Geometry - Geometric Solids - Quick Guides

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Study GuideGeometry–Geometric Solids1. Right Circular Cylinders1.1What is a Cylinder?Acylinderis a 3D shape that has two identical, circular bases and straight sides connecting them. Ifthe segment joining the centers of the circles is perpendicular to the planes of the bases, we have aright circular cylinder.•Figure 1shows two types of cylinders:o(a) Aright circular cylinder, where the sides are straight and the two circles arealigned.o(b) Anoblique circular cylinder, where the sides are slanted.Figure 11.2Lateral Area, Total Area, and VolumeThe lateral area (LA), total area (TA), and volume (V) of aright circular cylinderare calculated insimilar ways to right prisms.When a cylinder is pictured like a soup can, the lateral area is the area of the label. If the label ispeeled off, it becomes a rectangle. This rectangle has the same height as the cylinder and the samewidth as the circumference of the base (the lid).

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Study GuideFigure 21.3Theorems for Right Circular CylindersTotal Area (TA)The total area of a right circular cylinder is the sum of the lateral area and the area of the two circularbases.Where (B) is the area of one of the bases.Volume (V)The volume of a right circular cylinder is the area of the base times the height.Where (B) is the area of the base and (h) is the height.Example1:Finding the Lateral Area, Total Area, and VolumeInFigure 3, we are given a right circular cylinder with:•Radius(r = 7)cm•Height(h = 10)cmNow, let’s calculate the following:

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Study Guide(a) Lateral Area (LA)The lateral area is the area of the side of the cylinder.(b) Total Area (TA)The total area is the lateral area plus the area of the two bases.(c) Volume (V)The volume is the area of the base times the height.Figure 32. PyramidsApyramidis a 3D shape with some special features. Let's break it down:Base•A pyramid hasone base, and this base is always a polygon (like a triangle, square, etc.).Vertex•Theverticesof the base are connected to a single point. This point is not on the base itself,and it is called thevertexof the pyramid.

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Study GuideLateral Faces•Thelateral facesof the pyramid are the triangular sides. All these triangular faces meet atthe vertex.Lateral Edges•Thelateral edgesare the segments where the lateral faces (the triangles) come together.Altitude•Thealtitudeof the pyramid is the perpendicular segment from the vertex straight down to thebase. It tells you how tall the pyramid is.3. Regular Pyramids3.1What is a Regular Pyramid?Aregular pyramidis a 3D shape that has a base which is a regular polygon (all sides and angles arethe same). Thelateral edgesof a regular pyramid are all the same length, and each lateral face is atriangle that is congruent to the others.3.2Different Types of Regular PyramidsRegular pyramids can have different polygonal bases. For example:•Aregular triangular pyramidhas a triangle as its base.•Aregular square pyramidhas a square as its base.•Aregular hexagonal pyramidhas a hexagon as its base.

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Study Guide3.3Slant HeightThe lateral faces of a regular pyramid are congruent isosceles triangles. Theslant heightof a regularpyramid is the height of any of these triangles, measured from the base to the apex (vertex).Figure 23.4Theorems for Regular Pyramids1.Lateral Area (LA)2.The lateral area of a regular pyramid can be found using this equation:Where(p)is the perimeter of the base, and(l)is the slant height.

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Study GuideExample1:Finding the Lateral Area of a Square PyramidFigure 3Let’s look at a square pyramid with the following measurements:•Base side:(16in)•Slant height:(10in)Step 1: Find the perimeter of the square base.Since the base is a square, the perimeter is:Step 2: Use the lateral area formula.Now, we can use the lateral area formula:So, the lateral area is(320in2).3.5Total Area (TA) of a PyramidExample2:Finding the Total Area of the Regular PyramidA pyramid only has one base, so its total area is the sum of the lateral area and the area of the base.

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Study GuideWhere(B)is the area of the base. For a square pyramid, the area of the base is:So, the total area is:Now that we know the lateral area, we can find the total area:3.6Volume of a PyramidThevolume (V)of a regular pyramid is given by the formula:Where(B)is the area of the base, and(h)is the height (altitude).Example3:Finding the Volume of a Regular PyramidFor the same pyramid, we know the base area is(B = 256in2), and the height(h = 6in). The volumeis:4. Right Circular Cones4.1What is a Right Circular Cone?Aright circular coneis a 3D shape that is similar to a pyramid, but with a circular base instead of apolygon. In this shape, the vertex (tip) is directly above the center of the base. The followingvocabulary and equations apply to right circular cones:•Theslant height((l)) is the distance from the vertex to any point on the edge of the base,along the side of the cone.

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Study Guide•Thealtitude((h)) is the perpendicular distance from the vertex to the center of the base(straight down).•Theradius((r)) is the distance from the center of the base to the edge of the base.Figure 1A right circular cone.4.2Key Equations for Right Circular Cones1.Lateral Area (LA)2.The lateral area of a right circular cone with base circumference (C) and slant height (l) isgiven by:3.Total Area (TA)4.The total area of a right circular cone, which includes the lateral area and the area of thebase, is given by:5.Volume (V)6.The volume of a right circular cone with base area (B) and height (h) is:

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