ACT Math - Pyramids | Practice Hub

Find the surface of a isoceles pyramid whose height is , base side length is , and slant height is .

Explanation

To find surface area of a pyramid, simply find the surface area of each of the faces and add them together.

$\textup{Base}=s^2=6^2=36$

$\textup{Triangle face}=\frac{1}{2}bh=\frac{1}{2}*6*4=12$

Since there are 4 trangular faces, we have to multiply that surface area by 4. Thus,

Find the surface of a isoceles pyramid whose height is , base side length is , and slant height is .

Explanation

To find surface area of a pyramid, simply find the surface area of each of the faces and add them together.

$\textup{Base}=s^2=6^2=36$

$\textup{Triangle face}=\frac{1}{2}bh=\frac{1}{2}*6*4=12$

Since there are 4 trangular faces, we have to multiply that surface area by 4. Thus,

A square pyramid has a slant height of 8. The edge of a face is 12. What is the measure of an edge of its base?

Explanation

In this kind of a problem, it's helpful to draw out all information that's given.

Pyramid

Because the triangular faces of a pyramid are isosceles triangles, the slant height can be seen to divide it into two right triangles. The goal is to find the measure of the edge of the square base. This quickly becomes a problem revolving around the use of the Pythagorean theorem.

If we take the measures 12 and 8 to be the hypotenuse and one of the legs, respectively, we can use the Pythagoream theorem to solve for the "base leg." Then, realizing that this measure if only half of the the total length of the square base edge, that value must be multiplied by 2.

$a=\sqrt{80}$

$a={\color{Green} 8.9}$

Therefore,

A square pyramid has a slant height of 8. The edge of a face is 12. What is the measure of an edge of its base?

Explanation

In this kind of a problem, it's helpful to draw out all information that's given.

Pyramid

$a=\sqrt{80}$

$a={\color{Green} 8.9}$

Therefore,

A square pyramid has a volume of and a height of $\small 9cm$ . What is the perimeter of the base of the pyramid?

Explanation

The formula for the volume of a pyramid is:

$\small V=\frac{lwh}{3}$

We know that and ; however, this still leaves us with two variables in the equation: and . By definition, a square pyramid's base has sides of equal length, meaning that and are the same. Therefore, we can substitute $\small lw$ for $\small l\cdot l$ , or $\small l^2$ .

This gives us a new equation of:

$\small V=\frac{l^2h}{3}$

We then plug in the variables we know:

$\small 363=\frac{9l^2}{3}$

Multiply both sides by 3:

$\small 1089=9l^2$

Divide both sides by 9:

Therefore, we now know that the length of the pyramid's base is $\small 11cm$ . The question, however, asks for the perimeter of the pyramid's base. Since all of the sides of the base are the same, they must all be $\small 11cm$ . So we multiply $11cm\cdot 4=44cm$ . Therefore, the perimeter of the pyramid's base is $\small 44cm$ .

A square pyramid has a volume of and a height of $\small 9cm$ . What is the perimeter of the base of the pyramid?

Explanation

The formula for the volume of a pyramid is:

$\small V=\frac{lwh}{3}$

This gives us a new equation of:

$\small V=\frac{l^2h}{3}$

We then plug in the variables we know:

$\small 363=\frac{9l^2}{3}$

Multiply both sides by 3:

$\small 1089=9l^2$

Divide both sides by 9:

What is the surface area of a square pyramid with a height of 12 in and a base side length of 10 in?

$360 \textup{ in}^{2}$

$260 \textup{ in}^{2}$

$150 \textup{ in}^{2}$

$100 \textup{ in}^{2}$

$420 \textup{ in}^{2}$

Explanation

The surface area of a square pyramid can be broken into the area of the square base and the areas of the four triangluar sides. The area of a square is given by:

$A = s^{2} = 100 in^{2}$

The area of a triangle is:

$A = \frac{1}{2} bh$

The given height of 12 in is from the vertex to the center of the base. We need to calculate the slant height of the triangular face by using the Pythagorean Theorem:

$SH^{2} = H^{2} + B^{2}$

where and (half the base side) resulting in a slant height of 13 in.

So, the area of the triangle is:

$A = \frac{1}{2} bh = \frac{1}{2} \cdot 10 \cdot 13 = 65 in^{2}$

There are four triangular sides totaling $260 in^{2}$ for the sides.

The total surface area is thus $360 in^{2}$ , including all four sides and the base.

What is the surface area of a square pyramid with a height of 12 in and a base side length of 10 in?

$360 \textup{ in}^{2}$

$260 \textup{ in}^{2}$

$150 \textup{ in}^{2}$

$100 \textup{ in}^{2}$

$420 \textup{ in}^{2}$

Explanation

The surface area of a square pyramid can be broken into the area of the square base and the areas of the four triangluar sides. The area of a square is given by:

$A = s^{2} = 100 in^{2}$

The area of a triangle is:

$A = \frac{1}{2} bh$

The given height of 12 in is from the vertex to the center of the base. We need to calculate the slant height of the triangular face by using the Pythagorean Theorem:

$SH^{2} = H^{2} + B^{2}$

where and (half the base side) resulting in a slant height of 13 in.

So, the area of the triangle is:

$A = \frac{1}{2} bh = \frac{1}{2} \cdot 10 \cdot 13 = 65 in^{2}$

There are four triangular sides totaling $260 in^{2}$ for the sides.

The total surface area is thus $360 in^{2}$ , including all four sides and the base.

An architect wants to make a square pyramid and fill it with 12,000 cubic feet of sand. If the base of the pyramid is 30 feet on each side, how tall does he need to make it?

$40\ ft$

$13.3\ ft$

$26.7\ ft$

$30\ ft$

$50\ ft$

Explanation

Volume of Pyramid = 1/3 * Area of Base * Height

12,000 ft3 = 1/3 * 30ft * 30ft * H

12,000 = 300 * H

H = 12,000 / 300 = 40

H = 40 ft

An architect wants to make a square pyramid and fill it with 12,000 cubic feet of sand. If the base of the pyramid is 30 feet on each side, how tall does he need to make it?

$40\ ft$

$13.3\ ft$

$26.7\ ft$

$30\ ft$

$50\ ft$

Explanation

Volume of Pyramid = 1/3 * Area of Base * Height

12,000 ft3 = 1/3 * 30ft * 30ft * H

12,000 = 300 * H

H = 12,000 / 300 = 40

H = 40 ft

Pyramids

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