Surface Area of Simple 3D Shapes

Surface Area of Simple 3D Shapes exercise

• Find the surface area of each prism.

  Hints

  The area of the top and bottom of the rectangular prism is,

  • $\text{Area}=2(16\cdot 12)~\text{in}^2$
  • $\text{Area}=2(192)~\text{in}^2$
  • $\text{Area}=384~\text{in}^2$

  The perimeter, $P$, of an isosceles triangle with leg lengths $15~\text{in}$ and base $10~\text{in}$ is,

  • $P=15~\text{in}+15~\text{in}+10~\text{in}$
  • $P=40~\text{in}$

  The area for the circular bases of the cylinder is,

  • $A=2(\pi \cdot 3^2)~\text{in}^2$
  • $A=18\pi ~\text{in}^2$
  • $A\approx18(3.14)~\text{in}^2$
  • $A\approx56.52~\text{in}^2$

  Solution

  Rectangular Prism

  To find the area of the rectangular prism, use the following steps:

  1. Recall that the area of a rectangle is found by multiplying the length by the width.

  2. Find the area of the following faces:

  • Let length $l=22$, width $w=19$, and height $h=6$.
  • Top: $A=lw=22~\text{in}\cdot 19~\text{in}=418~\text{in}^2$
  • The areas for the top and bottom of the rectangular prism are the same. So, we can just multiply the area of the top by $2$ to get the area of both the top and bottom. Now we can find the area of the six faces.
  • Top & Bottom: $A=2(lw)=2(22~\text{in}\cdot 19~\text{in})=2(418)~\text{in}^2=836~\text{in}^2$
  • Front & Back: $A=2(wh)=2(19~\text{in}\cdot 6~\text{in})=2(114)~\text{in}^2=228~\text{in}^2$
  • Left & Right:$A=2(lh)=2(22~\text{in}\cdot 6~\text{in})=2(132)~\text{in}^2=264~\text{in}^2$

  3. Add all of the areas together:

  • $\text{Surface Area}=836~\text{in}^2+228~\text{in}^2+264~\text{in}^2$
  • $\text{Surface Area}=1,328~\text{in}^2$

  Therefore, we can use the formula $\text{Surface Area}=2lw+2wh+2lh$ to represent surface area of a rectangular prism.

  $~$

  Triangular Prism

  To find the area of the triangular prism, use the following steps:

  1. Recall that the area of a triangle is found by multiplying the base and height by $\frac{1}{2}$.

  2. Find the area of the front and back triangles:

  • Use the area of a triangle formula, $\frac{1}{2}bh$, and multiply it by $2$ to represent both the front and back.
  • Let base $b=23$, height $h=29$.
  • Front & Back:$A=2(\frac{1}{2}bh)=2(\frac{1}{2}\cdot 23~\text{in}\cdot 29~\text{in})=2(333.5)~\text{in}^2=667~\text{in}^2$

  3. Find the area of the lateral face by unfolding the sides to create a long rectangle. The height of this new rectangle is $2~\text{in}$ and the length is found by finding the perimeter, $P$, of all three sides of the isosceles triangle.

  • Lateral Face: $A=hP=2~\text{in}(31~\text{in}+31~\text{in}+23~\text{in})=170~\text{in}^2$

  4. Add both areas together:

  • $\text{Surface Area}=667~\text{in}^2+170~\text{in}^2=837~\text{in}^2$

  Therefore, we can use the formula $\text{Surface Area}=2A_{\triangle}+hP$ to represent surface area of a triangular prism.

  $~$

  Cylinder

  To find the area of the cylinder, use the following steps:

  1. Recall that the area of a circle is found by using the formula $A=\pi r^2$.

  2. Find the area of the two circular bases:

  • Use the area of a circle formula and multiply it by $2$ to represent both bases.
  • Bases: $A=2(\pi \cdot 4^2)~\text{in}^2=2(16\pi)~\text{in}^2=32\pi~\text{in}^2$

  3. Find the area of the lateral face by unrolling the face of the cylinder. This creates a rectangle with a height of $16~\text{in}$ and a length that is the same as the circumference, $C$, of the circular base. Recall that $C=2\pi r$.

