Ronaldo is carving a sculpture from clay. The sculpture is made up of two figures, a rectangular prism and a square pyramid. A diagram of the sculpture is shown below.
7 1/4 in. 20 in. 4 1/2 in. 3 in.
The surface area of the sculpture is _____ in2.
To find the surface area of the sculpture, we need to calculate the surface area of each individual shape and then add them together.
Surface area of rectangular prism:
2lw + 2lh + 2wh
(7.25 * 20) + (7.25 * 4.5) + (20 * 4.5) = 145 + 32.63 + 90 = 267.63 square inches
Surface area of square pyramid:
Area of base + 4 * (1/2 * base perimeter * slant height)
(3 * 3) + 4 * (1/2 * (3 + 3 + 4.5) * 3) = 9 + 4 * (1/2 * 10.5 * 3) = 9 + 4 * 15.75 = 9 + 63 = 72 square inches
Total surface area:
267.63 + 72 = 339.63 square inches
So, the surface area of the sculpture is 339.63 square inches.
If a = 20 cm, b = 13 cm, c = 12 cm, d = 5 cm, and e = 8 cm, what is the surface area of the figure
To find the surface area of the figure, we need to first calculate the surface area of each individual shape (rectangle, triangle, and trapezoid) and then add them together.
1. Surface area of the rectangle with sides a and b:
2ab + 2bc + 2cd + 2da
= 2(20 * 13) + 2(13 * 12) + 2(12 * 5) + 2(5 * 20)
= 2(260) + 2(156) + 2(60) + 2(100)
= 520 + 312 + 120 + 200
= 1152 square cm
2. Surface area of the triangle with sides c, d, and e:
Using Heron's Formula to find the area first:
s = (c + d + e) / 2
s = (12 + 5 + 8) / 2
s = 25 / 2
s = 12.5
Area of triangle = √(s(s-c)(s-d)(s-e))
= √(12.5 * (12.5 - 12) * (12.5 - 5) * (12.5 - 8))
= √(12.5 * 0.5 * 7.5 * 4.5)
= √(28.125)
= 5.3077 square cm
3. Surface area of the trapezoid with bases b and e, and height a:
Area = (1/2) * (b + e) * a
= (1/2) * (13 + 8) * 20
= (1/2) * 21 * 20
= 210 square cm
Total surface area:
1152 + 5.3077 + 210 = 1367.31 square cm
Therefore, the surface area of the figure is approximately 1367.31 square cm.
If w = 8 inches, x = 5 inches, y = 20 inches, and z = 11.3 inches, what is the total surface area of the exterior faces of the figure?
To find the total surface area of the exterior faces of the figure, we need to calculate the surface area of each individual shape (three rectangles and two triangles) and then add them together.
Given:
- w = 8 inches
- x = 5 inches
- y = 20 inches
- z = 11.3 inches
1. Surface area of the rectangle with sides w and x:
2wx = 2(8 * 5) = 80 square inches
2. Surface area of the rectangle with sides x and y:
2xy = 2(5 * 20) = 200 square inches
3. Surface area of the rectangle with sides y and z:
2yz = 2(20 * 11.3) = 452 square inches
4. Surface area of the triangle with sides w, x, and y:
Using Heron's Formula to find the area first:
s = (w + x + y) / 2 = (8 + 5 + 20) / 2 = 33 / 2 = 16.5
Area of triangle = √(s(s-w)(s-x)(s-y))
= √(16.5 * (16.5 - 8) * (16.5 - 5) * (16.5 - 20))
= √(16.5 * 8.5 * 11.5 * 3.5)
= √(6590.25)
= 81.19 square inches
5. Surface area of the triangle with sides x, y, and z:
Using Heron's Formula to find the area first:
s = (x + y + z) / 2 = (5 + 20 + 11.3) / 2 = 36.3 / 2 = 18.15
Area of triangle = √(s(s-x)(s-y)(s-z))
= √(18.15 * (18.15 - 5) * (18.15 - 20) * (18.15 - 11.3))
= √(18.15 * 13.15 * 1.85 * 6.85)
= √(1812.441)
= 42.56 square inches
Total surface area:
80 + 200 + 452 + 81.19 + 42.56 = 855.75 square inches
Therefore, the total surface area of the exterior faces of the figure is 855.75 square inches.
