Triangles – ASVAB Math Review

1.General Triangle

A three-sided polygon where the sum of all internal angles is always 180 degrees. Area is the space enclosed by the three sides.

$$P = a + b + c$$; $$A = 0.5 \times \text{base} \times \text{height}$$

a, b, and c are the three side lengths. base is the bottom edge, and height is the perpendicular distance from the opposite vertex to the base.

Example

A triangle has sides of 5 cm, 6 cm, and 7 cm. Its base is 6 cm and its vertical height is 4 cm. Find its perimeter and area.

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A triangle has sides of 5 cm, 6 cm, and 7 cm. Its base is 6 cm and its vertical height is 4 cm. Find its perimeter and area.

Practice problems

Problem 1
Find the area of a triangular sail that has a base of 14 feet and a perpendicular height of 9 feet.

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Answer: 63 square feet

Step 1: Identify the given dimensions: base = 14 ft, height = 9 ft
Step 2: Use the standard area formula: Area = 0.5 × base × height
Step 3: Perform the calculation: $$0.5 \times 14 \times 9 = 7 \times 9 = 63$$ square feet
Problem 2
A triangle has side lengths of 11 cm, 14 cm, and 18 cm. What is the perimeter of this triangle?

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Answer: 43 cm

Step 1: Locate the three outer bounding side lengths: 11, 14, 18
Step 2: Apply the perimeter formula: $$P = a + b + c$$
Step 3: Sum the lengths together: $$11 + 14 + 18 = 43$$ cm
Problem 3
The area of a triangular sign is 30 square inches. If its perpendicular height is 5 inches, what is the length of its base?

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Answer: 12 inches

Step 1: Write down the area formula: Area = 0.5 × base × height. Rearrange to find base: base = (2 × Area) / height
Step 2: Substitute the known values into the equation: base = $$\frac{(2 \times 30)}{5}$$
Step 3: Solve for the base: $$\frac{60}{5} = 12$$ inches

Remember

The height must always be perpendicular (at a 90-degree angle) to the chosen base line.

2.Right Triangle

A triangle containing exactly one 90-degree (right) angle. The two sides forming the right angle act as the base and height.

$$P = \text{leg}_1 + \text{leg}_2 + \text{hypotenuse}$$; $$A = 0.5 \times \text{leg}_1 \times \text{leg}_2$$

The legs are the two shorter sides that meet at the right angle. The hypotenuse is the longest side opposite the right angle.

Example

A right triangle has legs measuring 6 inches and 8 inches, and a hypotenuse of 10 inches. Find its area.

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Step 1: Identify the two perpendicular legs: leg1 = 6, leg2 = 8

Step 2: Apply the right triangle area formula: $$A = 0.5 \times \text{leg}_1 \times \text{leg}_2$$

Step 3: Calculate the product: $$A = 0.5 \times 6 \times 8 = 3 \times 8 = 24$$ square inches

Practice problems

Problem 4
A right-angled metal bracket has perpendicular legs measuring 5 cm and 12 cm. What is the area of the bracket?

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Answer: 30 square cm

Step 1: Identify the two side lengths that create the right angle: 5 cm and 12 cm
Step 2: Use the right triangle area formula: Area = 0.5 × leg1 × leg2
Step 3: Complete the math: $$0.5 \times 5 \times 12 = 0.5 \times 60 = 30$$ square cm
Problem 5
Calculate the perimeter of a right triangle with legs of 9 inches and 12 inches, and a hypotenuse of 15 inches.

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Answer: 36 inches

Step 1: Group the three bounding side boundaries: 9, 12, 15
Step 2: Sum all side lengths up: $$P = \text{leg}_1 + \text{leg}_2 + \text{hypotenuse}$$
Step 3: Find the total length: $$9 + 12 + 15 = 36$$ inches
Problem 6
A right triangle has an area of 28 square meters. If one of its perpendicular legs is 7 meters, what is the length of the other leg?

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Answer: 8 meters

Step 1: Use the area formula: Area = 0.5 × leg1 × leg2. Rearrange it: leg2 = (2 × Area) / leg1
Step 2: Plug in the target values: leg2 = $$\frac{(2 \times 28)}{7}$$
Step 3: Evaluate the expression: $$\frac{56}{7} = 8$$ meters

Remember

You can directly use the two perpendicular legs as the base and height values to calculate the area.

