Hemispheres, Rectangular Parallelepiped, Regular Right Pyramid with Triangular or Square Base

Hemisphere

A hemisphere is exactly half of a sphere. Imagine slicing a sphere perfectly through its center; the two resulting pieces are hemispheres. Each hemisphere has a curved surface and a flat circular base.

Key Properties and Formulas:

Let 'r' be the radius of the sphere from which the hemisphere is formed.

• Radius: The radius of the hemisphere is the same as the radius of the original sphere.
• Curved Surface Area (CSA): This is the area of the curved part of the hemisphere. It is exactly half the surface area of the sphere.

  CSA = 2πr²

• Area of the Base: The base of a hemisphere is a circle.

  Area of Base = πr²

• Total Surface Area (TSA): This is the sum of the curved surface area and the area of the base.

  TSA = CSA + Area of Base = 2πr² + πr² = 3πr²

• Volume: The volume of a hemisphere is exactly half the volume of the sphere.

  Volume = (2/3)πr³

Example:

Calculate the total surface area and volume of a hemisphere with a radius of 7 cm.

Given: radius (r) = 7 cm

Using π ≈ 22/7:

Total Surface Area (TSA):

TSA = 3πr² = 3 * (22/7) * (7 cm)²

TSA = 3 * (22/7) * 49 cm²

TSA = 3 * 22 * 7 cm²

TSA = 66 * 7 cm²

TSA = 462 cm²

Volume:

Volume = (2/3)πr³ = (2/3) * (22/7) * (7 cm)³

Volume = (2/3) * (22/7) * 343 cm³

Volume = (2/3) * 22 * 49 cm³

Volume = (2 * 1078) / 3 cm³

Volume = 2156 / 3 cm³

Volume ≈ 718.67 cm³

Rectangular Parallelepiped (Cuboid)

A rectangular parallelepiped, commonly known as a cuboid, is a three-dimensional solid object bounded by six rectangular faces. All its angles are right angles. Think of a brick or a matchbox.

Key Properties and Formulas:

Let the length, breadth, and height of the rectangular parallelepiped be 'l', 'b', and 'h' respectively.

• Faces: It has 6 faces, all of which are rectangles. Opposite faces are identical.
• Edges: It has 12 edges. There are 4 edges of length 'l', 4 edges of length 'b', and 4 edges of length 'h'.
• Vertices: It has 8 vertices (corners).
• Lateral Surface Area (LSA): This is the sum of the areas of the four side faces (excluding the top and bottom faces).

  LSA = 2(l + b)h

• Total Surface Area (TSA): This is the sum of the areas of all six faces.

  TSA = 2(lb + bh + hl)

• Volume: This is the space occupied by the cuboid.

  Volume = l * b * h

• Diagonal: The longest distance between any two opposite vertices of the cuboid.

  Diagonal (d) = √(l² + b² + h²)

Example:

A rectangular room is 10 meters long, 8 meters wide, and 5 meters high. Find its total surface area and volume.

Given: l = 10 m, b = 8 m, h = 5 m

Total Surface Area (TSA):

TSA = 2(lb + bh + hl)

TSA = 2((10 m * 8 m) + (8 m * 5 m) + (5 m * 10 m))

TSA = 2(80 m² + 40 m² + 50 m²)

TSA = 2(170 m²)

TSA = 340 m²

Volume:

Volume = l * b * h

Volume = 10 m * 8 m * 5 m

Volume = 400 m³

Regular Right Pyramid with Triangular or Square Base

A pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. In a 'right pyramid', the apex is directly above the centroid of the base. A 'regular' pyramid has a regular polygon as its base.

Types to focus on:

1. Regular Right Pyramid with a Square Base
2. Regular Right Pyramid with an Equilateral Triangular Base

1. Regular Right Pyramid with a Square Base

The base is a square, and the apex is directly above the center of the square. The four triangular faces are congruent isosceles triangles.

Key Terms and Formulas:

• Base Edge (a): The side length of the square base.
• Height (h): The perpendicular distance from the apex to the center of the base.
• Slant Height (l): The height of each triangular face, measured from the midpoint of the base edge to the apex.
• Lateral Edge (e): The length of the edges connecting the base vertices to the apex.

