Surface area and Volume Ch Sol: G.10,12,13,14

Area of a Parallelogram: If a parallelogram has an area of A square units, a base of b units, and a height of h units, then A = bh. Example: Find the area of parallelogram STAR. Area of a Triangle: If a triangle has an area of A square units, a base of b units, and a corresponding height of h units, then A = ½bh. Example: Find the area of  THE.

Area of a Trapezoid: If a trapezoid has an area of A square units, bases of b 1 units and b 2 units, and height of h units, then A = ½h(b 1 + b 2 ). Example: Find the area of trapezoid TRAP. Area of a Rhombus: If a rhombus has an area of A square units and diagonals of d 1 and d 2 units, then A = ½d 1 d 2. Example: Find the area of rhombus SANE, if AE = 12 and SN = 8.

Homework Area of a Parallelogram

Area of a Triangle, Trapezoid, and a Rhombi

Lesson 9-1: Area of 2-D Shapes6 Areas of Regular Polygons Perimeter = (6)(8) = 48 apothem = Area = ½ (48)( ) = sq. units 8 If a regular polygon has an area of A square units, a perimeter of P units, and an apothem of a units, then A = ½ (a)(p).

Surface Area and Volume of Prisms Two faces, called bases, are formed by congruent polygons that lie in parallel planes. The lateral faces, the faces that are not bases are formed by parallelograms The intersections of two adjacent lateral faces are called lateral edges and are parallel segments. A segment perpendicular to the places containing the two bases, with an endpoint in each plane is called an altitude of the prism. The length of an altitude of a prism is called the height of the prism. If a lateral edge is also an altitude then the prism is called a right prism. If a prism is not right, then it is an oblique prism. The figure is a _________________ prism Prisms and Nets

Volume of a Right Prism: V = Bh. B – area of base Lateral Area of a Right Prism: L.A. = Ph. P- perimeter of base Surface Area of a Right Prism: S.A.= Ph + 2B. Example: Find the lateral area and the surface area of a right triangular prism with a height of 20 inches and a right triangular base with legs of 8 and 6 inches. First, use Pythagorean Theorem to Find the measure of the hypotenuse, c. Next, use the value of c to find the perimeter Now find the area of a base. B = ½ bh Find the Volume. V=Bh Finally, find the surface area. S.A.= Ph + 2B

Surface Area

Volume of Prisms

Lesson 9-3: Cylinders and Cones 12 Surface Area (SA) = 2B + LA = 2πr ( r + h ) Cylinders are right prisms with circular bases. Therefore, the formulas for prisms can be used for cylinders. Volume (V) = Bh = The base area is the area of the circle: The lateral area is the area of the rectangle: 2πrh h 2πr h

Lesson 9-3: Cylinders and Cones13 Example For the cylinder shown, find the lateral area, surface area and volume. L.A.= 2πrh L.A.= 2π(3)(4) L.A.= 24π sq. cm. 4 cm 3 cm S.A.= 2πr 2 + 2πrh S.A.= 2π(3) 2 + 2π(3)(4) S.A.= 18π +24π S.A.= 42π sq. cm. V = πr 2 h V = π(3) 2 (4) V = 36π

Surface Area of a Cylinder

Volume of a Cylinder

Lesson 9-3: Cylinders and Cones 16 Cones Surface Area (SA) = B + LA = π r (r + l) Cones are right pyramids with a circular base. Therefore, the formulas for pyramids can be used for cones. Volume (V) = Lateral Area (LA) = π r l, where l is the slant height. The base area is the area of the circle: Notice that the height (h) (altitude), the radius and the slant height create a right triangle. l r h Formulas: S.A. = π r ( r + l ) V =

Lesson 9-3: Cylinders and Cones17 Example: For the cone shown, find the lateral area surface area and volume. L.A.= πrl Note: We must use the Pythagorean theorem to find l. L.A.= π(6)(10) L.A.= 60π sq. cm. 6 cm = l 2 10 S.A.= πr (r + l ) S.A.= π6 (6 + 10) S.A.= 6π (16) S.A.= 96π sq. cm. V= 96π cubic cm.

Surface Area of Cones

Volume of a cone

A sphere is a three – dimensional circle Center: point that is the same distance from any point on the sphere Radius: distance from the center to any point on the sphere Chord: segment with both endpoints on the sphere Diameter: chord that passes through the center Tangent: line that only touches outer part of the sphere at one point Hemisphere: half of a sphere Great Circle: circle formed by looking at the flat part of a hemisphere Surface Area: Example: Find the volume and surface area of a sphere whose great circle has a circumference of cm. Volume:

Surface area of Spheres

Volume of a Sphere