VOLUME OF SOLIDS By: SAMUEL M. GIER

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.

COMMON SOLIDS 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. Video 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