A pyramid has a rectangular base 3.60 cm by 5.40 cm . Determine the volume and total surface are

A pyramid has a rectangular base 3.60 cm by 5.40 cm ....

Key Concepts

Pyramid Geometry

A pyramid is a polyhedron with a polygonal base and triangular faces that converge at a single point called the apex. In pyramids with a rectangular base, the symmetry of the base and the equal sloping edges implies that the apex is located directly above the center of the base. This geometric structure allows for specific calculations of volume and surface area using known relationships.

Volume Calculation

The volume of a pyramid is determined by the formula V = (1/3) × (base area) × (height). Calculating the volume requires knowing the area of the base, which for a rectangle is the product of its side lengths, and the perpendicular height from the base to the apex. In problems where the height is not directly given, additional geometric relationships are used to compute it.

Surface Area Calculation

The total surface area of a pyramid includes both the area of the base and the lateral surface area, which is the sum of the areas of the triangular faces. Calculating the lateral area typically involves finding the slant heights of the triangles, which can be determined using geometric relations. This comprehensive calculation takes into account all the visible faces of the pyramid.

Pythagorean Theorem

The Pythagorean theorem is a fundamental tool in geometry that relates the sides of a right triangle. It is used to determine unknown distances in a pyramid, such as the perpendicular height from the apex to the base, by relating the length of the sloping edge to the horizontal distance from the base's center to its vertices. This theorem is essential for connecting the given edge lengths with the dimensions of the base.

Transcript

00:01 In this question we are given with a regular pyramid okay it's given in the question that we are given with a regular pyramid and the base has four sides so for a regular pyramid to have a base with four sides then the basis square okay, that means all the base edges are of the same length and we are given with two values first is the slant height so what is slant height? slant height is nothing but the height of the slant face so the slant phase is a trial triangle and we are given with the height of this triangle so if you in this figure this particular line right this particular line that passes through the middle of the space is the slant height it is it is given as excess so i'm denoted i have denoted us it's us and then the lateral edge lateral edge is nothing but the this particular ends length that is the edge of the lateral face that is the lateral edge so i denoted as e and it is given as 15 so this means that this particular blue line is 12 and this particular black line is 50 and we don't know this this length so we'll make it as b so we're asked to find the perimeter of this base so we know that from this from the question we know that the base is a square because it's a regular four -sided polygon it's it's a square if that is against then it's a square and we did not have the base at just b so this these two will be the same but we don't know the we don't know the value of b right in order to find that what we can do we are given with the this triangle that is we're given with this particular triangle and this it's this is or this edge is perpendicular to this base so we can take this particular half section of this triangle and find half of b this is half of b and then multiplied by 2 and we'll get b so let's do that so that particular half section of this triangle this is 15 this is 12 right and this is b by 2 so that means applying by 2 in theorem we'll get this is base this is all due this is hypotenuse b by 2 is nothing but root of 15 square minus 12 square this 15 square is 225 minus 144 so 15 square minus 12 square right that will be a root of 81 that is 801 is a perfect square of which is where it is 9 hence we got b by 2 is equal to nine which gives us b to b to be nine multiplied by two right this 18 so we got the value from b as 80 and for a square the perimeter is given by perimeter of the base right for a square the perimeter is given by four times its side length that is four multiplied by b this multiplied by 80 this is 72 so we got the base perimeter of this pyramid next one we need to find the lateral area in order to find the lateral area just need to find the area of…

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