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: a² + b² = c²

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

You can use the Pythagorean Theorem, a² + b² = c², 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.
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Areas of parallelograms and trapezoids

Area of a Parallelogram

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

Formula: A=bh

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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:

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Area of a trapezoid

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 = πr²

Example:

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Three-Dimensional Figures

Surface Areas of Prisms and Cylinders

Surface Area of a Rectangular Prism Using a Net
Rectangular Prism Net - Finding The Surface Area
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

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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

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Surface Areas of Pyramids and Cones

Surface Area of a Pyramid

The surface area of a rectangular pyramid is the sum of the area of the base, B, and one half the product of the base perimeter, P, and the slant height, l.


Equation: SA = B + 1/2Pl

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

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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.

Volume and Surface Area of a Cone
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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.

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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

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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

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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

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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

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Surface Area of a Sphere

Equation: A=4πr²

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