9.4 – Perimeter, Area, and Circumference Perimeter of a Polygon The perimeter of any polygon is the sum of the measures of the line segments that form its sides. Perimeter is measured in linear units. Perimeter of a Triangle a b c The perimeter P of a triangle with sides of lengths a, b, and c is given by the formula: P = a + b + c.
9.4 – Perimeter, Area, and Circumference Perimeter of a Rectangle w l The perimeter P of a rectangle with length l and width w is given by the formula: P = 2l + 2w or P = 2(l + w). s Perimeter of a Square s The perimeter P of a square with all sides of length s is given by the formula: P = 4s.
9.4 – Perimeter, Area, and Circumference Area of a Polygon The amount of plane surface covered by a polygon is called its area. Area is measured in square units. l Area of a Rectangle w The area A of a rectangle with length l and width w is given by the formula: A = l w. s Area of a Square s The area A of a square with all sides of length s is given by the formula: P = s2.
9.4 – Perimeter, Area, and Circumference Area of a Parallelogram h b The area A of a parallelogram with height h and base b is given by the formula: A = bh. b1 Area of a Trapezoid h b2 The area A of a trapezoid with parallel bases b1 and b2 and height h is given by the formula: A = (1/2) h (b1 + b2)
9.4 – Perimeter, Area, and Circumference Area of a Triangle h b The area A of a triangle with base b and height h is given by the formula: A = (1/2) b h
9.4 – Perimeter, Area, and Circumference Find the perimeter and area of the rectangle. 15 ft 7 ft Perimeter P = 2l + 2w 2(15) + 2(7) P = 44 ft Area A = lw A = (15)(7) A = 105 ft2 Find the area of the trapezoid. 7 cm. A = (1/2) h (b1 + b2) 6 cm. 6 cm. 5 cm. A = (1/2) (5) (7 + 13) 13 cm. A = (1/2) (5) (20) A = 50 cm2
9.4 – Perimeter, Area, and Circumference Find the area of the shaded region. Area of square – Area of triangle 4 in. s2 – (1/2) b h 4 in. 42 – (1/2) (4)(4) 16 – 8 8 in2
9.4 – Perimeter, Area, and Circumference Circumference and Area of a Circle r d The circumference C of a circle of diameter d is given by the formula: or where r is a radius. The area A of a circle with radius r is given by the formula:
9.4 – Perimeter, Area, and Circumference Find the area and circumference of a circle with a radius that is 6 inches long (use 3.14 as an approximation for ). Circumference ( 3.14) Circumference () C = 2 r C = 2 r C = 2 (3.14) 6 C = 2 6 C = 37.68 in C = 37.699 in Area ( 3.14) Area () A = r2 A = r2 A = (3.14) 62 A = 62 A = 113.04 in2 A = 113.097 in2
9.5 – Space Figures, Volume, and Surface Area Space figures: Figures requiring three dimensions to represent the figure. Polyhedra: Three dimensional figures whose faces are made only of polygons. Regular Polyhedra: A polyhedra whose faces are made only of regular polygons (all sides are equal and all angles are equal.
9.5 – Space Figures, Volume, and Surface Area Other Polyhedra Pyramids are made of triangular sides and a polygonal base. Prisms have two faces in parallel planes; these faces are congruent polygons.
9.5 – Space Figures, Volume, and Surface Area Other Space Figures Right Circular Cones have a circle as a base and the surface tapers to a point directly above the center of the base. Right Circular Cylinders have two circles as bases, parallel to each other and whose centers are directly above each other.
Volume and Surface Area of a Rectangular solid (box) 9.5 – Space Figures, Volume, and Surface Area Volume and Surface Area Volume is a measure of capacity of a space figure. It is always measured in cubic units. Surface Area is the total region bound by two dimensions. It is always measured in square units. Volume and Surface Area of a Rectangular solid (box) The volume V and surface area S of a box with length l, width w, and height h is given by the formulas: V = lwh w h l and S = 2lw + 2lh + 2hw
9.5 – Space Figures, Volume, and Surface Area Find the volume and surface area of the box below. 2 in. 3 in. 7 in. V = 7(2)(3) S = 2(7)(2) + 2(7)(3) + 2(3)(2) V = 42 in.3 S = 28 + 42 + 12 S = 82 in.2
Volume and Surface Area of a Cube 9.5 – Space Figures, Volume, and Surface Area Volume and Surface Area Volume and Surface Area of a Cube The volume V and surface area S of a cube with side lengths of s are given by the formulas: V = s3 s and S = 6s2 Find the volume and surface area of the cube below. V = 53 S = 6(5)2 V = 125 ft.3 S = 625 S = 150 ft.2 5 ft.
Volume of Surface Area of a Right Circular Cylinder 9.5 – Space Figures, Volume, and Surface Area Volume and Surface Area Volume of Surface Area of a Right Circular Cylinder The volume V and surface area S of a right circular cylinder with base radius r and height h are given by the formulas: V = r2h h r and S = 2rh + 2r2 Find the volume and surface area of the cylinder below. V = (2)2(10) S = 2(2)(10) + 2(2)2 10 m 2 m V = 40 S = 40 + 8 = 48 V = 125.6 m3 S = 150.72 m2
Volume of Surface Area of a Sphere 9.5 – Space Figures, Volume, and Surface Area Volume and Surface Area Volume of Surface Area of a Sphere The volume V and surface area S of a sphere radius r are given by the formulas: r V = (4/3) r3 and S = 4 r2 Find the volume and surface area of the sphere below. V = (4/3)(9)3 S = 4 (9)2 9 in. V = 972 S = 324 V = 3052.08 in.3 S = 1017.36 in.2
Volume of Surface Area of a Right Circular Cone 9.5 – Space Figures, Volume, and Surface Area Volume and Surface Area Volume of Surface Area of a Right Circular Cone The volume V and surface area S of a right circular cone with base radius r and height h are given by the formulas:
9.5 – Space Figures, Volume, and Surface Area Find the volume and surface area of the cone below. h = 4 m r = 3 m V = (1/3)(3)2(4) V = 12 V = 37.68 m3 S = 15 + 9 S = 24 S = 75.36 m2
9.5 – Space Figures, Volume, and Surface Area Volume of a Pyramid The volume V of a pyramid with height h and base of area B is given by the formula: Note: B represents the area of the base (l w). Find the volume of the pyramid (rectangular base) below. 7 cm 6 cm 3 cm cm3