CP1 Math 2 Unit 8: S.A., Volume, Trigonometry: Day 7 Name Surface Area Objectives: Define important vocabulary for 3-dimensional figures Find the surface area for various prisms Generalize a formula for surface area of cylinders Vocabulary PRISM: a prism is a solid figure whose two end faces are congruent and parallel polygons, and whose sides are parallelograms. Which shapes below are prisms?? CYLINDER: a cylinder is a 3-dimensional object with 2 circular, parallel faces. Right cylinder Oblique cylinder PYRAMID: a 3-dimensional figure with a polygon for a base and with a single vertex (the apex ), which is not in the same plane as the base Triangular pyramid Square pyramid Oblique square pyramid Pentagonal pyramid Hexagonal Pyramid CONE: a 3-dimensional figure with a cicle for a base and with a single vertex point not in the same plane as the base Right cone: Oblique cone: **3-dimensional figures can be right (perpendicular) or oblique as shown above. SPHERE: a 3-dimensional figure in which every point on the surface is equidistance from the center
SURFACE AREA is the total area of the surfaces (or faces) of a 3-dimensional object. Sometimes we flatten a 3- dimensional figure into a 2-dimensional net to find its surface area. Here is a net of a cube (prism with all sides congruent): We measure surface area in square units (cm 2, in 2, ft 2, etc) because we are measuring in 2 dimensions. To find the surface area of prisms and pyramids, consider the net of all the faces. Find the area of all the faces separately then add the areas together. To find the surface area of cylinders, cones, and spheres, we usually rely on the formulas given below. The last page of this packet shows you where some of the formulas come from and are left as an optional challenge. Surface Area of a Cylinder Surface Area of a Cone Surface Area of a Sphere SA = 2πr! + 2πrh SA = πr! + πrl SA = 4πr! SLANT HEIGHT (l): the distance measured along the lateral face (one of the sides) from the base to the apex (In the images on the right, h is the height and l is the slant height) Practice 1. Find the surface area of the following shapes. Be sure to include units in your answers when appropriate. a. A rectangular prism that measures 1 cm by 2cm by 4 cm 1 cm 4 cm 2 cm b. A triangular prism with the given dimensions:
c. An open-top triangular prism, as shown below. Give an exact answer (no decimals). d. The oblique cylinder shown below. 2. Find the total surface area (including the rectangular face) of a half cylinder with a radius of 5 inches and a height of 2 inches, as shown below. 3. Charlie s new hat has the dimensions shown. Find the amount of cloth needed to make the hat. 30 cm 32 cm open
4. A cone fits perfectly inside a cylinder with a height of 7 cm, as shown below. a. Find the surface area of the cylinder. 7 cm 55 b. Find the surface area of the cone. 5. Find the surface area of the solid shown below. Answers 1. a. 28 cm 2 b. 120 in 2 c. 144 + 66 3 d. 1170π 2. 20 + 35π in 2 3. 544π cm 2 4. a. 116.7π 366.5 cm 2 b. 65.9π 207 cm 2 5. 93π
Optional Challenge Problems Finding the formulas for cylinders and cones 1. Here you are given the net of a general cylinder with base radius r and height h. Notice that it is made up of two circles and a rectangle. a. What is the area of one of the circles (the base of the cylinder)? b. We know the height of the rectangle is h. What is the width of the rectangle? c. What is the area of the rectangle? d. What is the total surface area of the cylinder? 2. Below is the net of a right circular cone. Notice that it is made up of a circle and a sector of a circle. a. What is the area of the circle? b. What is the radius of the sector? c. What is the arc length of the sector, in terms of r? d. Knowing the arc length and the radius of the sector, find the angle measure or θ of the sector. e. Now that we know the radius and angle measure of the sector, find the area of the sector. f. Since the cone is made up of these two shapes, what is the surface area formula for a right circular cone?