Bradley made a house for his dog, Bowser, out of wood with a cube base and a triangular prism top. The dimensions of the dog house are a = 5 feet, b = 1 foot, and c = 2.7 feet.
If Bradley plans to paint the outside of the dog house blue, not including the bottom, how many square feet of paint will he use
To find the total surface area of the dog house that Bradley needs to paint blue (not including the bottom), we need to calculate the surface area of each individual shape (cube and triangular prism) and then add them together.
Given:
- a = 5 feet (side length of the cube base)
- b = 1 foot (base of the triangular prism)
- c = 2.7 feet (height of the triangular prism)
1. Surface area of the cube:
The cube has 6 faces that are all squares, each with an area of a^2.
Total surface area of the cube = 6 * a^2 = 6 * 5^2 = 6 * 25 = 150 square feet
2. Surface area of the triangular prism:
There are three rectangular faces and two triangular faces on a triangular prism.
Rectangular faces:
- Two faces: b * c = 1 * 2.7 = 2.7 square feet (each)
- One face: a * c = 5 * 2.7 = 13.5 square feet
Triangular faces:
Each triangular face is a right triangle with sides b, c, and a.
Using the formula for the area of a triangle (1/2 * base * height):
Area of a triangular face = (1/2) * b * c = (1/2) * 1 * 2.7 = 1.35 square feet
Total surface area of the triangular prism = 2 * 2.7 + 13.5 + 2 * 1.35 = 5.4 + 13.5 + 2.7 = 21.6 square feet
Total surface area to paint blue:
150 (cube) + 21.6 (triangular prism) = 171.6 square feet
Therefore, Bradley will need to use 171.6 square feet of paint to paint the outside of the dog house blue, not including the bottom.
If X = 3 inches, Y = 10 inches, W = 5 inches, and Z = 5 inches, what is the area of the object?
To find the total surface area of the object with dimensions X, Y, W, and Z, we need to determine the areas of the individual faces and add them together. The object consists of a rectangular prism and a triangular prism.
Given:
- X = 3 inches
- Y = 10 inches
- W = 5 inches
- Z = 5 inches
1. Surface area of the rectangular prism:
The rectangular prism has 6 faces, and each pair of opposite faces are congruent rectangles. The formula for the surface area of a rectangular prism is:
Surface area = 2lw + 2lh + 2wh
Where:
l = length, w = width, h = height
Substitute the values:
Surface area = 2(5*3) + 2(5*10) + 2(3*10)
Surface area = 2(15) + 2(50) + 2(30)
Surface area = 30 + 100 + 60
Surface area = 190 square inches
2. Surface area of the triangular prism:
A triangular prism has 2 triangular faces and 3 rectangular faces.
a. Surface area of the triangular faces:
The area of a triangular face is given by the formula: 0.5 * base * height.
Area of a triangular face = 0.5 * X * Y = 0.5 * 3 * 10 = 15 square inches (each)
Total area of two triangular faces = 2 * 15 = 30 square inches
b. Surface area of the rectangular faces:
These are the three rectangular faces on the prism, in which two adjacent faces are congruent rectangles.
Substitute the values:
Surface area = 2(10*5) + 2(3*5) + 2(3*5)
Surface area = 2(50) + 2(15) + 2(15)
Surface area = 100 + 30 + 30
Surface area = 160 square inches
Total surface area of the triangular prism = 30 (triangular faces) + 160 (rectangular faces) = 190 square inches
Total area of the object = Surface area of the rectangular prism + Surface area of the triangular prism
Total area = 190 (rectangular prism) + 190 (triangular prism) = 380 square inches
Therefore, the area of the object is 380 square inches.