VOLUME OF SOLIDS

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?

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.

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.

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

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

VOLUME OF A RECTANGULAR PRISM

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

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

VOLUME OF A SQUARE PRISM

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

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

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

VOLUME OF A TRIANGULAR PRISM

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

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 = 24.57 cm³ 3.9 cm 4.5cm 2.8 cm

ANOTHER KIND OF POLYHEDRON PYRAMIDS

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

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

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.

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

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

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

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

CYLINDERS

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

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

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

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

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

REFLECTIVE TRAFFIC CONE VOLUME OF A CONE REFLECTIVE TRAFFIC CONE

. 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

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

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

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= 261.67 cm³ 10 cm 5 cm

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 (113.04 cm³) V= 37.68 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

VOLUME OF A SPHERE THE EARTH BALLS

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

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?

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.

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

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

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

1. Find the volume of a sphere. Use π = 3.14 Solution: V = 4/3 πr³ =4/3(3.14)(10 cm)³ = 12.56 (1000 cm³) 3 = 12,560 cm³) V= 4,186.67 cm³ 10 cm

2. Find the volume of a sphere. Use π = 3.14 Solution: V = 4/3 πr³ =4/3(3.14)(7.8 cm)³ = 12.56 (474.552 cm³) 3 = 5960.37312 cm³ V= 1,986.79 cm³ 7.8 cm