Geometry 10 and 11 Notes Area and Volume Name Per Date 10.1 Area is the amount of space inside of a two dimensional object. When working with irregular shapes, we can find its area by breaking it up into shapes we know how to find the area of. The area of a region is equal to the sum of the areas of its non-overlapping parts. To understand this, we have to remember how to find the areas of different shapes. Area of a rectangle can be found by multiplying the base of the rectangle by its height (or in the past has been said length times width) 10.1 Area of a parallelogram: When working with a parallelogram, if we cut off a right triangle from the end of the parallelogram and move it to the other end of the parallelogram: We find that the area of a parallelogram is also base times height (just remember the height is not the side of the parallelogram but the length of the segment perpendicular to the base) 10.1 Find the area of the parallelogram: Find the height of a rectangle that has a base of 5 and an area of 5x 5 x. 10.1 Area of a triangle: To understand the formula for area of a triangle, let s look at what happens when we take the triangle and flip it on itself.... We have just created a parallelogram! So the area of the triangle is ½ the area of the parallelogram formed by the two triangles. Area triangle b h = 10.1 Area of a trapezoid: If we flip the trapezoid on itself, just like we did for the triangle. So again we have created a triangle with base ( b b ) + and height h 1 So the area is half of the parallelogram s area Area trapezoid ( + b) hb = 1
10.1 Find the area of a trapezoid Find the area of the triangle: 10.1 A kite or a rhombus can be divided into two congruent triangles 10.1 Find the area of the kite: with a base of d 1 and a height of d. Since there are two triangles with the same area, 1 1 1 1 Areakite = dd = dd 1 1 Since a rhombus is both a kite and a parallelogram, either formulas can be used to find the area of rhombus: Area 1 d d = OR Arearhombus rhombus 1 = bh 10.A A circle is a locus (group) of points in a plane that are a fixed distance (radius or radii) from a fixed point (center). A circle is named by the symbol and its center. Name: C Radii: AB, AC, and AD Diameter: CD The distance around the circle (perimeter) is called the circumference. The irrational number π is defined as the ratio of the circumference of the circle to the diameter: C π =, solving for C gives us a formula for the circumference (perimeter of a circle) C = π d d And since the diameter is really two radii, we can also use the formula C = πr = πr 10.A Name the circle and find the circumference of the circle:
10.A Area of a Circle: You can also use the circumference of a circle to find its area. Divide the circle and arrange the pieces to make a shape that resembles a parallelogram. The more pieces you divide the circle into, the more accurate your estimate will be... 1 The base of your parallelogram is about half the circumference πr = πr and the height is close to the radius so Area = b h = πr r = πr 10.A Find the area of the circle: circle 10.A (a) Find the radius if the circumference is 4π (b) Find the circumference if A = 169π 10.B A regular polygon is a polygon whose angles are all congruent and sides are all congruent. It is equilateral and equiangular. The center of a regular polygon is the center of the circle inscribed in the polygon AND the center of the circle circumscribed about the polygon. The apothem is the segment that connects the center of the regular polygon to the midpoint of any side of the polygon. It will be the perpendicular bisector of the side of the polygon. The radius of a regular polygon is a radius to the circumscribed circle of the polygon. It will also be the angle bisector of any angle in the regular polygon. The central angle of a regular polygon has a vertex of the center of the polygon and whose sides are radii of the polygon. Each central 360 angle has a measure of where n is the number of sides the polygon has. n Each interior angle of a regular polygon has a measure of I = ( n ) 180 n
10.B To find the area of a regular polygon with n sides that each measure s and an apothem that measures a : we divide the polygon into n congruent triangles the triangles have a base of s and sides that are the radii of the polygon. This allows are apothem a to be the height of the triangle. 1 1 Area of 1 triangle: bh = as Since there are n triangles, the area of the regular polygon is: 1 1 Arearegular n = as = ans 10.B Find the area of each regular polygon: 10.C Now that we know how to find the area of regular polygons, we are going to bring together all of our knowledge to create formula s to find the areas of regular polygons... 10.C Area of a regular triangle:
10.C Area of a regular quadrilateral: 10.C Area of a regular pentagon: 10.C 1.) Find the area of a regular triangle with sides measuring 6..) Find the area of a regular quadrilateral with sides measuring 11. 3.) Find the area of a regular pentagon with sides measuring 18. 10.D Now we have created formulas for the shapes we will use for the castle project, let s use our methods to find the area of any regular polygon... (a) Find the area of a regular hexagon with sides measuring 6m (b) Find the area of a regular octagon with a side length of 4cm
10.D Find the area of a regular heptagon with side length of 8 ft. 10.3 Finding the area of composite figures: 11.1 Three dimensional figures (3D), or solids, can be made up of flat or curved surfaces. Each flat surface is called a face. An edge is the segment that is the intersection of two faces. A vertex is the point that is the intersection of three or more faces. 11.1 Types of 3D figures: 1) A prism is formed by two parallel congruent polygonal faces called bases connected by faces that are parallelograms. Each prism is named by its polygonal base: Triangular prism, rectangular prism, pentagonal prism... A prism with six square faces is called a cube. ) A cylinder is formed by two parallel congruent circular bases and a curved surface that connects those bases. 3) A pyramid is formed by a polygonal base and triangular faces that meet at a common vertex. Each pyramid is named by its polygonal base: Triangular pyramid, rectangular pyramid, pentagonal pyramid 4) A cone is formed by a circular base and a curved surface that connects the base to the vertex. 5) A sphere is a locus (group) of points in space that are a fixed distance (radius) from a given point called the center. A hemisphere is half of a sphere. A great circle divides a sphere into two hemispheres.
