?

40. Archimedes of Syracuse (287-212 BC)
is regarded as the greatest of Greek mathematicians, and was also an inventor of many mechanical devices (including the screw, the pulley, and the lever).
He perfected integration using Eudoxus' method of exhaustion, and found the areas and volumes of many objects.

41. Archimedes of Syracuse (287-212 BC)
A famous result of his is that the volume of a sphere is two-thirds the volume of its circumscribed cylinder, a picture of which was inscribed on his tomb.

42. radius r r
H = d
The height (H)of the cylinder is equal to the diameter (d) of the sphere.

43. Volume (Sphere)= ⅔ the volume of a circumscribed cylinder
radius r r
H = d
Volume (Sphere)= ⅔ the volume of a circumscribed cylinder

44. Volume (Sphere)= ⅔ r²h = ⅔ r² (2r) = r³
radius r r
H = d= 2r
Volume (Sphere)= ⅔ r²h
= ⅔ r² (2r)
= r³

45. 1. Find the volume of a sphere. Use π = 3.14
Solution:
V = 4/3 πr³
=4/3(3.14)(10 cm)³
= (1000 cm³) 3
= 12,560 cm³)
V= 4, cm³
10 cm

46. 2. Find the volume of a sphere. Use π = 3.14
Solution:
V = 4/3 πr³
=4/3(3.14)(7.8 cm)³
= ( cm³) 3
= cm³
V= 1, cm³
7.8 cm