  • Base:$C=2\pi \cdot 4~\text{in}=8\pi ~\text{in}$
  • Lateral Face: $A=16~\text{in}(8\pi ~\text{in})=128\pi~\text{in}^2$

  4. Add both areas together and use $3.14$ to approximate $\pi$:

  • $\text{Surface Area}\approx 32(3.14)~\text{in}^2+128(3.14)~\text{in}^2\approx 100.48~\text{in}^2+401.92~\text{in}^2=502.4~\text{in}^2$

  Therefore, we can use the formula $\text{Surface Area}=2A_{\circ}+hC$ to represent surface area of a cylinder.

• Identify the formula for the surface area of each prism.

  Hints

  The surface area for all prisms can be found by doubling the area of the base and adding the lateral area.

  The area of the base can be found by identifying the shape, and using the area formula for that particular shape.

  The base of a cylinder is a circle.

  Solution

  The surface area for all prisms can be found by doubling the area of the base, the shape that clarifies the 3D object, and adding the lateral area.

  This is represented by the formula, $SA=2A_{\text{base}}+Ph$, where $P$ is the perimeter of the lateral face, and $h$ is the height of the lateral face.

  1. The formula for the surface area of a cylinder is, $SA=2(\pi r^2)+Ph$.

  • The base of a cylinder is a circle.
  • The area of a circle is, $A=\pi r^2$.
  • The lateral face is a rectangle represented by $Ph$, where $P$ is actually the circumference of the circular base, $C=2 \pi r$, and $h$ is the height of the cylinder.

  2. The formula for the surface area of a rectangular prism is, $SA=2lw+2wh+2lh$.

  • The base of a rectangular prism is a rectangle.
  • The area of a rectangle is $A=lw$
  • There are four lateral faces that are all rectangles: the front, back, left, and right of the prism, which represent $2wh$ and $2lh$.

  3. The formula for the surface area of a triangular prism is, $SA=2(\frac{1}{2}bh)+Ph$.

  • The base of a triangular prism are two triangles.
  • The area of a triangle is $A=\frac{1}{2}bh$, where $b$ represents the base, and $h$ represents the height.
  • The lateral face is a rectangle formed from unfolding the sides and is represented by $Ph$, where $P$ is the perimeter of all $3$ sides, and $h$ is the height of the unfolded side.

  4. Another formula for the surface area of a cylinder is, $SA\approx2(3.14\cdot r^2)+Ph$.

  • The base of a cylinder is a circle.
  • The area of a circle is, $A=\pi r^2$.
  • $3.14$ is used an approximation for $\pi$.
  • The lateral face is a rectangle represented by $Ph$, where $P$ is the circumference of the circular base and $h$ is the height of the cylinder.

• Determine the surface area of each prism.

  Hints

  $3.14$ can be used as an approximation for pi, $\pi$.

  The surface area of the cylinder is found by solving the following expression,

  • $SA \approx 2(\pi\cdot 2^2)~\text{in}^2+(2\pi \cdot 2)(10)~\text{in}^2$

  The surface area of the triangular prism is found by solving the following expression,

  • $SA=2\left( \frac{1}{2}(16\cdot 15)~\text{in}^2 \right)+(16+17+17)(1)~\text{in}^2$

  Solution

  1. Cylinder with radius $7~\text{in}$ and height $17~\text{in}$.

  • The surface area is found by using the formula, $SA=2\text{Area}_{\text{circle}}+Ph$.
  • $Ph$ represents the lateral face of the cylinder found by unrolling the face of the cylinder into a rectangle.
  • The rectangle is composed of the circumference, which we use the perimeter, $P$, of the circular base, and the height $h$ of the cylinder.
  • The area of a circle is $\pi r^2$, and the circumference of a circle is $2\pi r$.
  • $SA=2(\pi \cdot 7^2)+(2\pi \cdot 7)\cdot 17$
  • $SA \approx 2(3.14\cdot 49)+(2\cdot 3.14\cdot 7)\cdot 17$
  • $SA \approx 307.72+747.32$
  • $SA \approx 1,055.04~\text{in}^2$