3.Equilateral Triangle

A specialized triangle where all three sides are equal in length and all three internal angles are exactly 60 degrees.

$$P = 3 \times s$$; $$A = \frac{(s^2 \times \sqrt{3})}{4}$$

s is the uniform side length of the triangle. All sides are perfectly congruent.

Example

An equilateral triangle has a side length of 10 cm. Find its perimeter.

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Step 1: Identify the congruent side length: $$s = 10$$

Step 2: Apply the uniform perimeter formula: $$P = 3 \times s$$

Step 3: Compute the product: $$3 \times 10 = 30$$ cm

Practice problems

Problem 7
A warning traffic sign is an equilateral triangle with a side length of 24 inches. Find its perimeter.

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Answer: 72 inches

Step 1: Identify the single given uniform side value: $$s = 24$$ in
Step 2: Use the equilateral perimeter rule: $$P = 3 \times s$$
Step 3: Multiply the parameters: $$3 \times 24 = 72$$ inches
Problem 8
An equilateral triangular patch has a perimeter of 45 cm. What is the length of each individual side?

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Answer: 15 cm

Step 1: Recall that an equilateral triangle has three identical sides: Perimeter = 3 × side
Step 2: Isolate the side variable: side = Perimeter / 3
Step 3: Complete the division: $$\frac{45}{3} = 15$$ cm
Problem 9
An equilateral triangle has a side length of 6 meters and a vertical height measured at approximately 5.2 meters. What is its estimated area?

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Answer: 15.6 square meters

Step 1: Use the base side (6 m) and the provided perpendicular height (5.2 m)
Step 2: Set up the standard triangle area formula: Area = 0.5 × base × height
Step 3: Solve the math: $$0.5 \times 6 \times 5.2 = 3 \times 5.2 = 15.6$$ square meters

Remember

For basic ASVAB application, if height is provided directly, you can still fall back on the standard 0.5 × base × height formula.

4.Isosceles Triangle

A triangle with at least two sides of equal length. The angles opposite those equal sides are also equal.

$$P = 2 \times a + b$$; $$A = 0.5 \times \text{base} \times \text{height}$$

a represents the length of one of the two twin legs. b represents the distinct base edge line.

Example

An isosceles triangle has two equal legs of 7 cm each and a base of 4 cm. Find its perimeter.

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Step 1: Identify the matching legs $$(a = 7)$$ and the distinctive base $$(b = 4)$$

Step 2: Set up the calculation: $$P = (2 \times 7) + 4$$

Step 3: Solve the expressions: $$14 + 4 = 18$$ cm

Practice problems

Problem 10
An isosceles triangle has a base of 10 inches and two congruent legs that each measure 13 inches. Find its perimeter.

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Answer: 36 inches

Step 1: Identify the parts: two legs are 13 in each, base is 10 in
Step 2: Set up the perimeter addition: $$P = 2 \times (\text{leg}) + \text{base}$$
Step 3: Solve: $$P = (2 \times 13) + 10 = 26 + 10 = 36$$ inches
Problem 11
An isosceles triangle frame has an area of 48 square feet and a base tracking line of 12 feet. What is its perpendicular height?

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Answer: 8 feet

Step 1: Use the area formula framework: Area = 0.5 × base × height -> height = (2 × Area) / base
Step 2: Insert the parameters: height = $$\frac{(2 \times 48)}{12}$$
Step 3: Calculate the value: $$\frac{96}{12} = 8$$ feet
Problem 12
The perimeter of an isosceles triangle is 32 cm. If the unique base side measures 12 cm, what is the length of each matching leg?

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Answer: 10 cm

Step 1: Subtract the base value from the total perimeter: $$32 - 12 = 20$$ cm
Step 2: Divide the remaining total length by 2 to isolate one matching leg
Step 3: Calculate the individual leg value: $$\frac{20}{2} = 10$$ cm

Remember

When calculating perimeter, make sure to multiply the repeating leg measurement by two before adding the base.