OPQRS is a right pyramid whose base is a square of sides 12cm each. Given that the slant height of the pyramid is 15cm. Find the height of the pyramid. The volume of the pyramid and the total surface of the pyramid.

Part(a)

Base of square =12cm
Slant height of pyramid =15cm

[Applying the theorem]

(Height)^2+6^2=15^2
(Height)^2=225-36
(Height)^2=189
Height=189^2
Height= (square root of189)
= 13.74cm

Part(b). :. volume of pyram
=1/3×base area×h
Volume of pyramid =1/3×12^2×13.74cm^3
=659.52cm^3

Since we cannot draw the figure here, you can draw the figure yourself.

Note that a right pyramid is a pyramid with a square base and the apex (the tip of the pyramid) is aligned directly at the center of the base.
Now, draw an line from the apex to the center of the base, which serves as the height of the pyramid. Then connect the center of the base to one of the midpoints of the side of the square. The slant height is from that midpoint to the apex.
Note that you have formed a right triangle, where the height is unknown, and the hypotenuse (which is also the slant height) is 15 cm. You can solve for the base of the right triangle by getting the half of the side of the square, which is 6 cm.
Solving for height, recall pythagorean theorem. For any right triangle,
a^2 + b^2 = c^2
where
a = base of right triangle
b = height of right triangle
c = hypotenuse
Substituting,
6^2 + b^2 = 15^2
36 + b^2 = 225
b^2 = 189
b = 3*sqrt(21) cm [height]

Recall that the volume of ANY pyramid is just
V = (Area of the base)*(height)/3
*Area of the base = (length of side of square)^2
V = (12^2)*[3*sqrt(21)]/3
V = 144*sqrt(21) cm^3

Recall that the SA of a SQUARE pyramid is just
SA = 2bs + b^2
where
s = slant height
b = length of side of square
SA = 2*12*15 + 12^2
SA = 504 cm^2

Hope this helps~ :)

To find the height of the pyramid, we can use the Pythagorean theorem. The slant height of the pyramid is the hypotenuse of a right triangle formed by the height, half of the base, and the slant height.

Let's label the height of the pyramid as 'h'.

Using the Pythagorean theorem, we have:

(h/2)^2 + 12^2 = 15^2

Simplifying the equation:

(h/2)^2 + 144 = 225

(h/2)^2 = 225 - 144

(h/2)^2 = 81

Taking the square root of both sides:

h/2 = sqrt(81)

h/2 = 9

Multiplying both sides by 2:

h = 18 cm

Therefore, the height of the pyramid is 18 cm.

To find the volume of the pyramid, we can use the formula for the volume of a pyramid:

Volume = (1/3) * base area * height

Since the base is a square with sides of 12 cm each, the area of the base is:

Base Area = 12^2 = 144 cm^2

Substituting the values into the formula:

Volume = (1/3) * 144 * 18

Volume = 864 cm^3

Therefore, the volume of the pyramid is 864 cm^3.

To find the total surface area of the pyramid, we need to find the area of each face and add them up.

The base of the pyramid is a square, so the area of the base is 12 * 12 = 144 cm^2.

There are 4 triangular faces in the pyramid. Each face is an isosceles triangle with base length 12 cm and height 15 cm (the slant height of the pyramid).

The area of each triangular face can be calculated using the formula:

Triangle Area = (1/2) * base * height

Substituting the values into the formula:

Triangle Area = (1/2) * 12 * 15 = 90 cm^2

The total surface area is the sum of the areas of the base and the 4 triangular faces:

Total Surface Area = Base Area + 4 * Triangle Area

Total Surface Area = 144 + 4 * 90

Total Surface Area = 144 + 360

Total Surface Area = 504 cm^2

Therefore, the total surface area of the pyramid is 504 cm^2.

To find the height of the pyramid, the Pythagorean theorem can be used since the pyramid is a right pyramid. The slant height, height, and base of the pyramid form a right triangle.

Using the Pythagorean theorem, we can write:

\((\text{slant height})^2 = (\text{height})^2 + (\text{base}/2)^2\)

Substituting the given values:

\(15^2 = (\text{height})^2 + (12/2)^2\)

Simplifying:

\(225 = (\text{height})^2 + (6)^2\)

\(225 = (\text{height})^2 + 36\)

\(225 - 36 = (\text{height})^2\)

\(189 = (\text{height})^2\)

To find the height, we can take the square root of both sides:

\(\sqrt{189} = \text{height}\)

Simplifying:

\(\text{height} \approx 13.74 \, \text{cm}\)

Now, let's calculate the volume of the pyramid. The formula for the volume of a pyramid is:

\(\text{Volume} = \frac{1}{3} \times \text{base area} \times \text{height}\)

The base area of the pyramid is given as a square with sides of 12 cm, so its area is \(12 \times 12 = 144 \, \text{cm}^2\).

Substituting the values into the formula:

\(\text{Volume} = \frac{1}{3} \times 144 \, \text{cm}^2 \times 13.74 \, \text{cm}\)

\(\text{Volume} \approx 783.24 \, \text{cm}^3\)

Finally, let's calculate the total surface area of the pyramid. The total surface area includes the area of the base and the areas of the four triangular faces.

The area of the base is already calculated as \(144 \, \text{cm}^2\).

To find the areas of the triangular faces, we can use the formula for the area of a triangle:

\(\text{Triangle Area} = \frac{1}{2} \times \text{base} \times \text{height}\)

Substituting the given values:

\(\text{Triangle Area} = \frac{1}{2} \times 12 \, \text{cm} \times 15 \, \text{cm}\)

\(\text{Triangle Area} = 90 \, \text{cm}^2\)

Since there are four triangular faces, the total area of the triangular faces is \(4 \times 90 = 360 \, \text{cm}^2\).

Adding the base area and the areas of the triangular faces, we get the total surface area:

\(\text{Total Surface Area} = 144 \, \text{cm}^2 + 360 \, \text{cm}^2\)

\(\text{Total Surface Area} = 504 \, \text{cm}^2\)