  2. Triangular Prism with base $18~\text{in}$, height $12~\text{in}$, legs $15~\text{in}$, and depth $3~\text{in}$.

  • The surface area is found by using the formula, $SA=2\text{Area}_{\text{triangle}}+Ph$.
  • $Ph$ represents the lateral faces of the prism found by placing 1 triangular face on ground and unfolding the sides into 1 long rectangle.
  • The rectangle is composed of the perimeter $P$ of the isosceles triangle, and the depth $h$ of the triangular prism.
  • $SA=2\cdot \frac{1}{2}(18\cdot 12)+(15+15+18)(3)$
  • $SA=216+144$
  • $SA=360~\text{in}^2$

  3. Rectangular Prism length $21~\text{in}$, width $12~\text{in}$, and height $17~\text{in}$.

  • The surface area is found by using the formula, $SA=2lw+2wh+2lh$, where $l$ represents the length, $w$ represents the width, and $h$ represents the height.
  • The surface area is basically adding the areas of all 6 rectangular faces: front, back, top, bottom, left and right.
  • $SA=2(21\cdot 12)+2(12\cdot 17)+2(21\cdot 17)$
  • $SA=2(252)+2(204)+2(357)$
  • $SA=504+408+714$
  • $SA=1,626~\text{in}^2$

  4. Cylinder with radius $5~\text{in}$ and height $10~\text{in}$.

  • The surface area is found by using the formula, $SA=2\text{Area}_{\text{circle}}+ph$.
  • $Ph$ represents the lateral face of the cylinder found by unrolling the face of the cylinder into a rectangle.
  • The rectangle is composed of the circumference, which we use the perimeter, $P$, of the circular base, and the height $h$ of the cylinder.
  • The area of a circle is $\pi r^2$ and the circumference of a circle is $2\pi r$.
  • $SA=2(\pi \cdot 5^2)+(2\pi \cdot 5)\cdot 10$
  • $SA \approx 2(3.14\cdot 25)+(2\cdot 3.14\cdot 5)\cdot 10$
  • $SA=157+314$
  • $SA=471~\text{in}^2$

  5. Triangular Prism with base $10~\text{in}$, height $12~\text{in}$, legs $13~\text{in}$, and depth $2~\text{in}$.

  • The surface area is found by using the formula, $SA=2\text{Area}_{\text{triangle}}+Ph$.
  • $Ph$ represents the lateral faces of the prism found by unfolding the sides into 1 long rectangle.
  • The rectangle is composed of the perimeter $P$ of the isosceles triangle, and the depth $h$ of the triangular prism.
  • $SA=2\cdot \frac{1}{2}(10\cdot 12)+(13+13+10)(2)$
  • $SA=120+72$
  • $SA=192~\text{in}^2$

• Calculate the surface area of the given composite 3D object.

  Hints

  The surface area for all prisms can be found by doubling the area of the base and adding the lateral area.

  Remember that the radius of a circle is half the diameter.

  The total surface area is represented by four cylinders, one triangular prism, and one rectangular prism. Don't forget to subtract the area of any shared sides that are not on the surface.

  Solution

  To find the surface area of the 3D figure, we need to calculate the surface of the individual prisms and cylinders, then add them all together. There are 4 cylinders, 1 triangular prism, and 1 rectangular prism.

