Unit 8: Applying Formulas Sections: 10-3, 10-5, 10-6, , 11-4, 11-5, and 11-6

Learning Target: I will be able to identify the appropriate formula for a figure and find the area, volume, surface area, or perimeter.

Area of Parallelograms THEOREM 10.1 – Area of a Rectangle The area of a rectangle is the product of its base and height. THEOREM 10.2 – Area of a Parallelogram The area of a parallelogram is the product of a base and a corresponding height.

Area of Triangle A triangle is half of a parallelogram. THEOREM 10.3 – Area of a Triangle The area of a triangle is half the product of a base and the corresponding height.

Area of Trapezoid THEOREM 10.4 – Area of a Trapezoid The area of a trapezoid is half the product of the height and the sum of the bases.

Area of Rhombus and Kite THEOREM 10.5 – Area of a Rhombus or a Kite The area of a rhombus or kite is half the product of the lengths of the diagonals.

Regular Polygons radius – the distance from the center to a vertex We can center any regular polygon inside of a circle: Regular polygons: -all sides congruent-all angles congruent-convex apothem – perpendicular distance from the center to a side

THEOREM 10.6 – Area of a Regular Polygon The area of a regular polygon is half the product of the apothem and the perimeter Areas of Regular Polygons

Regular decagon:

Words… Circle: the set of all points equidistant from a given point called the center (name a circle by its center) P radius diameter Central angle – an angle whose vertex is the center of the circle.

Arcs… Semicircle – an arc that is half the circle Minor arc – smaller than a semicircle Major arc – greater than a semicircle Minor Arc: Defined as the same as the measure of its corresponding central angle. Major Arc: This is 360° minus the degree measure of the minor arc that has the same endpoints as the major arc..

Arc Length… THEOREM 10.9: Circumference of a Circle The circumference of a circle is times the diameter. THEOREM 10.10: Arc Length

Area… THEOREM 10.11: Area of a Circle The area of a circle is the product of and the square of the radius. THEOREM 10.12: Area of a Sector of a Circle

Arc Length… Find the length of each arc shown in red. Leave your answer in terms of.

Area… Find the area of sector ZOM. Leave your answer in terms of.

Prisms Prism – A 3-dimensional figure with two congruent, parallel faces, called bases. Lateral Faces – Faces that are not bases

Surface Area of Prisms THEOREM 11.1 – Surface Area of Prism The surface area of a prism is the sum of the lateral area and the area of the two bases. 4 in. 12 in. Regular Pentagon

Surface Area of Cylinder THEOREM 11.2 – Surface Area of Cylinder The surface area of a cylinder is the sum of the lateral area and the area of the two bases.

Volume of Prisms 4 cm 6 cm 12 cm THEOREM 11.6 – Volume of Prism The volume of a prism is the product of the area of a base and the height of the prism. Area of the base times the height.

Volume of Cylinders THEOREM 11.7 – Volume of Cylinder The volume of a cylinder is the product of the area of the base and the height of the cylinder. Find the volume of the following cylinder in terms of π.

Pyramids Pyramid – A 3-D figure with one face (the base) that is any polygon and the other faces (the lateral faces) are triangles that meet at a common vertex. Regular Pyramid – Base is a regular polygon. Slant Height ( l ) – The length of an altitude of a lateral face of the pyramid. How are the height and slant height related to the edge of the base of a pyramid???

Surface Area of Pyramids THEOREM 11.3 – Surface Area of Regular Pyramid The surface area of a regular pyramid is the sum of the lateral area and the area of the base.

Cones Cone – A 3-D figure with a circular base and a curved lateral surface How are the radius, height, and slant height related???

Surface Area of Cones THEOREM 11.4 – Surface Area of Cone The surface area of a cone is the sum of the lateral area and the area of the base. Find the surface area of the following cone.

Volume of Pyramids THEOREM 11.8 – Volume of Pyramid The volume of a pyramid is one third the product of the area of a base and the height of the pyramid. The Pyramid arena in Memphis, TN has a base of area 300,000 ft 2. Its height is 321 ft. What is the volume of the pyramid?

Volume of Cones THEOREM 11.9 – Volume of Cone The volume of a cone is one third the product of the area of the base and the height of the cone. Find the volume of the following cone in terms of π.

THEOREM – Surface Area of a Sphere The surface area of a sphere is four times the product of pi and the square of the radius of the sphere. Surface Area and Volume of a Sphere THEOREM 11.11– Volume of a Sphere The volume of a sphere is four thirds the product of pi and the cube of the radius of the sphere.

Four Types of Rigid Motion A rigid motion is the action of taking an object and moving it to a different location without altering its shape or size. 1.Translation (slide) 2.Reflection (about a line or an axis) 3.Dilation (little/big) 4.Rotation (clockwise or counterclockwise)