How to find the surface area of a polyhedron - PSAT Math

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Question

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle.

The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

Answer

The total surface area of the icosahedron is 400 square centimeters; since the icosahedron comprises twenty congruent faces, each has area $400 \div 20 = 20$ square centimeters.

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

Set and solve for :

$\frac{s^{2}\sqrt{3}}{4} = 20$

$\frac{s^{2}\sqrt{3}}{4} \cdot \frac{4}{\sqrt{3}}= 20 \cdot \frac{4}{\sqrt{3}}$

$s^{2} = \frac{80}{ \sqrt{3}} \approx \frac{80}{1.732} \approx 46.19$

$s \approx \sqrt{46.19} \approx 6.8$ centimeters.

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Question

A regular octahedron has eight congruent faces, each of which is an equilateral triangle.

A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

Answer

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

Since there are eight equilateral triangles that comprise the surface of the octahedron, the total surface area is

$SA = 8 A = 8 \cdot \frac{s^{2}\sqrt{3}}{4} = 2 s^{2}\sqrt{3}$

Substitute :

$SA =2 \cdot 5^{2}\sqrt{3} = 50\sqrt{3}$ square inches.

Compare your answer with the correct one above

Question

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle.

The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

Answer

The total surface area of the icosahedron is 400 square centimeters; since the icosahedron comprises twenty congruent faces, each has area $400 \div 20 = 20$ square centimeters.

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

Set and solve for :

$\frac{s^{2}\sqrt{3}}{4} = 20$

$\frac{s^{2}\sqrt{3}}{4} \cdot \frac{4}{\sqrt{3}}= 20 \cdot \frac{4}{\sqrt{3}}$

$s^{2} = \frac{80}{ \sqrt{3}} \approx \frac{80}{1.732} \approx 46.19$

$s \approx \sqrt{46.19} \approx 6.8$ centimeters.

Compare your answer with the correct one above

Question

A regular octahedron has eight congruent faces, each of which is an equilateral triangle.

A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

Answer

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

Since there are eight equilateral triangles that comprise the surface of the octahedron, the total surface area is

$SA = 8 A = 8 \cdot \frac{s^{2}\sqrt{3}}{4} = 2 s^{2}\sqrt{3}$

Substitute :

$SA =2 \cdot 5^{2}\sqrt{3} = 50\sqrt{3}$ square inches.

Compare your answer with the correct one above

Question

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle.

The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

Answer

The total surface area of the icosahedron is 400 square centimeters; since the icosahedron comprises twenty congruent faces, each has area $400 \div 20 = 20$ square centimeters.

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

Set and solve for :

$\frac{s^{2}\sqrt{3}}{4} = 20$

$\frac{s^{2}\sqrt{3}}{4} \cdot \frac{4}{\sqrt{3}}= 20 \cdot \frac{4}{\sqrt{3}}$

$s^{2} = \frac{80}{ \sqrt{3}} \approx \frac{80}{1.732} \approx 46.19$

$s \approx \sqrt{46.19} \approx 6.8$ centimeters.

Compare your answer with the correct one above

Question

A regular octahedron has eight congruent faces, each of which is an equilateral triangle.

A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

Answer

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

Since there are eight equilateral triangles that comprise the surface of the octahedron, the total surface area is

$SA = 8 A = 8 \cdot \frac{s^{2}\sqrt{3}}{4} = 2 s^{2}\sqrt{3}$

Substitute :

$SA =2 \cdot 5^{2}\sqrt{3} = 50\sqrt{3}$ square inches.

Compare your answer with the correct one above

Question

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle.

The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

Answer

The total surface area of the icosahedron is 400 square centimeters; since the icosahedron comprises twenty congruent faces, each has area $400 \div 20 = 20$ square centimeters.

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

Set and solve for :

$\frac{s^{2}\sqrt{3}}{4} = 20$

$\frac{s^{2}\sqrt{3}}{4} \cdot \frac{4}{\sqrt{3}}= 20 \cdot \frac{4}{\sqrt{3}}$

$s^{2} = \frac{80}{ \sqrt{3}} \approx \frac{80}{1.732} \approx 46.19$

$s \approx \sqrt{46.19} \approx 6.8$ centimeters.