  1. Find the surface area of the 4 cylinders with diameter $2~\text{in}$ and height $5~\text{in}$.

  • The surface area is found by using the formula, $SA=2(\pi r^2)+Ph$.
  • $Ph$ represents the lateral face of the cylinder found by unrolling the face of the cylinder into a rectangle.
  • The rectangle is composed of the circumference $P$ of the circular base, and the height $h$ of the cylinder.
  • The area of a circle is $\pi r^2$ and the circumference of a circle is $2\pi r$.
  • Since the diameter is twice the radius, we know that $r=1~\text{in}$.
  • $SA=2(\pi \cdot 1^2)~\text{in}^2+(2\pi \cdot 1)\cdot 5~\text{in}^2$
  • $SA\approx 2(3.14\cdot 1)~\text{in}^2+(2\cdot 3.14\cdot 1)\cdot 5~\text{in}^2$
  • $SA\approx 6.28~\text{in}^2+31.4~\text{in}^2$
  • $SA\approx 37.68~\text{in}^2$
  • The surface area of 1 cylinder is $37.68~\text{in}^2$, which makes the surface area of 4 cylinders, $4(37.68~\text{in}^2)=150.72~\text{in}^2$.
  • Since the top and bottom of all four cylinders are being shared with the other objects and therefore not on the surface, subtract the area of $8$ circles from the total surface area.
  • $\text{Area}=8\pi r^2=8(\pi \cdot (1~\text{in})^2)=8\pi~\text{in}^2\approx 8(3.14)~\text{in}^2\approx 25.12~\text{in}^2$.
  • The new surface area is $150.72~\text{in}^2-25.12~\text{in}^2=125.60~\text{in}^2$.

  2. Find the surface area of the triangular prism with base $10~\text{in}$, height $12~\text{in}$, legs $13~\text{in}$, and depth $3~\text{in}$.

  • The surface area is found by using the formula, $SA=2(\frac{1}{2}bh_{1})+Ph_{2}$.
  • $Ph_{2}$ represents the lateral faces of the prism, found by placing 1 triangular face on ground and unfolding the sides into 1 long rectangle.
  • The rectangle is composed of the perimeter $P$ of the isosceles triangle, and the depth $h_{2}$ of the triangular prism.
  • $SA=2\cdot \frac{1}{2}(10\cdot 12)~\text{in}^2+(13+13+10)(3)~\text{in}^2$
  • $SA=120~\text{in}^2+36(3)~\text{in}^2$
  • $SA=120~\text{in}^2+108~\text{in}^2$
  • $SA=228~\text{in}^2$

  3. Find the surface area of the rectangular prism length $10~\text{in}$, width $3~\text{in}$, and height $4~\text{in}$.

  • The surface area is found by using the formula, $SA=2lw+2wh+2lh$, where $l$ represents the length, $w$ represents the width, and $h$ represents the height.
  • The surface area is basically adding the areas of all 6 rectangular faces: front, back, top, bottom, left and right.
  • $SA=2(10\cdot 3)~\text{in}^2+2(3\cdot 4)~\text{in}^2+2(10\cdot 4)~\text{in}^2$
  • $SA=2(30)~\text{in}^2+2(12)~\text{in}^2+2(40)~\text{in}^2$
  • $SA=60~\text{in}^2+24~\text{in}^2+80~\text{in}^2$
  • $SA=164~\text{in}^2$

  4. Add the surface areas together to find the total surface area of the 3D object.

  • $SA=125.60~\text{in}^2+228~\text{in}^2+164~\text{in}^2$
  • $SA=517.60~\text{in}^2$

• Compute the area of each shape.

  Hints

  $3.14$ can be used as an approximation for pi, $\pi$.

  The radius, $r$, is a straight line from the center to the circumference of a circle.

  The area for a triangle with a base of $4~\text{in}$ and height of $8~\text{in}$, is $16~\text{in}^2$.

