Skip to content
Quick Guides

Geometry - Geometric Solids

Preview 8 of 25 pages

See the first 8 pages exactly as they appear in the document.

  1. Page 1
    Read page 1 as text

    Study Guide Geometry – Geometric Solids 1. Right Circular Cylinders 1.1 What is a Cylinder? A cylinder is a 3D shape that has two identical, circular bases and straight sides connecting them. If the segment joining the centers of the circles is perpendicular to the planes of the bases, we have a right circular cylinder . • Figure 1 shows two types of cylinders: o (a) A right circular cylinder , where the sides are straight and the two circles are aligned. o (b) An oblique circular cylinder , where the sides are slanted. Figure 1 1.2 Lateral Area, Total Area, and Volume The lateral area (LA), total area (TA), and volume (V) of a right circular cylinder are calculated in similar 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 is peeled off, it becomes a rectangle. This rectangle has the same height as the cylinder and the same width as the circumference of the base (the lid).

  2. Page 2
    Read page 2 as text

    Study Guide Figure 2 1.3 Theorems for Right Circular Cylinders Total Area (TA) The total area of a right circular cylinder is the sum of the lateral area and the area of the two circular bases. 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. Example 1 : Finding the Lateral Area, Total Area, and Volume In Figure 3 , we are given a right circular cylinder with: • Radius ( r = 7 ) cm • Height ( h = 10 ) cm Now, let’s calculate the following:

  3. Page 3
    Read page 3 as text

    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 3 2. Pyramids A pyramid is a 3D shape with some special features. Let's break it down: Base • A pyramid has one base , and this base is always a polygon (like a triangle, square, etc.). Vertex • The vertices of the base are connected to a single point. This point is not on the base itself, and it is called the vertex of the pyramid.

  4. Page 4
    Read page 4 as text

    Study Guide Lateral Faces • The lateral faces of the pyramid are the triangular sides. All these triangular faces meet at the vertex. Lateral Edges • The lateral edges are the segments where the lateral faces (the triangles) come together. Altitude • The altitude of the pyramid is the perpendicular segment from the vertex straight down to the base. It tells you how tall the pyramid is. 3. Regular Pyramids 3.1 What is a Regular Pyramid? A regular pyramid is a 3D shape that has a base which is a regular polygon (all sides and angles are the same). The lateral edges of a regular pyramid are all the same length, and each lateral face is a triangle that is congruent to the others. 3.2 Different Types of Regular Pyramids Regular pyramids can have different polygonal bases. For example: • A regular triangular pyramid has a triangle as its base. • A regular square pyramid has a square as its base. • A regular hexagonal pyramid has a hexagon as its base.

  5. Page 5
    Read page 5 as text

    Study Guide 3.3 Slant Height The lateral faces of a regular pyramid are congruent isosceles triangles. The slant height of a regular pyramid is the height of any of these triangles, measured from the base to the apex (vertex). Figure 2 3.4 Theorems for Regular Pyramids 1. 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.

  6. Page 6
    Read page 6 as text

    Study Guide Example 1 : Finding the Lateral Area of a Square Pyramid Figure 3 Let’s look at a square pyramid with the following measurements: • Base side: ( 16 in) • Slant height: ( 10 in) 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 ( 320 in 2 ) . 3.5 Total Area (TA) of a Pyramid Example 2 : Finding the Total Area of the Regular Pyramid A pyramid only has one base, so its total area is the sum of the lateral area and the area of the base.

  7. Page 7
    Read page 7 as text

    Study Guide Where ( 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: 3.6 Volume of a Pyramid The volume (V) of a regular pyramid is given by the formula: Where ( B ) is the area of the base, and ( h ) is the height (altitude). Example 3 : Finding the Volume of a Regular Pyramid For the same pyramid, we know the base area is ( B = 256 in 2 ) , and the height ( h = 6 in). The volume is: 4. Right Circular Cones 4.1 What is a Right Circular Cone? A right circular cone is a 3D shape that is similar to a pyramid, but with a circular base instead of a polygon. In this shape, the vertex (tip) is directly above the center of the base. The following vocabulary and equations apply to right circular cones: • The slant height ((l)) is the distance from the vertex to any point on the edge of the base, along the side of the cone.

  8. Page 8
    Read page 8 as text

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

17 more pages in the full document

Unlock it once and it stays in your library, ready to chat with or turn into flashcards.

Unlock the full document

More Mathematics study materials

See all

Study this document with CramX

Once it is in your library, every tool can work from it.