Compare your answer with the correct one above

Question

A regular octahedron has eight congruent faces, each of which is an equilateral triangle.

A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

Answer

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

Since there are eight equilateral triangles that comprise the surface of the octahedron, the total surface area is

$SA = 8 A = 8 \cdot \frac{s^{2}\sqrt{3}}{4} = 2 s^{2}\sqrt{3}$

Substitute :

$SA =2 \cdot 5^{2}\sqrt{3} = 50\sqrt{3}$ square inches.

Compare your answer with the correct one above

Question

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle.

The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

Answer

The total surface area of the icosahedron is 400 square centimeters; since the icosahedron comprises twenty congruent faces, each has area $400 \div 20 = 20$ square centimeters.

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

Set and solve for :

$\frac{s^{2}\sqrt{3}}{4} = 20$

$\frac{s^{2}\sqrt{3}}{4} \cdot \frac{4}{\sqrt{3}}= 20 \cdot \frac{4}{\sqrt{3}}$

$s^{2} = \frac{80}{ \sqrt{3}} \approx \frac{80}{1.732} \approx 46.19$

$s \approx \sqrt{46.19} \approx 6.8$ centimeters.

Compare your answer with the correct one above

Question

A regular octahedron has eight congruent faces, each of which is an equilateral triangle.

A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

Answer

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

Since there are eight equilateral triangles that comprise the surface of the octahedron, the total surface area is

$SA = 8 A = 8 \cdot \frac{s^{2}\sqrt{3}}{4} = 2 s^{2}\sqrt{3}$

Substitute :

$SA =2 \cdot 5^{2}\sqrt{3} = 50\sqrt{3}$ square inches.

Compare your answer with the correct one above

Question

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle.

The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

Answer

The total surface area of the icosahedron is 400 square centimeters; since the icosahedron comprises twenty congruent faces, each has area $400 \div 20 = 20$ square centimeters.

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

Set and solve for :

$\frac{s^{2}\sqrt{3}}{4} = 20$

$\frac{s^{2}\sqrt{3}}{4} \cdot \frac{4}{\sqrt{3}}= 20 \cdot \frac{4}{\sqrt{3}}$

$s^{2} = \frac{80}{ \sqrt{3}} \approx \frac{80}{1.732} \approx 46.19$

$s \approx \sqrt{46.19} \approx 6.8$ centimeters.

Compare your answer with the correct one above

Question

A regular octahedron has eight congruent faces, each of which is an equilateral triangle.

A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

Answer

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

Since there are eight equilateral triangles that comprise the surface of the octahedron, the total surface area is

$SA = 8 A = 8 \cdot \frac{s^{2}\sqrt{3}}{4} = 2 s^{2}\sqrt{3}$

Substitute :

$SA =2 \cdot 5^{2}\sqrt{3} = 50\sqrt{3}$ square inches.

Compare your answer with the correct one above

Question

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle.

The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

Answer

The total surface area of the icosahedron is 400 square centimeters; since the icosahedron comprises twenty congruent faces, each has area $400 \div 20 = 20$ square centimeters.

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

Set and solve for :

$\frac{s^{2}\sqrt{3}}{4} = 20$

$\frac{s^{2}\sqrt{3}}{4} \cdot \frac{4}{\sqrt{3}}= 20 \cdot \frac{4}{\sqrt{3}}$

$s^{2} = \frac{80}{ \sqrt{3}} \approx \frac{80}{1.732} \approx 46.19$

$s \approx \sqrt{46.19} \approx 6.8$ centimeters.

Compare your answer with the correct one above

Question

A regular octahedron has eight congruent faces, each of which is an equilateral triangle.

A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

Answer

The area of an equilateral triangle is given by the formula

$A = \frac{s^{2}\sqrt{3}}{4}$

Since there are eight equilateral triangles that comprise the surface of the octahedron, the total surface area is

$SA = 8 A = 8 \cdot \frac{s^{2}\sqrt{3}}{4} = 2 s^{2}\sqrt{3}$

Substitute :

$SA =2 \cdot 5^{2}\sqrt{3} = 50\sqrt{3}$ square inches.

