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4.6: Area and Perimeter of Triangles

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    Area is half the base times the height while the perimeter is the sum of the sides.

    The formula for the area of a triangle is half the area of a parallelogram.

    f-d_4c0c681521c5908019784c1fbaa3d1c36c4afd60221641d471444e18+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    Area of a Triangle: \(A=\dfrac{1}{2} bh\) or \(A=\dfrac{bh}{2}\).

    f-d_909ec4625918d0fc0dabfd6820b19940ba1d7f9794a9469b6af39ab0+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    What if you were given a triangle and the size of its base and height? How could you find the total distance around the triangle and the amount of space it takes up?

    For Examples 1 and 2, use the following triangle.

    f-d_433cac41e518719b0c01b5bd620222c8574946a1ac01f3ff2651e2e4+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

    Example \(\PageIndex{1}\)

    Find the height of the triangle.

    Solution

    Use the Pythagorean Theorem to find the height.

    \(\begin{align*} 8^2+h^2 &=17^2 \\ h^2 &=225\\ h &=15 in \end{align*} \)

    Example \(\PageIndex{2}\)

    Find the perimeter.

    Solution

    We need to find the hypotenuse. Use the Pythagorean Theorem again.

    \(\begin{align*} (8+24)^2+15^2 &=h^2 \\ h^2 &=1249 \\ h &\approx 35.3 in \end{align*}\)

    The perimeter is \(24+35.3+17\approx 76.3\: in\).

    Example \(\PageIndex{3}\)

    Find the area of the triangle.

    f-d_4ddc6462941435dec5c611b3571f22c7c6d49f5f8ca8f2e56c41c0e8+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

    Solution

    To find the area, we need to find the height of the triangle. We are given two sides of the small right triangle, where the hypotenuse is also the short side of the obtuse triangle.

    \(\begin{align*} 3^2+h^2&=5^2 \\ 9+h^2&=25 \\ h^2&=16 \\h &=4 \\ A&=\dfrac{1}{2}(4)(7)=14 \: units^2 \end{align*}\)

    Example \(\PageIndex{4}\)

    Find the perimeter of the triangle in Example 3.

    Solution

    To find the perimeter, we need to find the longest side of the obtuse triangle. If we used the black lines in the picture, we would see that the longest side is also the hypotenuse of the right triangle with legs 4 and 10.

    \(\begin{align*} 4^2+10^2&=c^2 \\ 16+100&=c^2 \\ c &=\sqrt{116}\approx 10.77 \end{align*} \)

    The perimeter is \(7+5+10.77\approx 22.77\) units

    Example \(\PageIndex{5}\)

    Find the area of a triangle with base of length \(28 \: cm\) and height of \(15\: cm\).

    Solution

    The area is \(dfrac{1}{2}(28)(15)=210\: cm^2\).

    Review

    Use the triangle to answer the following questions.

    f-d_b00417194f7cf82e2346b42062316704c5870be37b8293377a9218e2+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{5}\)
    1. Find the height of the triangle by using the geometric mean.
    2. Find the perimeter.
    3. Find the area.

    Find the area of the following shape.

    1. f-d_30bcc55e0170103930d479f168441eb89d69ac1d6642f65a0c7e6805+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{6}\)
    2. What is the height of a triangle with area \(144\: m^2\) and a base of \(24\: m\)?

    In questions 6-11 we are going to derive a formula for the area of an equilateral triangle.

    f-d_10415d49f9724fa6930eabadd628ad7883728ed212373a3f7cfdb33c+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{7}\)
    1. What kind of triangle is \(\Delta ABD\)? Find \(AD\) and \(BD\).
    2. Find the area of \(\Delta ABC\).
    3. If each side is \(x\), what is \(AD\) and \(BD\)?
    4. If each side is \(x\), find the area of \(\Delta ABC\).
    5. Using your formula from #9, find the area of an equilateral triangle with 12 inch sides.
    6. Using your formula from #9, find the area of an equilateral triangle with 5 inch sides.

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 10.3.

    Vocabulary

    Term Definition
    Area Area is the space within the perimeter of a two-dimensional figure.
    Perimeter Perimeter is the distance around a two-dimensional figure.
    Perpendicular Perpendicular lines are lines that intersect at a 90∘ angle. The product of the slopes of two perpendicular lines is -1.
    Right Angle A right angle is an angle equal to 90 degrees.
    Right Triangle A right triangle is a triangle with one 90 degree angle.
    Area of a Parallelogram The area of a parallelogram is equal to the base multiplied by the height: \(A = bh\). The height of a parallelogram is always perpendicular to the base (the sides are not the height).
    Area of a Triangle The area of a triangle is half the area of a parallelogram. Hence the formula: \(A=\dfrac{1}{2} bh\) or \(A=\dfrac{bh}{2}\).

    Additional Resources

    Interactive Element

    Video: Area of a Triangle (Whole Numbers)

    Activities: Area and Perimeter of Triangles Discussion Questions

    Study Aids: Triangles and Quadrilaterals Study Guide

    Practice: Area and Perimeter of Triangles

    Real World: Perimeter

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