Volume of Solids

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

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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

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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

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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