Presentation1 math
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Types of pyramids include triangular, square, and pentagonal pyramids based on the shape of their base. A pyramid has triangular sides meeting at a single apex point directly above the center of its base. The surface area of a pyramid is the sum of the base area and half the perimeter multiplied by the slant height when the sides are equal triangles, or the sum of the base area and individual side areas when the sides are different shapes. The volume of any pyramid is one-third the area of the base multiplied by the height.
![• Definition Of Surface Area
• The surface area of a three-dimensional figure is
the sum of the areas of all its faces.
• The Surface Area of a Pyramid
• When all side faces are the same:
• [Base Area] + 1/2 × Perimeter × [Slant Length]
• When side faces are different:
• [Base Area] + [Lateral Area]
• Notes On Surface Area
• The Surface Area has two parts: the area of the
base (the Base Area), and the area of the side faces
(the Lateral Area).
• For Base Area :
• It depends on the shape, there are different
formulas for triangle, square, etc.
• For Lateral Area :
• When all the side faces are the same:
• Multiply the perimeter by the "slant length" and
divide by 2. This is because the side faces are
always triangles and the triangle formula is "base
times height divided by 2"
• But when the side faces are different (such as an
"irregular" pyramid) we must add up the area of
each triangle to find the total lateral area.
Surface Area](https://image.slidesharecdn.com/presentation1math-150130114907-conversion-gate02/85/Presentation1-math-3-320.jpg)
![V = volume of frustum
H = height of frustum
A = area of lower base
A’ = area of upper base
VOLUME OF PYRAMIDAL FRUSTUM
Formula :
FRUSTUM
the portion of a cone or pyramid which remains after its
upper part has been cut off by a plane parallel to its
base, or which is intercepted between two such planes.
13² = h² + 5²
h = √(13² - 5² ) = 12cm
V = 12/3 [ 576 + 196 + √(576 x 196)] =4432 cm ³
Ap = 13 cm](https://image.slidesharecdn.com/presentation1math-150130114907-conversion-gate02/85/Presentation1-math-12-320.jpg)
Presentation1 math
- 1. Types of Pyramids Triangular Pyramid : has a triangle as its base Square Pyramid : has a square as its base Pentagonal Pyramid : has a pentagon as its base Introduction of Pyramid Pyramid is a solid object where the base is a polygon (a straight-sided flat shape) and the sides are triangles which meet at the top (the apex). It is a polyhedron.
- 2. Right Pyramid : the apex of the pyramid is directly above the center of its base Right Pyramid : the apex of the pyramid is directly above the center of its base Regular Pyramid : the base of this pyramid is a regular polygon Irregular Pyramid : this type of pyramid has an irregular polygon as its base
- 3. • Definition Of Surface Area • The surface area of a three-dimensional figure is the sum of the areas of all its faces. • The Surface Area of a Pyramid • When all side faces are the same: • [Base Area] + 1/2 × Perimeter × [Slant Length] • When side faces are different: • [Base Area] + [Lateral Area] • Notes On Surface Area • The Surface Area has two parts: the area of the base (the Base Area), and the area of the side faces (the Lateral Area). • For Base Area : • It depends on the shape, there are different formulas for triangle, square, etc. • For Lateral Area : • When all the side faces are the same: • Multiply the perimeter by the "slant length" and divide by 2. This is because the side faces are always triangles and the triangle formula is "base times height divided by 2" • But when the side faces are different (such as an "irregular" pyramid) we must add up the area of each triangle to find the total lateral area. Surface Area
- 4. • A pyramidal frustum is a frustum made by chopping the top off a pyramid. • The lateral faces of a pyramidal frustum are trapezoids. • The height of the pyramidal frustum is the perpendicular distance between the bases. • The apothem is the height of any of its sides.
- 5. Total Surface Area of a Pyramid A pyramid is a three-dimensional figure made up of a base and triangular faces that meet at the vertex, V, which is also called the apex of the pyramid. The lower face ABCD is called the base and the perpendicular distance from the vertex, V, to the base at O is called the height of the pyramid. The total surface area of a pyramid is the sum of the areas of its faces including its base. The number of triangular faces depends on the number of sides of the base. For example, a pyramid with a rectangular base has four triangular faces, whereas a pyramid with a hexagonal face is made up of six triangular faces, and so on. Note: •A square pyramid has four equal triangular faces and a square base. •A pyramid does not have uniform (or congruent) cross-sections.
- 6. By Pythagoras' Theorem from right-triangle VOM, we have Example Find the total surface area of a square pyramid with a perpendicular height of 16 cm and base edge of 24 cm. Solution:
- 8. Volume of a Pyramid A pyramid has a base and triangular sides which rise to meet at the same point. The base may be any polygon such as a square, rectangle, triangle, etc. Volume of a Pyramid The volume, V, of a pyramid in cubic units is given by where A is the area of the base and h is the height of the pyramid. V=1/3 (area of base)(height) =1/3 Ah
- 9. Volume of a Square-based Pyramid The volume of a square-based pyramid is given by Example Example A pyramid has a square base of side 4 cm and a height of 9 cm. Find its volume. Solution:
- 10. Volume of a Rectangular-based Pyramid The volume of a rectangular-based pyramid is given by Example Example Find the volume of a rectangular-based pyramid whose base is 8 cm by 6 cm and height is 5 cm. Solution:
- 11. Volume of a Triangular Pyramid The volume of a triangular pyramid is given by Example Find the volume of the following triangular pyramid, rounding your answer to two decimal places. Solution:
- 12. V = volume of frustum H = height of frustum A = area of lower base A’ = area of upper base VOLUME OF PYRAMIDAL FRUSTUM Formula : FRUSTUM the portion of a cone or pyramid which remains after its upper part has been cut off by a plane parallel to its base, or which is intercepted between two such planes. 13² = h² + 5² h = √(13² - 5² ) = 12cm V = 12/3 [ 576 + 196 + √(576 x 196)] =4432 cm ³ Ap = 13 cm
- 13. Surface Area Of Frustum Calculate the lateral area, surface area and volume of the truncated square pyramid whose larger base edge is 24, smaller base edge is 14 cm and whose lateral edge is 13 cm. 13² = h² + 5² h = √(13² - 5² ) = 12cm Ap = 13 cm