  Solution

  Green Circle

  • The area formula for a circle is $A=\pi r^2$, where $r$ represents the radius.
  • The image shows that the radius is $3~\text{in}$.
  • $\text{A}=\pi (3^2)~\text{in}^2$
  • $\text{A}\approx 3.14(9)~\text{in}^2$
  • $\text{A}\approx 28.26~\text{in}^2$

  Orange Triangle

  • The area formula for a triangle is $A=\frac{1}{2}bh$, where $b$ represents the base and $h$ represents the height.
  • The image shows that the base is $6~\text{in}$ and the height is $4~\text{in}$.
  • $\text{A}=\frac{1}{2}(6)(4)~\text{in}^2$
  • $\text{A}=3(4)~\text{in}^2$
  • $\text{A}=12~\text{in}^2$

  Blue Rectangle

  • The area formula for a rectangle is $A=lw$, where $l$ represents the length and $w$ represents the width.
  • The image shows that the length is $10~\text{in}$ and the width is $6~\text{in}$.
  • $\text{A}=10(6)~\text{in}^2$
  • $\text{A}=60~\text{in}^2$

• Find the surface area of the composite 3D object.

  Hints

  The surface area for all prisms can be found by doubling the area of the base and adding the lateral area.

  Unraveling the lateral faces of a prism normally creates a rectangular figure.

  The total surface area of the 3D figure is represented by a rectangular prism and a triangular prism. Don't forget to subtract the area of any shared sides that are not on the surface.

  Solution

  To find the surface area of the 3D figure, we need to calculate the surface of the individual prisms, then add them all together, and subtract the area of any sides that are not on the surface. There is 1 triangular prism and 1 rectangular prism.

  1. Find the surface area of the rectangular prism length $8~\text{cm}$, width $10~\text{cm}$, and height $4~\text{cm}$.

  • The surface area is found by using the formula, $SA=2lw+2wh+2lh$, where $l$ represents the length, $w$ represents the width, and $h$ represents the height.
  • The surface area is basically adding the areas of all 6 rectangular faces: front, back, top, bottom, left and right.
  • $SA=2(8\cdot 10)~\text{cm}^2+2(10\cdot 4)~\text{cm}^2+2(8\cdot 4)~\text{cm}^2$
  • $SA=2(80)~\text{cm}^2+2(40)~\text{cm}^2+2(32)~\text{cm}^2$
  • $SA=160~\text{cm}^2+80~\text{cm}^2+64~\text{cm}^2$
  • $SA=304~\text{cm}^2$
  • Since the top of the rectangular prism is a shared side and not on the surface, subtract the rectangular area from the total surface area.
  • $\text{Area}=8~\text{cm}\cdot 10~\text{cm}=80~\text{cm}^2$
  • The new surface area is $304~\text{cm}^2-80~\text{cm}^2=224~\text{cm}^2$.

  2. Find the surface area of the triangular prism with base $8~\text{in}$, height $3~\text{in}$, legs $5~\text{in}$, and depth $10~\text{in}$.

  • The surface area is found by using the formula, $SA=2(\frac{1}{2}bh_{1})+Ph_{2}$.
  • $Ph_{2}$ represents the lateral faces of the prism, found by placing 1 triangular face on ground and unfolding the sides into 1 long rectangle.
  • The rectangle is composed of the perimeter $P$ of the isosceles triangle, and the depth $h_{2}$ of the triangular prism.
  • $SA=2\cdot \frac{1}{2}(8\cdot 3)~\text{cm}^2+(5+5+8)(10)~\text{cm}^2$
  • $SA=24~\text{cm}^2+18(10)~\text{cm}^2$
  • $SA=24~\text{cm}^2+180~\text{cm}^2$
  • $SA=204~\text{cm}^2$
  • Since the bottom of the triangular prism is a shared side and not on the surface, subtract the rectangular area from the total surface area.
  • $A=8~\text{cm}\cdot 10~\text{cm}=80~\text{cm}^2$
  • The new surface area is $204~\text{cm}^2-80~\text{cm}^2=124~\text{cm}^2$.

  3. Add the surface areas together to find the total surface area of the 3D object.

  • $SA= 224~\text{cm}^2+124~\text{cm}^2$
  • $SA=348~\text{cm}^2$