Transcript Geometry
Pre-College Math Classifying Solids Prisms Prisms are named for their base shape: Rectangular Prism Triangular Prism Hexagonal Prism Pentagonal Octagonal Prism Trapezoidal Square Prism Prism Prism Prisms Prisms are made up of two bases and rectangular faces connecting the edges of the bases two each other. Cylinders Cylinders have a circle for a base and one rectangular face curved around the circumference of the circle Pyramids Pyramids are named for their bases as well Rectangular Pyramid Triangular Pyramid Hexagonal Pyramid Pentagonal Octagonal Square Pyramid Pyramid Pyramid Pyramids Pyramids are made up of one base and triangles which converge at a point (vertex) opposite the base Spheres and Hemispheres Spheres are shaped like a round ball A hemi-sphere is half a sphere Cones Cones have a circle for a base and a curved face that converges at a vertex opposite the base Homework Assignment 10-6 Pre-College Math Volume of Prisms and Cylinders What is volume? The volume of a solid is the number of cubic units contained in its interior. Meaning how many cubic units it can “hold”. The volume of an object is measured in cubic units (i.e. 𝑐𝑚3 , 𝑚3 , 𝑚𝑖 3 , 𝑓𝑡 3 , etc). Volume of a Prism The Volume of a Prism is found by taking the area of the base shape and multiplying it by the height if the prism So the formula is: 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝐵𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡 This works because you can think of taking slices of the base shape and stacking them on top of each other Some extras… Volume of a cube: The volume of a cube is the cube of the length of its side: 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑠 3 Congruence: If two polyhedra (solid bounded by polygons) are congruent, then they have the same volume Volume Addition: The volume of a solid is the sum of the volumes of all its NON-OVERLAPPING parts. Helpful Hints Area 1 2 of a regular polygon: 𝑎𝑝𝑜𝑡ℎ𝑒𝑚 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑖𝑑𝑒𝑠 𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ Area of a rectangle: 𝑙𝑒𝑛𝑔𝑡ℎ 𝑤𝑖𝑑𝑡ℎ Area of a triangle: 1 (𝑏𝑎𝑠𝑒 2 𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒)(ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒) Volume of a Cylinder This is the same as the volume of a prism with one little difference, the base is ALWAYS a circle Therefore, the formula is: 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝐴𝑟𝑒𝑎 𝑜𝑓 𝐶𝑖𝑟𝑐𝑙𝑒 × 𝐻𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝐶𝑦𝑙𝑖𝑛𝑑𝑒𝑟 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 2 ℎ Homework Assignment 10-7 Pre-College Math Volume of a Pyramid, Cone, and Sphere Volume of a Pyramid The Volume of a Pyramid is given by the area of the base of the pyramid times the height of the pyramid all divided by three Thus, the formula is: 1 𝑉𝑜𝑙𝑢𝑚𝑒 = (𝑏𝑎𝑠𝑒 𝑎𝑟𝑒𝑎)(ℎ𝑒𝑖𝑔ℎ𝑡) 3 Volume of A Cone The volume of a cone is given by the area of the circular base multiplied by the height od the cone, the whole quantity dived by three Thus, the formula is: 1 2 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 ℎ 3 Volume of a Sphere The volume of a sphere is given by four thirds times pi times the radius of the sphere cubed (raised to a power of three). Thus, the formula is: 4 3 𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 3 Homework Assignment 10-8