Geometry: Surface Area and Volume of 3D Shapes

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10 Questions

A rectangular prism has a length of 8 cm, a width of 5 cm, and a height of 3 cm. Find the total surface area of the prism.

148 cm^2

A cylinder has a radius of 4 cm and a height of 10 cm. Find the volume of the cylinder.

502.65 cm^3

A cone has a radius of 6 cm and a height of 12 cm. Find the slant height of the cone.

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.

8 cm

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.

11.25 m^3

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.

56 cm²

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.

2 × π × 3² + 2 × π × 3 × 8 = 113.04 cm² (approx.)

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.

9/15 = 0.6

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.

π × 2² × 2.5 = 25π m³ (approx.)

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.

5/√(5² + 10²) = 5/√(125) = 1/5

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

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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