GEOMETRY UNIT

Perimeter

Vocabulary Perimeter : “peri” means around, so perimeter means the distance around a plane figure.

Perimeter Formula for finding perimeter: (look on back of math journal) Square = 4s means 4 times the side length Rectangle = 2l + 2w means the length times 2 plus the width times 2 Irregular figure = find the sum of all the sides

Perimeter Find the perimeter of the square:

Perimeter Find the perimeter of the rectangle:

Perimeter Find the perimeter of the rectangle:

Perimeter Find the perimeter of the square:

Perimeter Find the perimeter of the square:

Perimeter Find the perimeter of the rectangle:

Perimeter Find the perimeter of this irregular figure:

Perimeter Find the perimeter of this irregular figure:

Perimeter Find the perimeter of this irregular figure:

Perimeter Find the perimeter of this irregular figure:

Perimeter Find the missing side lengths of this rectangle using the given perimeter:

Area of a parallelogram Objective: Find the area of a parallelogram using the formula

Vocabulary Parallelogram: a four sided figure in which opposite sides are parallel and congruent Parallel: lines that never cross or touch Congruent: exact same; Complete Copy!

Area Formula for finding area of a parallelogram: Area = Base x Height ( a = bh ) Base: the bottom side length Height: the distance from the base straight up to the top. 2 ** Answer is always in units squared** Example = 43 in

Area Find the area of this parallelogram: A = B x H A = 6 x 6 2 A = 36 in.

Area Find the area of this parallelogram: A = bh A = 6 x 15 A = 90 ft. 2 A = 90 ft.

Area Find the area of the parallelogram: A = bh A = 7 x 11 A = 77 ft. 2 A = 77 ft.

Area Find the area of the parallelogram:

Area Find the area of the parallelogram:

Area of irregular shapes

Vocabulary Area: How much space is in the middle of a flat figure. Always answered in units squared

Area To find the area using units, just count up all the squares that are inside the flat shape. A = 15 units 2

Area Find the area of the irregular shape in units:

Area Find the area of the irregular figure using units:

Area Find the area of the irregular figure using units:

Area Now we need to find area of figures using other measurements (not units). To do this, we have to separate the irregular figure into separate parallelograms so that we can use that formula, do you remember what that formula is? If you said that Area = Base x Height, or A = bh; you are correct! We have to remember and use that formula to find area of irregular figures!

Area Steps Find the area of this irregular shape: 1. Separate into parallelograms. 2. Find the base and height of every parallelogram you made. 3. Find the area of each parallelogram and then add the answers together. 2 So…. 4 x 4 = 16 and 10 x 5 = 50 so my total answer is 66 in

Area Find the area of this irregular figure: A = base x height A = (1 x 1) + (10 x 10) 2 A = 101 yd

Area Find the area of this irregular figure: A = Base x Height A = (3 x 4) + (3 x 8) 2 A = 36 ft

Area Find the area of this irregular figure: A = 87 cm 2

Area Find the area of this irregular polygon:

Area of a Triangle

Vocabulary Area: The measure of how much space is inside an object. (example: how many feet are in your fenced backyard?) Triangle: A 3-sided, plane figure Height: The distance from the base (straight up or perpendicular to the base) to highest part of the triangle.

Area Before we can look at how to find the area of a triangle, we have to first figure out what a triangle is. What is a triangle??? A triangle (any triangle) is exactly HALF of a parallelogram.

Area Using what we know about what a triangle is, how can we find the area of a triangle? (We can use the formula for area of a parallelogram, but just have to add one extra simple step.) For a parallelogram we know that Area = Base x Height…. Therefore, for a triangle, we can do Area = Base x Height, and then cut the answer in half or divide that answer by 2. So…. Area of a triangle is Base x Height A = BH 2 2

Area Find the area of this triangle: Remember, for a triangle we use………. A = B x H 2 Let’s plug in our numbers into the formula now……. A = 8 x 10 2 So our Area = 80/2, which simplifies to 40 m. 2

Area Find the area of this triangle: A = 12 in. Remember, for a triangle we use…. A = B x H 2 A = 12 in. 2

Area Does this rule work for ALL triangles, or just right triangles??? Let’s try it…. YES, EVERY triangle is exactly half of a parallelogram, so this formula works for ALL triangles!! Remember, for the area of a triangle, the formula is A = BH 2

Area Find the area of this triangle: (Just don’t forget which number the height is) 2 A = 16 ft.

Area Find the area of the triangle: 2 A = 20 ft.

Area Find the area of the triangle: 2 A = 14 m.

Area Find the area of this equilateral triangle: 2 A = 15 yd.

Area Find the area of the triangle: 2 A = 20 cm.

Circumference

Circumference Objective: Find the circumference of a circle Vocab: Radius: a line segment that connects the center of the circle, to any point on the circle. Diameter: a line segment that passes through the center of the circle, and touches two points on the circle. Circumference: distance around the circle

Circumference Let’s see how the diameter and the circumference are related: Using the string and the circle, cut one string that is the diameter of the circle. Then, cut another string that is the circumference of the circle. How are they related?

Circumference How many times bigger is the circumference string than the diameter string? About 3 times (and a little more) longer! That distance is called pi (π) and it equals 3.14 or 22/7. This distance is the same for EVERY circle.

Circumference We need all that information to find pi (π). There are two formulas to find π. (They actually mean the exact same thing, it just depends on which information you already have on which formula you use.)

Circumference For which circle would we use which formula? C = 2πr or C = πd

Circumference Let’s try one, find the circumference of the circle: We’ll use C = 2πr this time. C = 2π3 C = 6π C = 18.84 M 3 M

Circumference Find the circumference of the circle: We’ll use C = 2πr this time. C = 2π10 C = 20π C = 62.8 ft 10 ft

Circumference Find the circumference of the circle: We’ll use C = 2πr this time. C = 2π8.4 C = 16.8π C = 52.752 in 8.4 in

Circumference Find the circumference of the circle: We’ll use C = πd this time. C = π15 C = 47.1 in 15 in

Circumference Find the circumference of the circle: We’ll use C = πd this time. C = π6.3 C = 19.782 yd 6.3 yd

Circumference Find the circumference of the circle: ¾ km We’ll use C = πd this time. C = π3/4 C = 33/7 Which simplifies to 4 5/7 km ¾ km

Area of a Circle

Area of a circle Topic: Calculate the area of a circle. Vocab: Area: The amount of space an object takes up Pi: 3.14 or 22/7 Diameter: Line that passes through center of circle Radius: Line segment from center to outside of circle

Area of a circle When finding the area of a circle, we are finding how much space the circle takes up: To find the area of a circle, there is a formula. It is, A = πr² Which means, pi x radius x radius

Area of a circle Practice: Let’s find the area of this circle together: A = πr² A = π3² A = π9 3 M A = 28.26 M ² (remember the ² because it’s AREA!)

Area of a circle Find the area of this circle: A = πr² A = π6² A = π36 A = 113.04 ft ²

Area of a circle Find the area of this circle: A = πr² A = π8² A = π64 8 yd ² A = 200.96 ft

Area of a circle Find the area of this circle: A = πr² A = π¼² ¼ yd ² A = 11/56 yd

Area of a circle Find the area of this circle: A = πr² A = π2² A = π4 For the formula, we need the radius but we only have the diameter; what can we do? If you have the diameter, you have to take half of it, which is the radius! A = πr² A = π2² A = π4 4 yd ² A = 12.56 yd

Area of a circle Find the area of this circle: A = πr² A = π5² A = π25 10 mm ² A = 78.5 mm

Area of a circle Find the area of this circle: A = πr² A = π15² 30 km ² A = 706.5 km

Area of a circle Find the area of this circle: A = πr² A = π7/8² 7/8 yd ² A = 2 13/32 yd

Solid Figures

Solid Figures Objective: Learn the names of solids. (Icosahedron)

Vocabulary Solid Figure: a three-dimensional figure in space. Prism: a solid figure that has two parallel congruent bases and parallelograms for faces. Pyramid: a solid figure whose base can be any polygon and whose faces are triangles that meet at a point called an apex. Cylinder: a solid with two circular faces that are congruent and parallel.

Solid Figures Organize your solids into those three groups which we just learned: Which ones are….. Prisms? Pyramids? Cylinders? Any that don’t fit into any of those three??

Solid Figures Let’s name all our solid figures that we made: Cylinder

Solid Figures Square Pyramid

Solid Figures Cone

Solid Figures Triangular Prism

Solid Figures Rectangular Prism

Solid Figures Cube (Can also be called a square prism)

Solid Figures Triangular Pyramid

Solids Objective: Learn the parts of a solid. Vocab Base: bottom side of a figure Face: flat surface of a figure Edge: segment where two faces meet Vertex: point at which three edges meet.

Solid Figures BASE: The bottom side of a solid figure. Parts of a solid figure: BASE: The bottom side of a solid figure.

Solid Figures Face: a flat surface of a solid figure. Parts of a solid figure: Face: a flat surface of a solid figure. How many faces does our rectangle prism have? The square pyramid?

Solid Figures Edges: the line segment where two faces meet. Parts of a solid figure: Edges: the line segment where two faces meet. How many edges do these shapes have?? How about a cylinder??

Solid Figures Vertex: a corner, or point, at which three edges meet. Parts of a solid figure: Vertex: a corner, or point, at which three edges meet. (The top of a pyramid is called an apex, but it can also be counted as a vertex) How many vertices do these solid figures have?

Solid Figures Count how many of each part are on the given solid figure: What shape is the base? rectangle How many faces? 6 How many edges? 12 How many vertices? 8

Solid Figures Count how many of each part are on the given solid figure: What shape is the base? square How many faces? 5 How many edges? 8 How many vertices? 5

Solid Figures Count how many of each part are on the given solid figure: What shape is the base? circle How many faces? 2 How many edges? How many vertices?

Solid Figures Count how many of each part are on the given solid figure: What shape is the base? No base How many faces? How many edges? How many vertices?

NETS

Vocabulary Net: a flat pattern that can be folded to make a solid figure.

Nets There can be more than one net that can make a solid figure. You’ve got to visualize what all the folding would make!

Nets

Nets Rectangular prism What solid figure will this net make? We can tell there is a rectangle base, and that the sides are parallelograms….what solid does that sound like? Rectangular prism

Nets Square Pyramid What solid figure will this net make? We can tell there is a square base and triangular sides….what solid does that sound like? Square Pyramid

Nets What solid figure will this net make? Rectangular Prism

Nets What solid figure will this net make? Triangular Prism

Nets What solid figure will this next make? Hexagonal Prism

Nets What solid figure will this net make? Pentagonal Pyramid

Nets Let’s go backwards now….given the solid figure, draw what you think the next would look like!

Nets ….given the solid figure, draw what you think the net would look like!

Volume Objective: Find the volume of a solid

Vocabulary Volume: the amount of space a solid figure takes up. Cubic units: what volume is measured in. (This means that all answers for volume are measured using units cubed, not squared like area.) Example = 5 cm. 3

Volume To find out why our answer is cubed, we have to look at the formula for finding the volume of a solid figure. If you see on your math journal, the formula for finding the volume of a solid figure is Volume = Length x Width x Height. ….or V = LWH We are multiplying planes of a figure together, therefore, our answer is in units CUBED!

Volume V = 5 x 3 x 8 Let’s do an example volume problem together… Remember Volume = Length x Width x Height. V = 5 x 3 x 8 Does it matter which order these numbers are in? 3 V = 120 in.

Volume Let’s practice one that uses units… Remember Volume = Length x Width x Height. What’s the length? 4 What’s the width? 4 What’s the height? 4 What’s the answer for the volume of this solid? 3 V = 64 units

Volume Volume = 70 ft. Remember Volume = Length x Width x Height. Find the volume of this solid! Remember Volume = Length x Width x Height. Volume = 70 ft. 3

Volume Find the volume of this solid. Remember Volume = Length x Width x Height. V = 2 x 5 x 10 3 V = 100 yd.

Volume Now, if the solid is a triangular prism, just like with area, we have to cut it in half! So….if and ONLY if it’s a triangular prism, we have this formula Volume = Base x Height x Width 2 Just remember to find the normal volume and then cut your answer in half because a triangle is half of a parallelogram!

Volume V = 2 x 4 x 3 2 Find the volume of this triangular prism: Remember we do the normal formula for volume, then since it’s a triangle, we have to cut our answer in half! V = 2 x 4 x 3 2 3 V = 12 m.

Volume Find the volume of this triangular prism: V = LWH 2 V = 160 2 V = 80 cm. 3

Volume Find the area of this triangular prism: 3 V = 135 in.

Volume Find the volume of this solid figure: Hint: Find the volume of the rectangle prism first, then find the volume of the triangular prism second. Finally, add both volume’s together for your final answer. Rectangular Prism….. V = 6 x 2 x 7 = 84 Triangular Prism….. V = 5 x 2 x 7 = 35 2 84 + 35 = 3 Total Solid Volume…. V = 119 ft.

Surface Area of a rectangular prism

Surface Area Objective: Find the surface area of a rectangular prism.

Vocabulary Surface Area: the total area of every face of a solid Face: a flat side of a solid Area: Base x Height Surface Area is measured in units squared!

Surface Area Finding the surface area isn’t hard; there are just many steps and if you don’t organize yourself, you will make it very hard. The first thing you need to do to find surface area is first, figure out how many faces you have. Second, label each face… STEPS 1) How many faces? 6 2) Label each face top left side bottom front right side back 3) Make sure you have the same amount of labels as faces!

Surface Area STEPS (continued): 4) Now make a chart of all the faces: Top Bottom Front Back Left side Right side

Area = Base x Height Top 6 9 54 Bottom 6 9 54 Front 6 4 24 Back 24 6 4 Then add all your face area’s together, and that will be your total solid figure surface area! Left side 9 4 36 Right side 9 4 36 2 Surface area = 228 cm.

Surface Area What are the steps we need to take to find the surface area?

A = B x H Now what do we do? Top 2 6 12 Bottom 2 6 12 Front 2 10 20 Back 2 10 20 Left side 6 10 60 Now what do we do? Right side 6 10 60 2 Surface Area = 184 cm.

Surface Area Go through your steps…….6 faces…....6 labels……..now I’ve got to make my chart….

Top Bottom Front Back Left side Right side

Surface area of a triangular prism

Vocabulary Surface area: The sum of the area of all the faces on a solid. Area of a rectangle = B x H (A=BH) Area of a triangle = B x H/2 (A=BH/2)

Surface Area The steps to find area of a triangular prism is the same with a rectangular prism…. What is step one? Count how many faces Step two? Label each face (top, bottom, etc…) Step three? Make sure you have the same amount of labels as faces! Step four? Make a chart of all the faces

Surface Area Surface Area = 201 ft. What are my steps to find the surface area of this triangular prism?? LEFT SIDE 6 9 54 BOTTOM 5 9 45 FRONT 5 6 15 (Remember, when the face is a triangle, you have to cut that area in half, or divide it by 2) BACK 5 6 15 2 Surface Area = 201 ft. RIGHT 8 9 72

Surface Area Find the surface area of the triangular prism LEFT SIDE 5 8 40 BOTTOM 3 8 24 FRONT 3 4 6 BACK 3 4 6 RIGHT SIDE 4 8 32 2 Surface Area = 108 in.

Surface Area Find the surface are of the triangular prism. LEFT SIDE 2 10 20 BOTTOM 8 10 80 FRONT 2 8 8 BACK Surface Area = 206 m. 2 2 8 8 RIGHT SIDE 9 10 90

Surface Area Find the surface area of this triangular prism: LEFT SIDE 4 10 40 BOTTOM 8 10 80 FRONT 4 4 8 BACK 4 4 8 2 Surface Area = 176 yd. RIGHT SIDE 4 10 40

Surface Area Find the surface area of this triangular prism: Surface Area = 140 cm. 2

Surface Area Surface Area = 238 m. Find the surface are of the triangular prism: Surface Area = 238 m. 2

Volume of a cylinder Objective: Find the Volume of a cylinder. Vocabulary: Volume: the amount of space a solid figure takes up. Pi or π: 3.14 Radius: distance from the center of a circle straight to the outside. Height: how tall a shape is Volume= πr² x h

Volume of a cylinder Volume = πr² x h 628 ft. π x r² x h 3.14 x 5² x 8 Let’s do a practice one together: 5 ft What’s the formula again? Volume = πr² x h π x r² x h 8 ft. 3.14 x 5² x 8 3.14 x 25 x 8 3.14 x 200 3 628 ft.

Volume of a cylinder Volume = πr² x h 602.88 in. π x r² x h Let’s do another one: 4 in What’s the formula again? Volume = πr² x h π x r² x h 12 in 3.14 x 4² x 12 3.14 x 16 x 12 3.14 x 192 3 602.88 in.

Volume of a cylinder Volume = πr² x h 7,850 mm. π x r² x h Let’s do another one: 10 mm What’s the formula again? Volume = πr² x h π x r² x h 25 mm 3.14 x 10² x 25 3.14 x 100 x 25 3.14 x 2,500 3 7,850 mm.

Volume of a cylinder Volume = πr² x h 863.5 m. π x r² x h Let’s do another one: 10 m What’s the formula again? Volume = πr² x h π x r² x h 3.14 x 5² x 11 11 m **Why is it 5 instead of 10?** 3.14 x 25 x 11 3.14 x 275 3 863.5 m.

Volume of a cylinder Volume = πr² x h 3818.24 miles π x r² x h Let’s do another one: 16 miles What’s the formula again? Volume = πr² x h π x r² x h 3.14 x 8² x 19 19 miles 3.14 x 64 x 19 3.14 x 1216 3 3818.24 miles

Surface Area of a Cylinder Objective: Find the surface area of a cylinder. Vocabulary: Surface Area: The area of all the sides added together. Pi or π: 3.14 Radius: distance from the center of a circle straight to the outside. (half the diameter) Height: how tall a shape is Surface Area= 2πr² + 2πrh

Surface Area of a cylinder Let’s do a practice one together: 5 ft What’s the formula again? Surface Area= 2πr² + 2πrh 2 x π x r² + 2 x π x r x h 8 ft. 2 x 3.14 x 5² + 2 x 3.14 x 5 x 8 2 x 3.14 x 25 + 2 x 3.14 x 40 3.14 x 50 + 80 x 3.14 157 + 251.2 408.2 ft. 2

Surface Area of a cylinder Let’s do a practice one together: What’s the formula again? 8 in. Surface Area= 2πr² + 2πrh 2 x π x r² + 2 x π x r x h 2 x 3.14 x 8² + 2 x 3.14 x 8 x 11 2 x 3.14 x 64 + 2 x 3.14 x 88 11 in. 3.14 x 128 + 176 x 3.14 401.92 + 552.64 2 954.56 in.

Surface Area of a cylinder Let’s do a practice one together: What’s the formula again? 9 yd. Surface Area= 2πr² + 2πrh 2 x π x r² + 2 x π x r x h 2 x 3.14 x 9² + 2 x 3.14 x 9 x 15 2 x 3.14 x 81 + 2 x 3.14 x 135 15 yd. 3.14 x 162 + 270 x 3.14 508.68 + 847.8 2 1,356.48 yd.

Surface Area of a cylinder Let’s do a practice one together: What’s the formula again? Surface Area= 2πr² + 2πrh 10 m. 2 x π x r² + 2 x π x r x h (What does r equal on this one since we have a diameter?) 2 x 3.14 x 5² + 2 x 3.14 x 5 x 12 2 x 3.14 x 25 + 2 x 3.14 x 60 12 m. 3.14 x 50 + 120 x 3.14 157 + 376.8 2 533.8 m.