# Chapter 10

## Formulas needed

Area of a Parallelogram: A= bh

Area of a Trapazoid: A= 1/2(b1 + b2)h

Area of a Circle: A= pi times radius squared

Circumfrence: pi times d times 2

Area of a triangle: A= 1/2bh

Surface area: S = 2B + Ph

Surface area of a cylinders: bases= A= pi times radius squared, curved surface= A= 2 times pi times rh. All together= both formulas together

Surface area of a pyramid: S= B + 1/2 Pl ( l stands for the slant height) to find the slant height of you don't already have it you use a2 + b2= c2

Volume of a prism: V= Bh or V= lwh

Volume of a cylinder= V=Bh = pi times radius squared for the base

Volume of a pyramid= V= 1/3 Bh

Lateral Surfaces area of a cone= (base) A = pi times r squared (cone part) A= pi times rl

Volume of a sphere= V= 4/3 times pi times the radius to the third

## 10.1 Areas of Parallelograms and Trapezoids

The base of a parallelogram is the space that the parallelogram most likely sits on, the height is the perpendicular distance between the 2 bases. Finding the area of a parallelogram can be confusing so to make it easier to see "cut" the parallelogram into a right triangle and a trapezoid and move the triangle to the right or left side to form a rectangle. To find the area of the rectangle you have now created use the formula A= bh or area equals base times height in words. Make sure after solving you add the units and the squared symbol over the units, repeat this step for all of the objects you find the area for. Finding the area of a trapezoid, use the formula A= (b1=b2)h or the area equals b1 plus b2 times the height of the trapezoid. Follow the same steps used to find the area of the parallelogram but insert this formula instead. https://www.youtube.com/watch?v=s8OPoot0IUg

## 10.2 Areas of Circles

To find the area of a circle, you should use the formula area= pi times the radius squared. If the problem stated fails to give you one of the measurements besides the area it is usually given in the diameter or is to be solved for by using the corresponding formula. The diameter of a circle is the distance across the circle crossing through the center. The radius of a circle is half the diameter. The circumference of a circle is the distance around the circle, the quotient of the diameter and the circumference is represented by pi.

## 10.3 Three-Dimensional Figures

A solid is a three-dimensional figure that encloses part of a space. A polyhedron is a solid that is enclosed by polygons. A polyhedron has only flat surfaces and all of the polygons that make up that specific polyhedron are called faces. A prism is a polyhedron that has two congruent bases that lie in parallel planes. In a prism all of the other faces are rectangles. A pyramid is a polyhedron that has one base and all of its other faces are triangles. A cylinder is a solid that is not a polyhedron. It has two congruent bases that lie on parallel planes that are also circles. A cone is another example of a non-polyhedron solid. It has one base and a rectangular shape wrapped around it. A sphere is a solid that is formed when all of the points inside it are exactly the same distance from the center of the object. A cylinder is not a polyhedron because it has two circular bases and circles are not classified as polygons so therefore it cannot be considered a polyhedron. The segments that form when the faces of a polygon meet are called edges and a vertex is a point where three or more of these edges meet. ALL POLYHEDRONS ARE NAMED AFTER THE SHAPE OF THEIR BASE.

## 10.4 Surface Areas of Prisms and Cylinders

One way for or finding the surface area of a prism/ cylinder is to use a net. A net is a two-dimensional pattern that forms a solid when it is folded. Another way to find the surface area is to use the apropret formula that corresponds with the shape of the base of the object you are calculating the base for. The formula is S= 2B + Ph for a cylinder the formula is S= 2 times pi times r squared + 2 times pi times rh. Another thing you need to know how to find is the Lateral Surface area, which is the surface area of the figure NOT including the area of its bases.

## 10.5 Surface Areas of Pyramids and Cones

The slant height (h) of a pyramid is the perpendicular distance between the vertex and the base. The slant height (l ) of a rectangular pyramid is the height of a lateral face that is any face that isn't the base. To find the slant height you can use half of one of the sides of the base times the height in the Pythagorean therm to find the slant height, or use the formula B + 1/2 (4s)l ( 4s stands for the product of the number of triangles and the side length of the base ) You can use the net of a cone to find its surface area by using the formula A= pi times r squared + pi times rl, r is the radius of the base of the cone.

## 10.6 Volumes of Prisms and Cylinders

The volume of a container is the measure of the amount of space it occupies. All volume is measured in cubic units, one cubic unit is the amount of space occupies by a cube that measures one unit on each side. The formula for volume is V=Bh. To find the volume for a cylinder it is almost the same process except this time you substitute the big B for the formula pi times radius squared. https://www.youtube.com/watch?v=y3TjAHV7esk

## 10.7 Volumes of Pyramids and Cones

The volume s of a pyramid and a prism with the same base and the same height are very similar, however, the volume of a pyramid is exactly 1/3 the volume of a prism. Therefore, the formula for finding the volume of a pyramid is V= 1/3Bh. To find the volume of a cone it is a bit more complicated. This time, the cone and the cylinder are being compared that in the sense that the base and the height of both are the same the cone will be exactly 1/3 the volume of the cylinder. The formula for the volume of a cone is V= 1/3Bh = 1/3 pi times r squared times the height. https://www.youtube.com/watch?v=e7-am8JtREI