HOW I FOUND THE VOLUME AND SURFACE AREA [an example of how Caroline found the volume and surface area of her chopsticks] FINDING VOLUME OF THE CHOPSTICKS To find the volume of my chopsticks, I imagined that the chopsticks were two trapezoidal prisms put together with the following measurements: One trapezoidal prism will have a height of 0.3 cm and the other trapezoidal prism on top will have a height of 0.1 cm. The trapezoidal prism on the top part of the chopsticks looks like this and has the following measurements: Height of prism = 0.1 cm Trapezoidal base is 20.2 cm long, and 0.6 cm wide on one side and 0.35 cm wide on the other side. To find the volume of a 3D object, you must find the area of the base first and then multiply it by the height of the object. To find the area of a trapezoidal base, I divided the trapezoid into two parts – a rectangle and a triangle – that looks like this: Work for finding the trapezoidal prism on the top part of the chopsticks: I’m finding the area of the trapezoid base first. The area of the triangle part is: Height of triangle is 0.25 cm, base of triangle is 20.2 cm. Area of triangle = 20.2
0.25
2.525 cm2 Area of rectangle = 20.2 cm x 0.35 cm = 7.175 cm2 Therefore, the area of the trapezoidal base if 2.525 cm2 + 7.175 cm2 = 9.7 cm2 If the area of the trapezoidal base is 9.7 cm2 and the height of the trapezoidal prism is 0.1 cm, then the volume is: Volume = (Area of trapezoidal base)(height) = (9.7 cm2)(0.1 cm) = 0.97 cm3 Therefore, the volume of the trapezoidal prism on the top part of the chopsticks is 0.97 cm3. The trapezoidal prism on the bottom part of the chopsticks looks like this: Height is 0.3 cm Trapezoidal base is the same as the base on the top part of the chopsticks. The base has a length of 20.2 cm and has sides that are 0.6 cm and 0.35 cm wide. Work for finding the trapezoidal prism on the bottom part of the chopsticks: I’m finding the area of the trapezoid base first. Since the trapezoid base is the same exact trapezoid base as the top part of the chopsticks, the area will be the same. The area of the trapezoidal base on the top part of the chopsticks was 9.7 cm2, therefore the area of the trapezoid base on the bottom part of the chopsticks will also be 9.7 cm2. If the area of the trapezoidal base is 9.7 cm2 and the height of the trapezoidal prism is 0.3 cm, then the volume is: Volume = (Area of trapezoidal base)(height) = (9.7 cm2)(0.3 cm) = 2.91 cm3 Therefore, the volume of the trapezoidal prism on the top part of the chopsticks is 2.91 cm3. The volume for one entire chopstick is the volume of the top trapezoidal prism and the bottom trapezoidal prism. Thus, the volume for one entire chopstick is Volume of top trapezoidal prism = 0.97 cm3 Volume of bottom trapezoidal prism = 2.91 cm3 Total volume of 1 chopstick is = 0.97 cm3 + 2.91 cm3 = 3.88 cm3. Since there are 2 chopsticks, I will double the total volume for one chopstick. 3.88 cm3 x 2 = 7.76 cm3 The total volume of the chopsticks is 7.76 cm3. FINDING THE SURFACE AREA OF THE CHOPSTICKS Surface area of an object is the sum of the areas of all the faces. Therefore, to find surface area of the chopsticks, I found the area of each face. Area of Top and Bottom Face The top and bottom faces look like the following picture. Therefore, I only need to find the area of one of the bases and then multiply it by two to find the surface area of both. These faces are in the shape of a trapezoid with the length of 20.2 cm and width of 0.6 cm and 0.35 cm. In the volume section, we already found the area of this base to be 9.7 cm2. Since there are two of them, the surface area for the top and bottom faces is 9.7 cm2 x 2 = 19.4 cm2. Area of Front Face The front base looks like the diagram below and has the following measurements: Length = 0.35 cm Width= 0.3 cm Area of front base is length x width, so Area of front base = (0.35 cm)(0.3 cm) = 0.105 cm2 So area of the front base is 0.105 cm2. Area of Back Face The back face looks like the following diagram and has the following measurements: Length of face is 0.6 cm and the width of the face is 0.4 cm. Area of the back base is (length)(width), so Area of the back base = (0.6 cm)(0.4 cm) = 0.24 cm2 so the area of the back base is 0.24 cm2. Area of Side Faces The last two faces are the sides of the chopsticks. Since they have equal measurements, then I will only find the area of one of them and multiply the answer by two to find the total surface area of the side faces. The side faces look like the following diagram and have the following measurements: Length of trapezoid is 20.2 cm and the widths of the sides are 0.4 cm and 0.3 cm. To find the area of the trapezoid, I divided it into two recognizable shapes – a rectangle and a triangle. The rectangle has a length of 20.2 cm and the width of 0.3 cm. So the area of the rectangle is Area of rectangle = (length)(width) Area of rectangle = (20.2 cm)(0.3 cm) = 6.06 cm2 The triangle has a base of 20.2 cm and a height of 0.1 cm, so the area of the triangle is Area of triangle = (length)(width) Area of triangle = (20.2cm)(0.1 cm) = 1.01 cm2 To find the area of the trapezoid, you just add the area of the rectangle and the area of the triangle. Area of trapezoid = 6.06 cm2 + 1.01 cm2 = 7.07 cm2 Since there are two side faces, I multiplied 7.07 cm2 by 2. 7.07 cm2 x 2 = 14.14 cm2. So the area of the side bases is 14.14 cm2. TOTAL SURFACE AREA OF ONE CHOPSTICK Now that I have found the area of all the faces, I am going to add them together. Area of top/bottom faces = 19.4 cm2 Area of front face = 0.105 cm2 Area of back face = 0.24 cm2 Area of side faces = 14.14 cm2 19.4 cm2 + 0.105 cm2 + 0.24 cm2 + 14.14 cm2 = 33.885 cm2 The total surface area of one chopstick is 33.885 cm2. The total surface area of both chopsticks is then… 33.885 cm2 x 2 = 67.77 cm2. TOTAL SURFACE AREA OF ORIGINAL OBJECT IS 67.77 cm2.