11.1 Classify each figure. Name the vertices, edges, and bases. I J B C K A 11.1 A net is a diagram of the surfaces of a 3-D figure. A net can be folded to form the 3D figure. Identify the name of the figure based on the net. 11.1 Drawing 3 dimensional figures: 11.1 A cross-section is the intersection of a 3 dimensional figure and a plane. Each cross-section will result in a dimensional shape. 11. Volume is the amount of space inside of a 3D figure. 11. Volumes of prisms and cylinders: B and multiplying The volume of a prism or cylinder can be found by finding the area of the base ( ) it by the height ( h ) of the prism or cylinder. So if the prism is a triangular prism, calculate the area of the triangular base, then multiply by the height. If the shape is a pentagonal prism, find the area of the pentagon and multiply by the height. V prism/ cylinder = Bh
11. The base of the prism is a regular polygon with sides that measure 1 and a height of 7. Find the volume of the prism. 11. Find the volume of the cylinder in exact form. 0ft 8ft 11. For a rectangular prism, any pair of parallel sides could be the bases. 11.3 Parts of Pyramids and Cones: Altitude The segment that begins at the vertex and is perpendicular to the base of the shape. Runs through the center of the figure. inside The height of a pyramid! 10 Slant Height The altitude of the lateral face. The height of the triangular face that is slanted. Apothem Runs from the center of the regular base (where the altitude touches the base) to the midpoint of the side (where the slant height touches) The altitude, slant height, and apothem create a right triangle and have a relationship of ( altitude) + ( apothem) = ( slant height) The altitude is important for calculating volume, the apothem is important for calculating volume, and the slant height is important for surface area and building pyramids. 11.3 Volumes of pyramids and cones: B and multiplying The volume of a pyramid or cone can be found by finding the area of the base ( ) it by the height ( h ) of the pyramid or cone then dividing by 3 (since it comes to a point instead of becoming a prism). So if the pyramid is a triangular pyramid, calculate the area of the triangular base, then multiply by the height, then divide by 3. If the shape is a pentagonal pyramid, find the area of the pentagon and multiply by the height, then divide by 3. Bh V pyramid / cone = 3
11.3 Find the volume of the square based pyramid. Calculate the slant height. 10 Altitude=1 11.3 Find the volume of the cone in exact form. Calculate the slant height. 11.4 altitude=8 diameter=16 The volume of a sphere with a radius of r is 11.4 Find the volume of the sphere in exact form. V 4 3 3 = π r. radius=6 Castle Building 3-Dimensional Figures Building Prisms Method 1: Build n (number of sides of the base) rectangles with sides measuring the side length of the base and the height of the prism. Method : Build 1 rectangle with one side equal to n times s (with markings every n) and other side the height of the prism. Fold the rectangle at every n marking to create the prism. Building Cylinders Build a rectangle with one side measuring the circumference of the circular base and the other side the height of the cylinder. Roll the height edges to each other and tape together. Castle Building Pyramids Calculate the apothem of the base. Use the apothem and the altitude to calculate the slant height. Build n triangles, with base measuring the length of the side and the height of the triangle being the slant height. Building Cones Use a compass to draw a circle with the radius measuring the same as the slant height. Draw a radius of the circle. Cut the radius and overlap the edges until the cone forms and fits on the cylinder.
Castle Perimeter cuts to the castle: Castle Volume cuts to the castle: