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Step-by-step solution for: Geometry Worksheets | Volume Worksheets
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Show Answer Key & Explanations
Step-by-step solution for: Geometry Worksheets | Volume Worksheets
Here are the step-by-step solutions for finding the surface area of each shape. We will use $\pi = 3.14$ as instructed.
Formula Reminder:
* Rectangular Prism: $2(lw + lh + wh)$
* Triangular Prism: $(2 \times \text{Area of Triangle}) + (\text{Perimeter of Triangle} \times \text{Length})$
* Cylinder: $2\pi r^2 + 2\pi rh$ (or $2\pi r(r+h)$)
---
* Dimensions: Length ($l$) = 5, Width ($w$) = 6, Height ($h$) = 10
* Step 1: Find the area of the three pairs of faces.
* Front/Back: $5 \times 10 = 50$
* Top/Bottom: $5 \times 6 = 30$
* Sides: $6 \times 10 = 60$
* Step 2: Add them up and multiply by 2.
* $50 + 30 + 60 = 140$
* $140 \times 2 = 280$
Surface Area = 280
---
* Triangle Dimensions: Base = 6, Height = 8. The sides of the triangle are 6, 8, and 10 (since it's a right triangle, we can check: $6^2+8^2=10^2$).
* Prism Length: 12
* Step 1: Area of the two triangular bases.
* Area of one triangle = $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 8 = 24$
* Two triangles = $24 \times 2 = 48$
* Step 2: Area of the three rectangular sides (Perimeter $\times$ Length).
* Perimeter of triangle = $6 + 8 + 10 = 24$
* Lateral Area = $24 \times 12 = 288$
* Step 3: Total Surface Area.
* $48 + 288 = 336$
Surface Area = 336
---
* Dimensions: Length = 7, Width = 4, Height = 9
* Step 1: Find the area of the face pairs.
* $7 \times 4 = 28$
* $7 \times 9 = 63$
* $4 \times 9 = 36$
* Step 2: Add and multiply by 2.
* $28 + 63 + 36 = 127$
* $127 \times 2 = 254$
Surface Area = 254
---
* Triangle Dimensions: Base = 8, Height = 6. The hypotenuse is 10 (checked via Pythagorean theorem: $\sqrt{6^2+8^2}=10$).
* Prism Length: 14
* Step 1: Area of the two triangular bases.
* Area of one triangle = $\frac{1}{2} \times 8 \times 6 = 24$
* Two triangles = $24 \times 2 = 48$
* Step 2: Area of the three rectangular sides.
* Perimeter of triangle = $6 + 8 + 10 = 24$
* Lateral Area = $24 \times 14 = 336$
* Step 3: Total Surface Area.
* $48 + 336 = 384$
Surface Area = 384
---
* Dimensions: Side length ($s$) = 8
* Step 1: A cube has 6 identical square faces.
* Area of one face = $8 \times 8 = 64$
* Step 2: Multiply by 6.
* $64 \times 6 = 384$
Surface Area = 384
---
* Dimensions: Radius ($r$) = 5, Height ($h$) = 12
* Step 1: Area of the two circular bases ($2\pi r^2$).
* $r^2 = 5 \times 5 = 25$
* $2 \times 3.14 \times 25 = 50 \times 3.14 = 157$
* Step 2: Area of the curved side ($2\pi rh$).
* Circumference = $2 \times 3.14 \times 5 = 31.4$
* Lateral Area = $31.4 \times 12 = 376.8$
* Step 3: Total Surface Area.
* $157 + 376.8 = 533.8$
Surface Area = 533.8
---
* Dimensions: Length = 10, Width = 4, Height = 6
* Step 1: Find the area of the face pairs.
* $10 \times 4 = 40$
* $10 \times 6 = 60$
* $4 \times 6 = 24$
* Step 2: Add and multiply by 2.
* $40 + 60 + 24 = 124$
* $124 \times 2 = 248$
Surface Area = 248
---
*Note: This shape is a cylinder cut in half lengthwise. It has 3 surfaces: two semi-circles, one curved rectangular surface, and one flat rectangular bottom.*
* Dimensions: Diameter = 10 (so Radius $r = 5$), Length ($h$) = 12
* Step 1: Area of the two semi-circular ends.
* Two semi-circles make one full circle.
* Area = $\pi r^2 = 3.14 \times 5^2 = 3.14 \times 25 = 78.5$
* Step 2: Area of the curved surface (half of a cylinder's side).
* Full cylinder side = $2\pi rh = 2 \times 3.14 \times 5 \times 12 = 376.8$
* Half of that = $376.8 / 2 = 188.4$
* Step 3: Area of the flat rectangular bottom.
* Width = Diameter = 10, Length = 12
* Area = $10 \times 12 = 120$
* Step 4: Total Surface Area.
* $78.5 + 188.4 + 120 = 386.9$
Surface Area = 386.9
---
* Triangle Dimensions: Base = 6, Height = 8. Hypotenuse = 10.
* Prism Length: 15
* Step 1: Area of the two triangular bases.
* Area of one triangle = $\frac{1}{2} \times 6 \times 8 = 24$
* Two triangles = $24 \times 2 = 48$
* Step 2: Area of the three rectangular sides.
* Perimeter of triangle = $6 + 8 + 10 = 24$
* Lateral Area = $24 \times 15 = 360$
* Step 3: Total Surface Area.
