volume of solids
TRANSCRIPT
VOLUME OF SOLIDSVOLUME OF SOLIDS
VOLUME OF SOLIDSVOLUME 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.
The volume V of a cube with edge s is the cube of s. That is,
V = s3
s=h
s
Find the volume of a cube whose sides 8 cm.
8 cm
Solution:V = s³ = (8cm) ³V = 512 cm³
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 = lwhl
w
h
Find the volume of a rectangular prism.
8 cm
Solution:V = lwh = (8cm)(4 cm) (5 cm)V = 160 cm³
4 cm5cm
It is a prism whose bases are squares and the other faces are rectangles.
Squarebase
Height (H)
ss
h
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
Find the volume of a square prism.
4 cm
Solution:V = s²H = (4 cm) ² (5 cm) = (16 cm²) 5 cmV = 80 cm³
4 cm
5cm
H
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 bBase
Solution: V = (½bh) H = ½(3.9 cm)(4.5 cm)(2.8 cm)
= ½ (49.14 cm³)V = 24.57 cm³
3.9 cm
4.5cm
2.8 cm
ANOTHER KIND OF ANOTHER KIND OF POLYHEDRONPOLYHEDRON
PYRAMIDSPYRAMIDS
Altitude Or Height-Height of thepyramid
Slant Height(height of theTriangular face)
TYPES OF PYRAMIDSTYPES OF PYRAMIDSPyramids are classified Pyramids are classified according to their base.according to their base.
1. SquarePyramid- 1. SquarePyramid- the base is the base is square.square. 2. Rectangular Pyramid- 2. Rectangular Pyramid- the base is the base is rectangle.rectangle.3. Triangular Pyramid- 3. Triangular Pyramid- the base is the base is triangle.triangle.
VOLUME of PYRAMIDS• 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.
w = 4 cm
l = 9 cm
h = 6 cm
base
Volume of pyramids Volume of pyramids 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 PYRAMIDSQUARE PYRAMID
V = V = ⅓(s²)H⅓(s²)HRECTANGULAR PYRAMIDRECTANGULAR PYRAMID
V = V = ⅓(lw)H⅓(lw)H TRIANGULAR PYRAMIDTRIANGULAR PYRAMID
V = V = ⅓(½bh)H⅓(½bh)H
EXAMPLE 5: FIND THE VOLUME OF A RECTANGULAR PYRAMID
Solution: V = ⅓(lwH) = ⅓(6 cm)(4 cm)(10 cm) = ⅓ (240 cm³)V = 80 cm³
W= 4 cm
Height ( 10 cm )
l= 6 cm
EXAMPLE 6: FIND THE VOLUME OF A SQUARE PYRAMID
Solution: V = ⅓(s²H) = ⅓(6 cm)²(8 cm) = ⅓ (36 cm²)(8 cm) = ⅓ (288 cm³)V = 96 cm³ 6 cm
Height ( 8 cm )
6 cm
EXAMPLE 7: Find the volume of a regular triangular pyramid.
8 cm
6 cm
Solution: V =⅓(½bh)H = ⅓[½(6 cm)(3 cm) (8 cm)] = ⅓(9 cm²) (8 cm) = ⅓ (72 )cm³V = 24 cm³
h=3 cm3
33
3
3
COMMON SOLIDSCOMMON SOLIDSCYLINDERSCYLINDERS
CYLINDER
-is a space figure with two circular bases that are parallel and
congruent.
Circular base
Circular base..
HeightRadius
CYLINDERSCYLINDERS
Guide questions:Guide questions: What is the What is the geometric geometric
figure figure represented by represented by the bases of the the bases of the
cylinder?cylinder?How do you How do you compute its compute its
area?area?
Volume of a Volume of a cylindercylinderANSWERSANSWERS Circles Circles
(circular (circular bases)bases)
A = A = r²r²
Height
radius
Volume of a Volume of a cylindercylinder How can the volume of a How can the volume of a
cylinder be computed?cylinder be computed? V = BhV = Bh, where , where BB is the is the
area of the base and area of the base and hh is the is the height of the cylinder.height of the cylinder.
by substitution,by substitution, V= V= ππr²r²hh
EXAMPLE 8: EXAMPLE 8: Find the volume of a Find the volume of a cylinder. cylinder.
