Geometry B: Unit 6

Using a Scale Factor

The lateral areas of two similar paint cans are 1019cm^2 and 425cm^2. The volume of the smaller can is 1157cm^3. What is the volume of the larger can? Step 1: Find the scale factor a:b. 1.a^2/b^2=1019/425 The ratio of the surface areas is a^2:b^2. 2. a/b=sqrt1019/sqrt425 Take the positive square root of each side. Step 2: Use the scale factor to find the volume. 1. V large/V small=(sqrt1019)^3/(sqrt425)^3 The ratio of the volumes is a^3/b^3. 2. V large/V small=(sqrt1019)^3/(sqrt425)^3 Solve for V small. 3. V large=1157*(sqrt1019)^3/(sqrt425)^3 Solve for V large. 4. V large=4295.475437 Use a calculator.

Finding the Scale Factor

The square prisms at the right are similar. What is the scale factor of the smaller prism to the larger prism. a^3/b^3=729/1331 The ratio for the volumes is a^3:b^3. a/b=9/11 Take the cube root of each side. The scale factor is 9:11.

Theorem 11-10 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. S.A.=4*pi*r^2

Two Similar Prisms

The two similar prisms shown here suggest two important relationships for similar solids. Smaller Prism: 3m(height), 1m(width), and 2m(length). S.A.=22m^2 Volume=6m^3 Larger prism: 6m(height) 2m(width), and 4m(length). S.A.=88m^2 V=48m^3 The ratio of the side lengths is 1:2. The ratio of the surface areas is 22:88, or 1:4. The ratio of the volumes is 6:48, or 1:8. The ratio of the surface areas is the square of the scale factor. The ratio of the volumes is the cube of the scale factor. These two facts apply to all similar solids.

Theorem 11-9 Volume of a Cone

The volume of a cone is one third the product of the area of the base and the height of the cone. V=1/3*B*h, or V=1/3*pi*r^2*h

Theorem 11-7 Volume of a Cylinder

The volume of a cylinder is the product of the area of the base and the height of the cylinder. V=B*h, or V=pi*r^2*h

Theorem 11-6 Volume of a Prism

The volume of a prism is the product of the area of the base and the height of the prism. V=B*h

Theorem 11-8 Volume of a Pyramid

The volume of a pyramid is one third the product of the area of the base and the height of the pyramid. V=1/3*B*h

Using Volume to Find Surface Area

The volume of a sphere is 5000 m^3. What is its surface area to the nearest square meter? Step 1. Find the radius of the sphere. 1. V=4/3*pi*r^3 Use the formula for volume of a sphere. 2. 5000=4/3*pi*r^3 Substitute. 3. 5000(3/4*pi)=r^3 Solve for r^3. ^3(?) sqrt 5000 (3/4*pi) = r Take the cube root of each side. r=10.60784418 Step 2. Find the surface area of the sphere. 1. S.A.=4*pi*r^2 Use the formula for surface area of a sphere. 2. =4*pi*ANS (radius from Step 1)(^2) Enter Use a calculator. 3. =1414.04792

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. V=4/3*pi*r^3

Cross Section

Two-dimensional drawing showing a plane intersecting with a three-dimensional object.

Euler's Formula

V-E+F=2

Volume

Volume is the space that a figure occupies. It is measured in cubic units such as cubic inches (in.^3), cubic feet (ft^3), or cubic centimeters (cm^3). The volume V of a cube is the cube of the length of its edge, or V=e^3.

Great Circle

When a plane and a sphere intersect in more than one point, the intersection is a circle. If the center of the circle is also the center of the sphere, it is called a great circle.

Finding the Volume of a Rectangular Prism

1. V=B*h Use the formula for the volume of a prism. 2. =480*10 The area of the base B is 24*20, or 480 cm^2, and the height is 10 cm. 3. =4800 Simplify.

Finding the Volume of a Cylinder

1. V=pi*r^2*h Use the formula for the volume of a cylinder. 2. =pi(3)^2(8) Substitute 3 for r and 8 for h. 3.=pi(72) Simplify.

