VWXYZ is a rectangle-based pyramid where WX=66 cm and XY=32cm. The vertex is vertically above the center of the base. Given that the slant heights VA and VB are 56 cm and 63 cm respectively, find the total surface area of the pyramid. Additionally, find the height and volume of the pyramid.

Well, well, well, we have ourselves a rectangular-based pyramid conundrum! It's time for Clan Bot to do some math tricks!

To find the surface area of the pyramid, we need to calculate the area of the base and the areas of the four triangular sides.

Now, we know that the base of the pyramid is a rectangle, with one side measuring 66 cm and the other measuring 32 cm. So, the area of the base is simply 66 multiplied by 32, which is 2112 cm².

As for the triangular sides, each side is a right-angled triangle. So, we can use the formula for the area of a triangle: 1/2 base times height.

The base of each triangle is either 66 cm or 32 cm (because the sides of the rectangle are the bases of the triangles), and the height is the slant height VA or VB. So, the area of each triangular side is simply 1/2 times the base times the height.

Now, we have four triangle sides, and each triangle has a different slant height, so we do the calculations for each side:

Triangle VAW: 1/2 times 66 times 56 = 1848 cm²
Triangle VAX: 1/2 times 32 times 56 = 896 cm²
Triangle VBW: 1/2 times 66 times 63 = 2079 cm²
Triangle VBX: 1/2 times 32 times 63 = 1008 cm²

Now, we add up the base area and the areas of the four triangular sides to get the total surface area:

2112 cm² + 1848 cm² + 896 cm² + 2079 cm² + 1008 cm² = 5943 cm²

So, the total surface area of the pyramid is 5943 cm².

Now, onto the next act! To calculate the height of the pyramid, we need to use some good old Pythagorean theorem:

We have the slant height VA as 56 cm and the height of the pyramid as h. Using Pythagoras, we can say that:
(h)² + (66/2)² = 56²

(h)² + 33² = 3136
(h)² = 3136 - 1089
(h)² = 2047

Taking the square root, we find the height (h) to be approximately 45.25 cm.

Finally, for the grand finale, let's find the volume of the pyramid! The volume of a pyramid is given by 1/3 times the base area times the height.

So, using the base area of 2112 cm² and the height of 45.25 cm, we can calculate the volume as follows:

Volume = 1/3 times 2112 times 45.25
Volume = 316,960 cm³

Ta-da! The volume of the pyramid is 316,960 cm³.

And there you have it, my friend! The surface area, height, and volume of the pyramid have been revealed. Clown Bot to the rescue!

To find the total surface area of the pyramid, we need to calculate the areas of all the individual faces and then add them together.

Let's start with the base of the pyramid. Since VWXYZ is a rectangle-based pyramid, the base is a rectangle with sides WX and XY. Given that WX = 66 cm and XY = 32 cm, we can calculate the area of the base as follows:

Area of the base = WX * XY = 66 cm * 32 cm = 2112 cm²

Now let's calculate the areas of the four triangular faces. Each triangular face has a base equal to the length of a side of the base rectangle and a height equal to the slant height.

Area of triangular face VA = (1/2) * XY * VA = (1/2) * 32 cm * 56 cm = 896 cm²
Area of triangular face VB = (1/2) * XY * VB = (1/2) * 32 cm * 63 cm = 1008 cm²
Area of triangular face VC = (1/2) * WX * VA = (1/2) * 66 cm * 56 cm = 1848 cm²
Area of triangular face VD = (1/2) * WX * VB = (1/2) * 66 cm * 63 cm = 2079 cm²

Next, we need to calculate the area of the triangular face connecting the vertex V to the centroid G of the base rectangle VWXY. The centroid of a rectangle is located at the center of the rectangle, so the distance from the centroid to any side is half of the length of that side.

The distance from the centroid G to the side WX is half of the length of WX, which is 66 cm / 2 = 33 cm. Similarly, the distance from G to the side XY is half of the length of XY, which is 32 cm / 2 = 16 cm.

Now, we can use the lengths of the sides VG, GX, and GY to calculate the area of the triangular face VGG'. Since VG is the slant height VB (63 cm) and G'Y is the distance from G to side XY (16 cm), we can calculate VG' using the Pythagorean theorem:

VG'² = VB² - G'Y²
VG'² = 63 cm² - 16 cm²
VG'² = 3969 cm² - 256 cm²
VG'² = 3713 cm²
VG' ≈ 60.95 cm

Therefore, the area of the triangular face VGG' is:

Area of triangular face VGG' = (1/2) * GY * VG' = (1/2) * 67 cm * 60.95 cm = 2049 cm²

Finally, we can calculate the total surface area by adding up all the areas:

Total surface area = Area of the base + Sum of areas of all triangular faces
Total surface area = 2112 cm² + 896 cm² + 1008 cm² + 1848 cm² + 2079 cm² + 2049 cm²
Total surface area ≈ 8992 cm²

To find the height of the pyramid, we can use the Pythagorean theorem. The height is the distance from the vertex V to the centroid G of the base rectangle VWXY. Since VG is the slant height VB (63 cm), and GX is the distance from G to side WX (33 cm), we can calculate the height VH using the Pythagorean theorem:

VH² = VG² - GX²
VH² = 63 cm² - 33 cm²
VH² = 3969 cm² - 1089 cm²
VH² = 2880 cm²
VH ≈ 53.67 cm

Finally, to find the volume of the pyramid, we can use the formula:

Volume = (1/3) * Area of the base * Height
Volume = (1/3) * 2112 cm² * 53.67 cm
Volume ≈ 38,249.28 cm³

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Surface area:

The points A and B are the mid-points of WX and XY respectively.

For triangle VWX, the surface area is half the base(66) multiplied by the slant height VA of 56.

For triangle VXY, the surface area is half the base(32) times the slant height of 63.

Add and double the two areas to get the total slant areas, add to the area of the base, a rectangle 66x32, to get the total surface area.

Height of Pyramid:
Drop a perpendicular from V to the centre of the base at point D. Then VAD and VBD are right triangles.
VA=56, AD=XY/2=16, VD=√2880 by Pythagoras theorem.
Similarly,
VB=63, BD=WX/2=33, and again, VD=√(63²-33²)=√2880.

Therefore height of pyramid=√2880.

Volume:
Volume of a pyramid is the area of the base multiplied by (1/3) of the height.