We can relate these using the Pythagorean theorem:

• Relationship between h, l, and a: The slant height 'l' is the hypotenuse of a right-angled triangle formed by the height 'h' and half the base edge (a/2).

  l² = h² + (a/2)²

• Area of the Base (B):

  B = a²

• Area of each Triangular Face:

  Area = (1/2) * base * height = (1/2) * a * l

• Lateral Surface Area (LSA): Sum of the areas of the four triangular faces.

  LSA = 4 * (1/2) * a * l = 2al

• Total Surface Area (TSA): Sum of the lateral surface area and the base area.

  TSA = LSA + B = 2al + a²

• Volume:

  Volume = (1/3) * Base Area * Height = (1/3)Bh = (1/3)a²h

Example:

A regular right pyramid has a square base of side 10 cm and a height of 12 cm. Calculate its volume and total surface area.

Given: a = 10 cm, h = 12 cm

First, find the slant height (l):

l² = h² + (a/2)²

l² = (12 cm)² + (10 cm / 2)²

l² = 144 cm² + (5 cm)²

l² = 144 cm² + 25 cm²

l² = 169 cm²

l = √169 cm² = 13 cm

Volume:

Volume = (1/3)a²h = (1/3) * (10 cm)² * 12 cm

Volume = (1/3) * 100 cm² * 12 cm

Volume = 100 cm² * 4 cm

Volume = 400 cm³

Total Surface Area (TSA):

TSA = 2al + a²

TSA = 2 * (10 cm) * (13 cm) + (10 cm)²

TSA = 260 cm² + 100 cm²

TSA = 360 cm²

2. Regular Right Pyramid with an Equilateral Triangular Base

The base is an equilateral triangle, and the apex is directly above the centroid of the triangle. The three triangular faces are congruent isosceles triangles.

Key Terms and Formulas:

• Base Edge (a): The side length of the equilateral triangular base.
• Height (h): The perpendicular distance from the apex to the centroid of the base.
• Slant Height (l): The height of each triangular face.

Properties of an Equilateral Triangle with side 'a':

• Area (B) = (√3 / 4) a²
• Height of base triangle (h_base) = (√3 / 2) a
• Distance from centroid to midpoint of a side = (1/3) * h_base = (1/3) * (√3 / 2) a = a / (2√3)
• Distance from centroid to a vertex = (2/3) * h_base = (2/3) * (√3 / 2) a = a / √3

Relationships in the Pyramid:

• Relationship between h, l, and a: The slant height 'l' is the hypotenuse of a right-angled triangle formed by the pyramid's height 'h' and the distance from the centroid to the midpoint of a base edge (a / (2√3)).

  l² = h² + (a / (2√3))²

• Area of the Base (B):

  B = (√3 / 4) a²

• Area of each Triangular Face:

  Area = (1/2) * base * slant height = (1/2) * a * l

• Lateral Surface Area (LSA): Sum of the areas of the three triangular faces.

  LSA = 3 * (1/2) * a * l = (3/2)al

• Total Surface Area (TSA): Sum of the lateral surface area and the base area.

  TSA = LSA + B = (3/2)al + (√3 / 4) a²

• Volume:

  Volume = (1/3) * Base Area * Height = (1/3)Bh = (1/3) * (√3 / 4) a²h

Example:

A regular right pyramid has an equilateral triangular base with side 6 cm and a height of 8 cm. Calculate its volume.

Given: a = 6 cm, h = 8 cm

Volume:

Volume = (1/3) * (√3 / 4) a²h

Volume = (1/3) * (√3 / 4) * (6 cm)² * 8 cm

Volume = (1/3) * (√3 / 4) * 36 cm² * 8 cm

Volume = (√3 / 12) * 36 * 8 cm³

Volume = √3 * 3 * 8 cm³

Volume = 24√3 cm³

Common Pitfalls and Exam Tips:

• Distinguish between height and slant height: Always check which one is given and which one is needed for the formula. Use Pythagoras theorem to find the missing one.
• Units: Ensure all measurements are in the same units before calculation. The final answer's units will be cubic for volume and square for area.
• Base Shape: Carefully identify the shape of the base (square, triangle, etc.) and use the correct area formula for it.
• Hemisphere vs. Sphere: Remember that hemisphere formulas are typically half of the sphere's formulas, with the addition of the base area for TSA.
• Cuboid Diagonal: Don't confuse the space diagonal (√(l²+b²+h²)) with face diagonals (e.g., √(l²+b²)).