Surface Area of a Rectangular Prism

2ab+2bc+2ac a, b, and c are the lengths of the 3 sides. a=length, b=height, c=width

Surface Area of a Cube

6a^2 a is the length of the side of each edge of the cube.

Hemispheres

A great circle divides a sphere into two hemispheres.

Polyhedron

A polyhedron is a space figure. A polyhedron has surfaces, called faces, which are polygons.

Prism

A prism is a polyhedron that has two congruent faces that are parallel. The congruent faces of a prism are called bases, and the other faces are called lateral faces.

Radius

A radius is a segment that has one endpoint at the center and the other endpoint on the sphere.

Diameter

A segment passing through the center with endpoints on the sphere.

Sphere

A sphere is the et of all points in space equidistant from a given point called the center.

Composite Space Figure

A three-dimensional figure that is the combination of two or more simpler figures. You can find the volume of a composite space figure by adding the volumes of the figures that are combined.

Vertex

A vertex is a point where three or more edges intersect.

Edge

An edge is a segment that is formed by the intersection of two faces.

Face

Each polygon is a face of the polyhedron

Theorem 11-12 Areas and Volumes of Similar Solids

If the scale factor of two similar solids is a:b, then -the ratio of their corresponding areas is a^2:b^2 -the ratio of their volumes is a^3:b^3.

Theorem 11-5 Cavalieri's Principle

If two space figures have the same height and the same cross-sectional area at every level, then they have the same volume.

Relationship Among the Faces, Vertices, and Edges

If you add 2 to the number of edges in a polyhedron, it will equal the sum of the number of faces and vertices. This relationship is called Euler's Formula because it was discovered by the Swiss mathematician Leonhard Euler in the 1700s.

Finding Surface Area (Circumference Known)

Q: Earth's equator is about 24,902 mi long. What is the approximate surface area of Earth? Round to the nearest thousand square miles. Step 1. Find the radius of Earth. 1. C=2*pi*r Use the formula for circumference. 2. 24,902=2*pi*r Substitute 24,902 for C. 3. 24,902/2*pi=r Divide each side by 2*pi. 4. r=3963.276393 Use a calculator. Step 2. Use the radius to find the surface area of Earth. 1. S.A.=4*pi*r^2 Use the formula for surface area. 2. =4*pi ANS (radius from Step 1) * (^2) Enter. 3. 197387017.5

Finding the Volume of a Triangular Prism

Step 1. Find the area of the base of the prism. Each base of the triangular prism is an equilateral triangle, as shown at the right. An altitude of the triangle divides it into tow 30-60-90 degree triangles. The height of the triangle is sqrt 3*shorter leg, or 4 sqrt 3. 1. B=1/2*b*h Use the formula for the area of a triangle. 2. =1/2(8)(4 sqrt 3) Substitute 8 for b and 4 sqrt 3 for h. 3. =16 sqrt 3 Simplify. Step 2. Find the volume of the prism. 1. V=b*h 2. =16 sqrt 3*10 Substitute 16 sqrt 3 for B and 10 for h. 3. =160 sqrt 3 Simplify. 4. =277.1281292 Use a calculator. Or V=l*w*h*1/2

Circumference

The circumference of a great circle is the circumference of the sphere.

Theorem 11-2 Lateral and Surface Aras of a Cylinder

The lateral area of a right cylinder is the product of the circumference of the base and the height of the cylinder. L.A.=2*pi*r *h, or L.A.=pi*d*h The surface area of a right cylinder is the sum of the lateral area and the areas of the two bases. S.A.=L.A.+2B, or S.A.=2*pi*r*h+2*pi*r^2

Theorem 11-1 Lateral and Surface Areas of a Prism

The lateral area of a right prism is the product of the perimeter of the base and the height of the prism. L.A.=p*h perimeter(p)= (2w+2l)*h The surface area of a right prism is the sum of the lateral area and the areas of the two bases. S.A.=L.A.+2B