* $48 + 360 = 408$
Surface Area = 408
Final Answer:
1. 280
2. 336
3. 254
4. 384
5. 384
6. 533.8
7. 248
8. 386.9
9. 408
Formula Reminder:
* Rectangular Prism: $2(lw + lh + wh)$
* Triangular Prism: $(2 \times \text{Area of Triangle}) + (\text{Perimeter of Triangle} \times \text{Length})$
* Cylinder: $2\pi r^2 + 2\pi rh$ (or $2\pi r(r+h)$)
---
1. Rectangular Prism
* Dimensions: Length ($l$) = 5, Width ($w$) = 6, Height ($h$) = 10
* Step 1: Find the area of the three pairs of faces.
* Front/Back: $5 \times 10 = 50$
* Top/Bottom: $5 \times 6 = 30$
* Sides: $6 \times 10 = 60$
* Step 2: Add them up and multiply by 2.
* $50 + 30 + 60 = 140$
* $140 \times 2 = 280$
Surface Area = 280
---
2. Triangular Prism
* Triangle Dimensions: Base = 6, Height = 8. The sides of the triangle are 6, 8, and 10 (since it's a right triangle, we can check: $6^2+8^2=10^2$).
* Prism Length: 12
* Step 1: Area of the two triangular bases.
* Area of one triangle = $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 8 = 24$
* Two triangles = $24 \times 2 = 48$
* Step 2: Area of the three rectangular sides (Perimeter $\times$ Length).
* Perimeter of triangle = $6 + 8 + 10 = 24$
* Lateral Area = $24 \times 12 = 288$
* Step 3: Total Surface Area.
* $48 + 288 = 336$
Surface Area = 336
---
3. Rectangular Prism
* Dimensions: Length = 7, Width = 4, Height = 9
* Step 1: Find the area of the face pairs.
* $7 \times 4 = 28$
* $7 \times 9 = 63$
* $4 \times 9 = 36$
* Step 2: Add and multiply by 2.
* $28 + 63 + 36 = 127$
* $127 \times 2 = 254$
Surface Area = 254
---
4. Triangular Prism
* Triangle Dimensions: Base = 8, Height = 6. The hypotenuse is 10 (checked via Pythagorean theorem: $\sqrt{6^2+8^2}=10$).
* Prism Length: 14
* Step 1: Area of the two triangular bases.
* Area of one triangle = $\frac{1}{2} \times 8 \times 6 = 24$
* Two triangles = $24 \times 2 = 48$
* Step 2: Area of the three rectangular sides.
* Perimeter of triangle = $6 + 8 + 10 = 24$
* Lateral Area = $24 \times 14 = 336$
* Step 3: Total Surface Area.
* $48 + 336 = 384$
Surface Area = 384
---
5. Cube
* Dimensions: Side length ($s$) = 8
* Step 1: A cube has 6 identical square faces.
* Area of one face = $8 \times 8 = 64$
* Step 2: Multiply by 6.
* $64 \times 6 = 384$
Surface Area = 384
---
6. Cylinder
* Dimensions: Radius ($r$) = 5, Height ($h$) = 12
* Step 1: Area of the two circular bases ($2\pi r^2$).
* $r^2 = 5 \times 5 = 25$
* $2 \times 3.14 \times 25 = 50 \times 3.14 = 157$
* Step 2: Area of the curved side ($2\pi rh$).
* Circumference = $2 \times 3.14 \times 5 = 31.4$
* Lateral Area = $31.4 \times 12 = 376.8$
* Step 3: Total Surface Area.
* $157 + 376.8 = 533.8$
Surface Area = 533.8
---
7. Rectangular Prism
* Dimensions: Length = 10, Width = 4, Height = 6
* Step 1: Find the area of the face pairs.
* $10 \times 4 = 40$
* $10 \times 6 = 60$
* $4 \times 6 = 24$
* Step 2: Add and multiply by 2.
* $40 + 60 + 24 = 124$
* $124 \times 2 = 248$
Surface Area = 248
---
8. Half-Cylinder (Semi-Cylinder)
*Note: This shape is a cylinder cut in half lengthwise. It has 3 surfaces: two semi-circles, one curved rectangular surface, and one flat rectangular bottom.*
* Dimensions: Diameter = 10 (so Radius $r = 5$), Length ($h$) = 12
* Step 1: Area of the two semi-circular ends.
* Two semi-circles make one full circle.
* Area = $\pi r^2 = 3.14 \times 5^2 = 3.14 \times 25 = 78.5$
* Step 2: Area of the curved surface (half of a cylinder's side).
* Full cylinder side = $2\pi rh = 2 \times 3.14 \times 5 \times 12 = 376.8$
* Half of that = $376.8 / 2 = 188.4$
* Step 3: Area of the flat rectangular bottom.
* Width = Diameter = 10, Length = 12
* Area = $10 \times 12 = 120$
* Step 4: Total Surface Area.
* $78.5 + 188.4 + 120 = 386.9$
Surface Area = 386.9
---
9. Triangular Prism
* Triangle Dimensions: Base = 6, Height = 8. Hypotenuse = 10.
* Prism Length: 15
* Step 1: Area of the two triangular bases.
* Area of one triangle = $\frac{1}{2} \times 6 \times 8 = 24$
* Two triangles = $24 \times 2 = 48$
* Step 2: Area of the three rectangular sides.
* Perimeter of triangle = $6 + 8 + 10 = 24$
* Lateral Area = $24 \times 15 = 360$
* Step 3: Total Surface Area.
* $48 + 360 = 408$
Surface Area = 408
Final Answer:
1. 280
2. 336
3. 254
4. 384
5. 384
6. 533.8
7. 248
8. 386.9
9. 408
Parent Tip: Review the logic above to help your child master the concept of surface area and volume worksheets.
