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# Geometry: Surface Area and Volume of 3D Shapes

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@SkillfulSulfur

148 cm^2

502.65 cm^3

13.42 cm

### In a right-angled triangle, the length of the hypotenuse is 10 cm and one of the other sides is 6 cm. Find the length of the third side.

<p>8 cm</p> Signup and view all the answers

### A water tank is in the shape of a rectangular prism with a length of 5 m, a width of 3 m, and a height of 2 m. If the tank is filled to a height of 1.5 m, find the volume of water in the tank.

<p>11.25 m^3</p> Signup and view all the answers

### A rectangular prism has a length of 12 cm, a width of 7 cm, and a height of 4 cm. Find the surface area of the front face of the prism.

<p>56 cm²</p> Signup and view all the answers

### A cylinder has a radius of 3 cm and a height of 8 cm. Find the total surface area of the cylinder, including the bases.

<p>2 × π × 3² + 2 × π × 3 × 8 = 113.04 cm² (approx.)</p> Signup and view all the answers

### In a right-angled triangle, the length of the hypotenuse is 15 cm and one of the other sides is 9 cm. Find the sine of the angle opposite the 9 cm side.

<p>9/15 = 0.6</p> Signup and view all the answers

### A water tank is in the shape of a cylinder with a radius of 2 m and a height of 3 m. If the tank is filled to a height of 2.5 m, find the volume of water in the tank.

<p>π × 2² × 2.5 = 25π m³ (approx.)</p> Signup and view all the answers

### A cone has a radius of 5 cm and a height of 10 cm. Find the cosine of the angle between the slant height and the base.

<p>5/√(5² + 10²) = 5/√(125) = 1/5</p> Signup and view all the answers

## Study Notes

### Surface Area of 3D Shapes

• The surface area of a 3D shape is the total area of its surface
• Formulas for surface area:
• Cube: 6s^2 (where s is the side length)
• Rectangular Prism: 2lh + 2lw + 2lh (where l, w, and h are the length, width, and height)
• Cylinder: 2πr(h + r) (where r is the radius and h is the height)
• Sphere: 4πr^2 (where r is the radius)

### Volume of 3D Shapes

• The volume of a 3D shape is the amount of space inside it
• Formulas for volume:
• Cube: s^3 (where s is the side length)
• Rectangular Prism: lwh (where l, w, and h are the length, width, and height)
• Cylinder: πr^2h (where r is the radius and h is the height)
• Sphere: (4/3)πr^3 (where r is the radius)

### Capacity of 3D Shapes

• Capacity is the amount of liquid a 3D shape can hold
• Measured in units such as milliliters (mL) or liters (L)

### Trigonometry

• Trigonometry involves the study of triangles and their relationships
• Key concepts:
• Angles: measured in degrees
• Sine (sin): opposite side / hypotenuse
• Cosine (cos): adjacent side / hypotenuse
• Tangent (tan): opposite side / adjacent side
• Right-angled triangles: used to solve problems involving trigonometry

### Surface Area of 3D Shapes

• Surface area is the total area of all surfaces of a 3D shape
• Formula for surface area of a rectangular prism: 2(lw + lh + wh)
• Formula for surface area of a cylinder: 2πr(h + r)
• Formula for surface area of a sphere: 4πr²

### Volume of 3D Shapes

• Volume is the amount of space inside a 3D shape
• Formula for volume of a rectangular prism: lwh
• Formula for volume of a cylinder: πr²h
• Formula for volume of a sphere: (4/3)πr³

### Capacity of 3D Shapes

• Capacity is the amount of fluid that can be held by a 3D shape
• Measured in units such as milliliters (mL) or liters (L)

### Trigonometry - Bearings

• Bearing is the direction or angle of a point from a reference point
• Measured in degrees, using a protractor or a compass
• 3-figure bearings: e.g. 045°, 135°, 225°, 315°

### Trigonometry - Sine, Cosine, and Tangent

• Sine (sin): opposite side / hypotenuse
• Cosine (cos): adjacent side / hypotenuse
• Tangent (tan): opposite side / adjacent side
• SOH-CAH-TOA: a mnemonic to remember the trig ratios

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## Description

Learn the formulas to calculate the surface area and volume of various 3D shapes, including cubes, rectangular prisms, cylinders, and spheres.

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