Use Use ππ = 3.14 = 3.14Solution:Solution: V =V = π πr²h r²h =(3.14)(5 cm)² 10 cm=(3.14)(5 cm)² 10 cm = 3.14( 25cm²) = 3.14( 25cm²)
(10 cm)(10 cm) = 3.14( 250 cm³)= 3.14( 250 cm³)V = 785 cm³V = 785 cm³
5 cm
10 cm
VOLUME OF A CONEVOLUME OF A CONE
REFLECTIVE TRAFFIC CONE
CONECONE-is a space figure with one circular baseand a vertex
Vertex
Circular base.HeightOf the cone
Radius
Slant HeightOf the cone
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
Volume of a Volume of a conecone How can the volume of a How can the volume of a
cone be computed?cone be computed? V = V = ⅓⅓BhBh, where , where BB is is
the area of the base and the area of the base and hh is is the height of the cone.the height of the cone.
by substitution,by substitution, V= V= ⅓ ⅓ ππr²r²hh
Find the volume of a cone. Find the volume of a cone. Use Use ππ = 3.14 = 3.14
Solution:Solution:V =V = ⅓ ⅓ ππr²hr²h == ⅓ ⅓(3.14)(5 cm) ²(10 (3.14)(5 cm) ²(10
cm)cm) = = ⅓ (3.14)(25 cm⅓ (3.14)(25 cm²²)) (10 (10
cm)cm) == ⅓ (785 cm³) ⅓ (785 cm³)V= 261.67 cmV= 261.67 cm³³
5 cm
10 cm
Find the volume of a cone. Find the volume of a cone. Use Use ππ = 3.14 = 3.14
Solution:Solution: Step 1. find h.Step 1. find h. Using Using
Pythagorean Pythagorean theorem,theorem,
h²= 5² - 3² h²= 5² - 3² =25-9=25-9 h² = 16h² = 16 h = 4 cmh = 4 cm
3 cm
5 cm
Solution:Solution:V =V = ⅓ ⅓ ππr²hr²h == ⅓ ⅓(3.14)(3 cm) ²(4 (3.14)(3 cm) ²(4
cm)cm) = = ⅓ (3.14)(9 cm⅓ (3.14)(9 cm²²))
(4 cm)(4 cm) == ⅓ (113.04 cm³) ⅓ (113.04 cm³)V= 37.68 cmV= 37.68 cm³³
4 cm
VOLUME OF A SPHEREVOLUME OF A SPHERE
THE EARTHBALLS
SPHERESPHERE
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
VOLUME OF A SPHEREVOLUME OF A SPHERE
The formula to find the VOLUMEof a sphere is V = πr³, where r is the length of its radius.
BALL
34
How can the volume of a sphere be computed?
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.
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.
The height (H)of the cylinder is equal to the diameter (d) of the sphere.
radius
radius r
H = dr
Volume (Sphere)= ⅔ the volume of a circumscribed cylinder
radius
radius r
H = dr
Volume (Sphere)= ⅔ r²h = ⅔ r² (2r) = r³
radius
radius r
H = d= 2rr
34
1. Find the volume of a sphere. 1. Find the volume of a sphere. Use Use ππ = 3.14 = 3.14
Solution:Solution: V =V = 4/3 4/3 ππr³ r³ ==4/3(3.14)(10 cm)³4/3(3.14)(10 cm)³ = = 12.56 (1000 cm³)12.56 (1000 cm³) 33 = = 12,560 cm³)12,560 cm³) 33 V= 4,186.67 cm³V= 4,186.67 cm³
10 cm
2. Find the volume of a sphere. 2. Find the volume of a sphere. Use Use ππ = 3.14 = 3.14
Solution:Solution: V =V = 4/3 4/3 ππr³ r³ ==4/3(3.14)(7.8 cm)³4/3(3.14)(7.8 cm)³ = = 12.56 (474.552 12.56 (474.552
cm³)cm³) 33 = = 5960.37312 cm³5960.37312 cm³ 33 V= 1,986.79 cm³V= 1,986.79 cm³
7.8 cm