# Chapter 10

### Alayna Miller

## List of all equations

**Area of a parallelogram:**A = b*h

**Area of a trapezoid:**A = ½ (b1 + b2) h

**Area of a circle:**A = πr²

**Surface area of a prism:**SA = 2B + Ph (B is the area of the base. P is the perimeter of the base).

**Surface area of a cylinder:**SA = 2B + Ch (B is the area of the base, πr². C is the circumference of the base, which would equal 2πr. So it would be SA = 2πr² + 2πrh ).

**Surface area of a pyramid:**SA = B + ½Pl (B is the area of the base. P is the perimeter of the base).

**Surface area of a cone:**SA = πr² + πrl

**Surface area of a sphere: **SA=4πr²**Volume of a prism: **V = Bh (B is the area of the base).**Volume of a cylinder: **V = Bh (B is the area of the base, which is πr². So it's V = πr²h).**Volume of a pyramid: **V = ⅓Bh (B is the area of the base).**Volume of a cone: **V = ⅓Bh (B is the area of the base, which would be πr². So, it's

V = ⅓πr²h

**Volume of a sphere: **V = 4/3πr³**Slant height: ** *a*² + *b*² = *c*²

## Vocabulary

**Area:** The number of square units covered by a figure.

**Base:** The surface a solid object stands on, or the bottom line of a shape such as a triangle or rectangle.

**Height:** Height is a measure of a polygon or solid figure, taken as a perpendicular from the base of the figure.

**Circle:** A 2-dimensional shape made by drawing a curve that is always the same distance from a center.

**Radius: **The distance from the center to the edge of a circle. It is half of the circle's diameter.

**Pi:** The ratio of a circle's circumference to its diameter. Equal to 3.14159265358979323846... (the digits go on forever without repeating)

**Trapezoid:** A 4-sided flat shape with straight sides that has a pair of opposite sides parallel.

**Parallelogram: **A 4-sided flat shape with straight sides where opposite sides are parallel.

**Rhombus: **A 4-sided flat shape with straight sides where all sides have equal length. Also opposite sides are parallel and opposite angles are equal.

**Diameter:** A straight line going through the center of a circle connecting two points on the circumference.

**Circumference: **The circle that passes through all vertices (corner points) of a regular polygon. Its radius is also the radius of the regular polygon.

**Solid:** A three dimensional (3D) object. The 3 dimensions are called width, depth and height. Examples include, spheres, cubes, pyramids and cylinders.

**Prism: **A solid object with two identical ends and flat sides. The sides are parallelograms (4-sided shape with opposites sides parallel). The cross section is the same all along its length. The shape of the ends give the prism a name, such as "triangular prism" It is also a polyhedron.

**Cylinder: **A solid object with two identical flat ends that are circular or elliptical and one curved side. It has the same cross-section from one end to the other.

**Cone:** A solid (3-dimensional) object with a circular base and one vertex.

**Sphere:** A 3-dimensional object shaped like a ball. Every point on the surface is the same distance from the center.

**Net: **A pattern that you can cut and fold to make a model of a solid shape.

**Surface Area: **The total area of the surface of a three-dimensional object. Example: the surface area of a cube is the area of all 6 faces added together.

**Slant Height:** The diagonal distance from the apex of a right circular cone or a right regular pyramid to the base.

**Volume:** The amount of 3-dimensional space an object occupies. Capacity.

**Pyramid: **A solid object where the base is a polygon (a straight-sided flat shape) and the sides are triangles which meet at the top (the apex).

**Polyhedron:** A solid that is enclosed with polygons.

**Faces of a polyhedron:**A polygon that is a side of the polyhedron.

**Edges of a polyhedron:** A line segment where two faces of a polyhedron meet.

**Vertices of a polyhedron:** Where the corners of two or more faces meet. (A corner).

## Finding Slant Height

*a*² +

*b*² =

*c*², to calculate the slant height. For both cones and pyramids,

*a*will be the length of the altitude and

*c*will be the slant height. For a cone,

*b*is the radius of the circle that forms the base. The radius is the distance from the center of the base to the edge of the base. For a pyramid,

*b*is half the length of the side of the square that forms the base. A and B are the "legs" of the triangle, and form the right angle.

## Areas of parallelograms and trapezoids

**Area of a P**

**arallelogram**

The area of a parallelogram is found when you multiply the base and the height.

Formula: * A=bh*

**Area of a Trapezoid**

The area of a trapezoid is one half the product of the sum of the bases and the height.

Equation: *A = 1/2( b1 + b2 )h*

Example:

## Areas of Circles

**Area of a Circle**

The area of a circle is the product of π and the square of the radius.

Formula: *A = *π*r*²

Example:

## Three-Dimensional Figures

## Surface Areas of Prisms and Cylinders

**Surface Area of a Rectangular Prism Using a Net**

**Surface Area of a Rectangular Prism using Algebra**

The surface area of a prism is the the sum of the twice the area of the base, B, and the product of the base's perimeter P and the height h.

Formula: S = 2B + Ph

----------The B changes depending on the base.

To the right: The Perimeter is the BASE'S perimeter

**Surface Area of a Cylinder**

The surface area of a cylinder is the sum of twice the area of __A__ base, B, and the product of the base's circumference, C, and the height, H.

Equation: SA = 2B + Ch. B is the area of the base, πr². C is the circumference of the base, which would equal 2πr. So it would be SA = 2πr² + 2πrh

## Surface Areas of Pyramids and Cones

**Surface Area of a Pyramid**

Equation: SA = B + 1/2Pl

B stands for the area of the base, and P stands for the perimeter of the base.

**Surface Area of a Cone**

The surface area of a cone is the sum of the area of the circular base with radius, r, and the product of pi, the radius r of the base, and the slant height l.

Equation: SA = πr² + πrl

In the video, s is the same as l in the equation above.

## Volumes of Prisms and Cylinders

**Volume of a Prism**

The volume of a prism is the product of the area of the base, B, and the height, h.

Equation: V = Bh

B varies based on the shape of the base.

**Volume of a Cylinder**

The volume of a cylinder is the product of the area of the base, B, and the height, H.

Equation: V = Bh

(B is really); πr²

Full equation: V = (πr²)h

## Volumes of Pyramids and Cones

**Volume of a Pyramid**

The volume, V, of a pyramid is one third the product of the area of the base, B, and the height, h.

Equation: V = 1/3Bh

**Volume of a Cone**

The volume of a cone, V, is one third the product of the area of the base, B, and the height, h.

Equation: V = 1/3Bh

*simplified: V = 1/3(πr²)h

## Volume of a Sphere

**Volume of a Sphere**

The volume, V, of a sphere is four thirds the product of π and the cube of the radius, r.

Equation: V = 4/3πr³

**Surface Area of a Sphere**

Equation: A=4πr²