Compare your answer with the correct one above

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How to find the surface area of a polyhedron - PSAT Math

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle. The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

The total surface area of the icosahedron is 400 square centimeters; since the icosahedron comprises twenty congruent faces, each has area square centimeters. The area of an equilateral triangle is given by the formula Set and solve for : centimeters.

A regular octahedron has eight congruent faces, each of which is an equilateral triangle. A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

The area of an equilateral triangle is given by the formula Since there are eight equilateral triangles that comprise the surface of the octahedron, the total surface area is Substitute : square inches.

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle. The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

The total surface area of the icosahedron is 400 square centimeters; since the icosahedron comprises twenty congruent faces, each has area square centimeters. The area of an equilateral triangle is given by the formula Set and solve for : centimeters.

A regular octahedron has eight congruent faces, each of which is an equilateral triangle. A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

The area of an equilateral triangle is given by the formula Since there are eight equilateral triangles that comprise the surface of the octahedron, the total surface area is Substitute : square inches.

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle. The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

The total surface area of the icosahedron is 400 square centimeters; since the icosahedron comprises twenty congruent faces, each has area square centimeters. The area of an equilateral triangle is given by the formula Set and solve for : centimeters.

A regular octahedron has eight congruent faces, each of which is an equilateral triangle. A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

The area of an equilateral triangle is given by the formula Since there are eight equilateral triangles that comprise the surface of the octahedron, the total surface area is Substitute : square inches.

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle. The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

The total surface area of the icosahedron is 400 square centimeters; since the icosahedron comprises twenty congruent faces, each has area square centimeters. The area of an equilateral triangle is given by the formula Set and solve for : centimeters.

A regular octahedron has eight congruent faces, each of which is an equilateral triangle. A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

The area of an equilateral triangle is given by the formula Since there are eight equilateral triangles that comprise the surface of the octahedron, the total surface area is Substitute : square inches.

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle. The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

The total surface area of the icosahedron is 400 square centimeters; since the icosahedron comprises twenty congruent faces, each has area square centimeters. The area of an equilateral triangle is given by the formula Set and solve for : centimeters.

A regular octahedron has eight congruent faces, each of which is an equilateral triangle. A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

The area of an equilateral triangle is given by the formula Since there are eight equilateral triangles that comprise the surface of the octahedron, the total surface area is Substitute : square inches.

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle. The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

The total surface area of the icosahedron is 400 square centimeters; since the icosahedron comprises twenty congruent faces, each has area square centimeters. The area of an equilateral triangle is given by the formula Set and solve for : centimeters.

A regular octahedron has eight congruent faces, each of which is an equilateral triangle. A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

The area of an equilateral triangle is given by the formula Since there are eight equilateral triangles that comprise the surface of the octahedron, the total surface area is Substitute : square inches.

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle. The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

The total surface area of the icosahedron is 400 square centimeters; since the icosahedron comprises twenty congruent faces, each has area square centimeters. The area of an equilateral triangle is given by the formula Set and solve for : centimeters.

A regular octahedron has eight congruent faces, each of which is an equilateral triangle. A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

The area of an equilateral triangle is given by the formula Since there are eight equilateral triangles that comprise the surface of the octahedron, the total surface area is Substitute : square inches.

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle.

The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

A regular octahedron has eight congruent faces, each of which is an equilateral triangle.

A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle.

The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

A regular octahedron has eight congruent faces, each of which is an equilateral triangle.

A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle.

The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

A regular octahedron has eight congruent faces, each of which is an equilateral triangle.

A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle.

The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

A regular octahedron has eight congruent faces, each of which is an equilateral triangle.

A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle.

The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

A regular octahedron has eight congruent faces, each of which is an equilateral triangle.

A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle.

The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

A regular octahedron has eight congruent faces, each of which is an equilateral triangle.

A given octahedron has edges of length five inches. Give the total surface area of the octahedron.

A regular icosahedron has twenty congruent faces, each of which is an equilateral triangle.

The total surface area of a given regular icosahedron is 400 square centimeters. To the nearest tenth of a centimeter, what is the length of each edge?

A regular octahedron has eight congruent faces, each of which is an equilateral triangle.

A given octahedron has edges of length five inches. Give the total surface area of the octahedron.